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Multi-component scalar dark matter from a spherical compactification Winslow, Peter Thomas 2009

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Multi-Component Scalar Dark Matter From A Spherical Compactification by Peter Thomas Winslow Hons. B.Sc., The University of Winnipeg, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in  THE FACULTY OF GRADUATE STUDIES  (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  September 2009 © Peter Thomas Winslow 2009  ABSTRACT  Current cosmological measurements of the various components of mass in the Universe indicate that a significant contribution to the total mass density is due to a previously undiscovered type of matter that must be both non-luminous and non-baryonic in nature, aptly named dark matter. The standard model of particle physics, while describing the results of collider experiments with unprecedented precision, does not include a suitable dark matter candidate. All of this is clear and strong evidence for the existence of new physics beyond both the standard models of particle physics and cosmology. One of the recent main areas of interest to physicists exploring beyond standard model physics is theories which incorporate extra dimensions. All of these theories postulate that the 3+1 spacetime that we experience exists as a localized subspace embedded within a larger (3+n)+1 spacetime. Among the many interesting motivations for exploring the phenomenology of extra dimensional models is the existence of a viable dark matter candidate, which has been found to arise somewhat naturally within the context of certain models. In the present work we consider a six dimensional model in which the two extra spatial dimensions are compactified onto a spherical geometry. An imposed parity and a subset of the spherical symmetry of the extra dimensions is then exploited to stabilize a number of massive four dimensional Kaluza Klein excitations, thereby leading to a self interacting multi-component theory of dark matter. The Boltzmann equations describing this system are then solved in order to simultaneously determine the relic densities of the stable dark matter particles. In the following thesis, we will describe this theory in detail along with its motivations, consequences, and compatibility with current observational data.  11  Contents  I  Abstract Table of Contents List of Tables List of Figures Acknowledgements Dedications  ii iii iv v vi vii  Introductory Material 1.1 Introduction to Extra Dimensions 1.2 Introduction to Kaluza Klein Theory 1.3 The Dark Matter Problem 1.3.1 Galactic Rotation Curves 1.3.2 X-Ray Emission in Nearby Clusters 1.3.3 Large Scale Structure Formation and the Cosmic Microwave Background. 1.3.4 Hot vs. Cold Dark Matter 1.4 Kaluza Klein Dark Matter in the Universal Extra Dimension Scenario  1 2 5 13 14 16 18 20 21  -  -  -  -  -  -  -  -  27 28 31 39 42  The Scalar Dark Matter Phenomenology 2.1 The Free Field Equations of Motion 2.2 The Action 2.3 The Feynman Rules 2.4 Decay Modes -  -  -  -  III  The Scalar Dark Matter Relic Abundance 3.1 The Boltzmann Equation 3.1.1 The Standard Case: One Particle With One Interaction 3.1.2 The Generalized Case: Many Particles With Many Interactions 3.2 The Thermalized Cross Section Calculations 3.2.1 The Magnetic Thermalized Cross Sections 3.2.2 The Non-Magnetic Thermalized Cross Sections. 3.3 Solving The Boltzmann Equations 3.4 General Conclusions  49 50 51 55 59 60 64 71 81  -  -  -  .  -  -  -  -  84  Bibliography Appendices Appendix A.1 A.2 Appendix Appendix  -  -  A Phase Spaces of Constant Amplitude The 2 Body Phase Space The 3 Body Phase Space B The Wigner 3-j Symbol C An Extremely Brief Introduction to the standard model of Particle Physics -  -  -  111  .  85 85 85 86 89 92  List of Tables 1 2  Example Freeze Out Temperature Calculations Example Decay Rate Calculations  iv  79 80  List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  The Braneworid Scenario The Galactic Rotation Curve Associated with M33 2 Orbifold The S’/Z The Higgs Propagator The Effective Scalar Propagator The Riggs Three Point Self Interaction The Riggs Four Point Self Interaction The Effective Scalar-Riggs Three Point Brane Interaction The Effective Scalar-Riggs Four Point Brane Interaction The Four Point Effective Scalar Bulk Interaction The Radiative Decay Modes The Riggs-Fermion Three Point Interaction The Riggs-Gauge Three Point Interactions Effective Scalar Annihilations into standard model End States The First Five Relic Densities from the Generalized Boltzmann Equations Compatible Regions of the 2 -g Parameter Space for mB = 1 TeV 1 Compatible Regions of the g-y Parameter Space for mB = .75 TeV Compatible Regions of the 2 -g Parameter Space for mB = .5 TeV 1 g Compatible Regions of the 2 -y Parameter Space for mB = .25 TeV 1 g 2 = .555 fib and j3 Relic Densities for mB = .75 TeV, 1/R = .5 TeV, g = .75, and g fib and j3 Relic Densities for m = .25 TeV, 1/R = 1 TeV, = .425, and 2 = .12685 2 = .505 fib and j3 Relic Densities for mB = 1 TeV, 1/R = 1 TeV, = .65, and y  v  .  .  2 15 24 39 39 39 39 40 40 41 45 65 65 65 74 75 76 76 77 77 78 78  Acknowledgements I would like to acknowledge here the supervision of Dr. John Ng of TRIUMF and Dr. Kris Sigurdson of UBC, without which this work would not have been possible. I would also like to express my gratitude to Kristian McDonald, Mark Van Raamsdonk, and Cliff Burgess for the helpful conversations which have contributed to my understanding of the following material. Finally, I would like to acknowledge the Natural Sciences and Engineering Research Council of Canada for their partial funding of this work.  vi  Dedications  In memory of Lajos Moldovan.  vii  Introductory Material  In this chapter we will introduce the reader to all background material needed to fully understand the main content of the present thesis. This will include general introductions to the concepts and ideas of extra dimensions, Kaluza Klein theory, dark matter, and one of the current theories of Kaluza Klein dark matter  -  the universal extra dimensional model. Our introduction to extra dimensions will place particular  emphasis on the original motivation and ideas behind the Arkani-Hamed, Dimopoulos, and Dvali (ADD) model, describing briefly how the model solves the hierarchy problem by lowering the Planck scale down to scales possibly detectable in present day accelerators. Kaluza Klein theory has been inexorably tied to the idea of extra dimensions since its birth and is, by now, quite old. In this introduction we will concentrate on the most interesting aspects of the modern view of the theory, i.e; its ability to provide a classical unification mechanism for gauge theories and General Relativity. Our introduction to dark matter will take the form of a sample list of the multitude of evidence for its existence. This sample was chosen specifically so as to guide us toward a better understanding of exactly what characteristics any particle theory of dark matter must exhibit. Finally, we will introduce the currently most popular theory which proposes a link between the concepts of extra dimensions and dark matter, the universal extra dimensional (TJED) model. This model provides a single unique dark matter candidate, the lightest Kaluza Klein particle (LKP), by manipulating the extra dimensional topology in a highly non-trivial way. This model will be only briefly introduced with the aim of introducing the methodology used to produce the LKP. In the subsequent chapters, we will then be concerned with contrasting our own methodology for producing multiple dark matter candidates with this one.  1  Introduction to Extra Dimensions Ever since Einstein’s theory of relativity compelled physicists to think in terms of four dimensions in stead of just three there have been speculations about the existence of even more dimensions. In order to avoid large disagreements with every day experience and experiments, the simplest solution is to compactify these extra dimensions so that their effects would have to be small enough to go unnoticed in present day accelerators. Since the advent of string theory, which needs the existence of extra dimensions for its own self consistency [1], there has been strong interest in the feasibility of localizing four dimensional gauge theories in higher dimensional spacetimes. This entails restricting some subset of the particle content of the standard model to 3 + 1 dimensional membranes, called branes for short, which are embedded within higher dimensional spacetime volumes generically called the bulk. The most exciting aspect of realistic models involving extra dimensions is the existence of experimental signatures which could be seen in cur rent or up-coming experiments such the Large Hadron Collider (LHC) or International Linear Collider (ILC).  caflone eape Into  bout  Figure 1: Matter is confined to branes, while gravity always propagates out into the full extra dimensional spacetime (the bulk).  There are many different extra dimensional scenarios, each with their own motivations. In this thesis, however, we will concentrate mostly on the Arkani-Hamed, Dimopoulos, and Dvali (ADD) type models [2] [3] [4]. In these models, the standard model gauge fields live on a brane consisting of 3 spatial dimensions which are embedded in 2 to 6 extra dimensions. Since ordinary particles do not propagate in the extra dimensions the bounds on the length scales are quite loose in the same way as above. The observed Planck scale is found to be a low energy effective coupling which arises from integrating over the extra dimensional volume.  2  At length scales smaller than the extra dimensional length scale gravity can become much stronger, allowing for the possibility of identifying the true Planck scale with scales on the order of the weak scale (r.TeV). The problem of explaining the seemingly massive difference between the Planck and weak scales, the hierarchy problem, is then translated into explaining why the extra dimensions are stabilized at such large radii. In this section we will review a simple illustrative example based on [5] which describes how the existence of small compactified extra dimensions give rise to a decrease in the fundamental gravitational scale.  Consider a sphere with uniform mass distribution with radius R and the total enclosed mass given by  Menc(r)  =  fp(r)dV  where dV is the volume element and the mass distribution, p, is given by  p(r)  =  Constant  p(r)=O  if r  R  ifr>R  The gravitational field can then be seen as a consequence of Gauss’s law by enclosing the sphere within a spherical Gaussian surface, S, with radius r  >  R. Gauss’s law then states that, for this surface,  dAMenc where  is the gravitational field, dA is the area element, and  is a proportionality constant. Due to  the spherical symmetry we can rewrite this as  In a 4 dimensional spacetime,  dIA  =  , so that the gravitational field takes on the familiar form 2 4xr  g—(4)  —  (4)  where  41 c  =  4ir  3  (D—1)/2  In aD dimensional spacetime [6], however,  jdAI  —  so that  =  1”“2” If the length scale of the extra dimensions is given by L then, on scales 2ir((’ 1)/2) 3 L, the gravitational field behaves as a full D dimensional field while, on scales large compared to L, the  where GCD) r  =  ‘‘  ‘.  field behaves like a scaled 4 dimensional gravitational field n(D) —  i,r  ) 4 LU  enc  2 r  In light of this result, we can then view the familiar gravitational coupling constant,  as an effective  coupling constant which is related to the D dimensional coupling constant by n(D) -  —  L(D4)  Writing the two coupling constants in terms of the effective Planck mass, Planck mass, G(D)  =  =  1/Mff, and the true  , we find 2 l/MF  Mf  ML”  where ci is the number of extra spatial dimensions. If the extra dimensional length scale is the true Planck length then Ld  =  M’ and Meff  =  M. On the other hand, if the extra dimensional length scale is much  larger than the Planck length then the true Planck mass scale can be much larger than the effective mass scale Meff. Physically speaking, we can then understand the fact that gravity is much weaker than the rest of the fundamental forces by assuming that gravity is allowed to leak out into the extra dimensions and is therefore diluted while the rest of the forces may be restricted to a brane. If this is the case, then the current experimental accuracy involving strictly the gravitational force implies the extremely loose bounds ofLO.1 mmandMp1 TeV[5].  Although the ADD type models are very useful for new phenomenology and a possible solution to the hierarchy problem, they have yet to produce a satisfying dark matter candidate. As yet, the only extra dimensional scenario which has found success in yielding a natural dark matter candidate is the Universal Extra Dimensional (UED) scenario. In this thesis, we will show that it is indeed possible for dark matter candidates to arise naturally in ADD type models through the use of continuous symmetries as opposed to the discrete parities currently used in the UED model.  4  Introduction to Kaluza Klein Theory In this section we briefly review Kaluza Klein theory, starting with its original form as an attempt at a unification of gravity and classical electromagnetism and culminating with its more modern interpretation as a method for unifying gravitation with gauge theories via the association of gauge groups with the isometries of the extra dimensional metric.  Very soon after the discovery and experimental confirmation of General Relativity (GR) in 1921, Theodore Kaluza proposed that the 4-dimensional spacetime of GR should be supplemented with a single extra spatial dimension [7]. His motivation was to give a unified account of the only two somewhat well understood fundamental interactions at the time, Gravity and Electromagnetism. Kaluza showed that both Electromagnetism and GR can be realized as different manifestations of a 5-dimensional GR. That is, point particles moving along geodesics in a 5-dimensional Riemannian spacetime appear in four dimensions as trajectories of particles subject to both Gravitational and Electromagnetic forces. One of the key underlying ideas of the Kaluza Klein idea is that the extra dimension allows for a unified description of particles with different spin [8]. In other words, a single representation of the D-dimensional Lorentz group, where D  >  4,  can be decomposed into several different representations of the 4-dimensional Lorentz group. Since different representations of the 4-dimensional Lorentz group correspond to distinct physical particles with different spin, we see that certain combinations of particles with different spin in 4 dimensions can be viewed as simply different components of a single particle with a single spin in higher dimensions. While formulating his theory, Kaluza faced the obvious question: why had this fifth dimension not been observed yet? Indeed, even today there is still no hard evidence for the existence of any new spatial dimensions of any kind. In order to avoid this question, Kaluza imposed his famous “cylinder condition”. This states that, for as yet unknown reasons, all physics takes place on a 4-dimensional hypersurface in a 5-dimensional Universe [7] [9]. In practice, this is equivalent to demanding that all derivatives with respect to the fifth coordinate vanish.  While Kaluza studied the classical structure of the 5-dimensional GR only, Oskar Klein, in 1926, provided the first investigation into its compatibility with Quantum Mechanics [10]. Klein assigned two properties to Kaluza’s extra dimension: i) that it be length-like with a circular topology and ii) that the length scale associated with the extra dimension be extremely small. Under the first property, any field which is a function of the full spacetime, x(x, y), becomes periodic in the extra dimensional coordinate (xP,y)  =  (x#,y + 2irR),  where B is the radius of the extra dimension.  5  These fields can then be Fourier expanded as  y(x,y)  =  where n labels the Fourier modes. Each of these modes then has a momentum in the extra dimension of order ii =  Klein saw that if R is taken to be extremely small then all of the modes (except, of course, the  0 mode) will have high enough momenta to put them beyond the reach of any experiment at the time.  In this case, only the n  =  0 modes are observable and each field should then be effectively independent of  the fifth coordinate. In this way, it was Klein’s work which provided the underlying physical explanation for Kaluza’s mysterious cylinder condition. In what follows, the 5-dimensional Kaluza Klein (KK) theory will be developed in detail and then a brief explanation of how to appropriately expand it for higher dimensions will be provided.  Before any calculations at the level of a Lagrangian are performed we must find a suitable parame terization for a 5-dimensional metric assuming that the ground state is factorizable and is given by M 4x where M 4 is the 4-dimensional Minkowski space and  81  81,  is the one-sphere. This choice of parameterization is  equivalent to the choice involved in how to decompose the representation of the 5-dimensional Lorentz group into different representations of the 4-dimensional Lorentz group. We choose our parameterization so that the 5-dimensional metric  §AB,  where A, B  =  0,1,2,3,4, can be written in block form with each different  4-dimensional representation comprising a different component. Letting for the coordinates  XA =  (x’, y) the 5-dimensional metric,  AB =  v  p4 9  Th  944  êA  =  (e, e ) 4  be the basis vectors  eAeB, takes on the general form  gAp =  The upper left component, and lower left components,  9ji4  ,,  with p, ii  and  =  0, 1,2,3, is a spin 2 symmetric tensor field, the upper right  are both spin 1 vector fields, and the lower right component is a  spin 0 scalar. We also assume that the structure of the 5-dimensional spacetime is such that, for each fixed point in the 4-dimensional spacetime, , tt there extends a 1-dimensional space with points labeled by y. Two x close-by points in the 1-dimensional space define a displacement, dye , exclusively in the extra dimension. 4 As usual, the square of the displacement, ds 2 terms of our parameterization g44  =  =  44 defines the metric for 81 which we can write in ‘ , 2 dy  In the same way, the metric for the 4-dimensional space can be  determined, in terms of the parameterization, by demanding that any displacement occurring exclusively in the 4-dimensional spacetime should be orthogonal to the internal 1-dimensional space. This orthogonality is needed in order to maintain consistency with our assumption of a factorizable spacetime. A displacement dxl’ê,, is not actually orthogonal since ê,ê 4  =  0. 6  The required displacement then must have the general form (da*, determined by demanding that (dxêfl +  yê4)ê4  =  =  y)  =  dxë + 23yê4 where y is  0 so that  dx” 4 44 —g  The metric for the 4-dimensional spacetime is then defined as usual by  2 ds  gdxbdx”  , Ay)AB(dx”, y) 1t (dx  =  =  —  Introducing the new notation  g44  =  4)  =  we can rewrite the 5-dimensional metric as  (g,u,  -  + 4)B,tBv  I  PAB  4)B  4)  If we let 4) represent the metric for S’, which just sets the value of 4) to —1, then the general form of the 5-dimensional metric becomes  gAB  =  (I  g  —  BB  —B  —B  —1  with an inverse given by  JLZ)  AB  -B”  -1+B”BA  The dynamics of the original KK theory are determined by the classical Einstein-Hubert action generalized to 5 dimensions  sPH  where  i%  =  is the 5-dimensional Gravitational coupling constant,  =  det(’AB), and fi’ is the 5-dimensional  Ricci scalar. To determine the determinant we use an identity for block matrices given by  (A  B(A  o(’i  A’B  C  D}C  i}o  D-CA’B  7  The determinant of the product is then the product of the determinants so that  det  (A  B =  det (A) det (D  —  CA’B)  D)  \‘  Applying this to our case leads to det (B)  =  det (gPj. Since det(B)  =  det(gAp)  and det(g’ ) t11  =  1 we can also equally conclude that det(ñAB) = det (g,,j. The components of the 5-dimensional d e (gflv) Ricci tensor can be written in terms of g, and B as [11]  =  +  (BMVrB: + BvVrBL) + B,LBvBArBAT + BBrv  + BMBArB  =  =  where Ba,.  =  All of this can be used to compute the Ricci scalar  —  =  ABft  =  R + BB1w  The Einstein-Hilbert action is then written as  =  -  z  f  x/ [R + BB1w] 5 d  In the original KK theory all the dynamical variables were assumed to be effectively independent of the coordinate y so that any integration over y simply lead to a redefinition of the Gravitational coupling To aid the correspondence between the B fields and the Electromagnetic vectors A we  constant ,c =  introduce a normalization constant, w  =  ‘../,  such that B  =  wAg,. This serves to correct the normalization  of the Maxwell action as well as adjust the mass dimension of the A field which now must have mass dimension 1 as required. We are then left with  = —--  f  x./R 4 d  —  f  x/F 4 d P 1 ”  sjj% + Sjzweti  The reduced 4-dimensional effective field theory then contains both Gravity and Electromagnetism, as promised. The procedure of starting with a higher dimensional action, expressing everything in terms of 4dimensional objects, and integrating over the extra dimensional coordinates is formally called a dimensional reduction and will be used extensively in the current thesis.  8  Recall that the above action was derived starting with the assumption that the ground state for the the ory is M 4 x S’. This form of the ground state can be understood in terms of the mechanism of spontaneous symmetry breaking [11]. That is, the symmetry group associated with the 5-dimensional Minkowski space is the 5-dimensional Poincare group F , while the symmetry group associated with the factorized spacetime, 5 4xS M 4 x U(1). The picture that we should have is that in extremely early times the , is the factor group, P 1  spacetime was based on the group P 5 which then spontaneously broke down to P 4 x U(1). Since S’ is a com pact space, the process which leads us to the ground state M 4 x S’ is called spontaneous compactification. It is now obvious that it was our choice of ground state that lead to the unification of Gravity and Electro magnetism. It has been known for a long time now that the correct field equations for General Relativity are obtained by assuming a local invariance with respect to P 4 transformations and, likewise, the correct field equations for Electromagnetism are obtained by enforcing a local invariance to U(1) transformations. The trick of Kaluza Klein theories is then revealed as reinterpreting the internal gauge symmetry that gives rise to Electromagnetism as a spacetime symmetry associated with the extra dimension compactified on a circle.  The above procedure can be generalized to higher dimensions as well by assuming a ground state of the form M 4 x ED where ED is a D-dimensional compact space with a Euclidean signature  (—, —, —,...).  Performing the same type of procedure to find a suitable parameterization, we can write  YAB =  ( g + &flBBe  I  5 Ba  k\  where 4 is the metric for the compactified space ED and g, defines the distance in the 4-dimensional spacetime between different copies of  ED.  There are, in principle, many symmetries associated with the  compact space ED. Each symmetry is associated with a set of transformations, called an isometry, which leave the metric,  invariant. The isometry transformations of a given metric are determined by the  vector solutions of Killing’s equation [12]  VaVas + V Vaa =0 0 where a, 3 are the space-like indices and a indexes the set of vectors that satisfy the equation. Each isometry constitutes a group so that any compact space ED can admit many different isometry groups. Since defined here as impose that  =  is a vector on ED, we can expand it in the Killing basis as  =  Vj”Bj and  Bflx). Two natural questions to ask are then how does it transform under the isometry  transformations and how are we to interpret it physically?  9  We’ll start by considering general coordinate transformations of the form  =  x’L(x1L)  y’  y’a(xL,yo)  =  where x 11 represents the coordinates of the 4-dimensional spacetime while y° represents the coordinates of the D-dimensional compactified space. Under these transformations the full metric transforms as C D  -,  AB 9  so that the off-diagonal elements and  -,  =  transform as  Ox  6 Di  -  -g  3 g  +  Di  6 7 Dy Dy  7Y6  respectively. We can then write the transformation law for B as  B,‘c  —  ‘cr’  —  8 Dy  =  Dx” 8y 6  DY ---c  Dycr =  D Dp D / 1 ---ö. -5-7;; gAo +  -  D =  +  Dy Dy 6 y6  Dya ----B +  In general, the isometry groups that we are interested in here are Lie groups. If the isometry group is a Lie group then all isometry transformations of  can be generated by a finite number of infinitesimal  coordinate transformations  such that 5ya  Capay, where  €a  are the infinitesimal parameters and  a  =  VfD are the generators of  the isometry transformations. The Lie Algebras associated with the isometry groups are then generated by the standard commutation rules  ri-’  1-’l_ 2cr’ Jabs-c  [‘a,’bJ  —  where fb are the totally anti-symmetric structure constants of the isometry group.  10  The Lie Algebra can also be defined in terms of the Killing basis vectors as  (vov,7  —  iqayj)  37  =  Under the isometry transformations, the variation of the Killing vectors can be determined by expanding in a Taylor series with the expansion parameter c’V,f  614  = v’a  -  va  =  ava 3 ebvI  Expanding B in the Killing basis, as before, allows us to determine its variation in terms of general coordinate transformations  —B  Varying B in terms of the isometry transformations then leads to  BdVf3flVfX + Vj6Bj  =  6 [(6 + eb3flV,)  B 3 vj  +  voAEa]  —  xvaBa  After solving for V,f6B  6B t V,f  = vaa a  +  v 0 .sze’ [vPa  —  0 va] 3 v/  =  vaaA? +  and rearranging the group indices, so as to factor out the Killing vector, we obtain the variation of B with respect to the isometry transformations  öB  = oea  + fBjeC  This is precisely the variation of a gauge field under a Non-Abelian gauge transformation!  11  This also makes the Kaluza Klein method abundantly clear: by assuming the local invariance of the theory under the infinitesimal coordinate transformations  =  =  x”(x”)  ya  + 3ya  The field Yjw is the metric of the 4-dimensional spacetime, the field 4 is the metric of the D-dimensional space ED, and the fields  are the gauge fields associated with the isometries of ED. Again, in the 5-  dimensional case, where the space E 1 is taken to be S , there is a single gauge field associated with the 1 isometry group, which is simply U(1). Since U( 1) is an abelian group, the Lie Algebra is trivial and the variation of the gauge field occurs as it should for the minimally coupled Electromagnetic field. Using the above method, we can then unify gravity with all of the familiar Non-Abelian gauge theories such as SU(2), SU(3), etc., by making the appropriate choice of extra dimensional geometry.  As a cautionary note, KK theory, at this point, may seem to be too good to be true. There are, however, many existing problems that continually keep it from providing a realistic description of the real world [11]. These are the inclusion of fermions and spontaneous symmetry breaking, anomaly cancellation, and the fact that, so far, there is absolutely no hard evidence of extra compactified dimensions to name just a few.  Although there is much more to be said about KK theory in general, any further discussion is beyond the scope of this thesis. For our purpose, which is to study an interacting particle theory within the context of an extra dimensional background, we need only know the ground state metric for a general spacetime, 4 x ED, and how to expand the fields which we allow to propagate out into the extra dimensional space. M Therefore, to conclude this section, I will simply introduce the general form of the ground state, given by  -  gAB  = (. I  0  4  where ‘q,,, is the standard 4-dimensional Minkowski metric and 4 is, again, the metric on the com pactified space ED.  12  The Dark Matter Problem Astronomers and cosmologists have, over the years, been very interested in the determination of what type of materials, and in what quantities, exist in the Universe. Some of these materials can be detected by their ability to emit or absorb electromagnetic radiation but all materials, in sufficient abundance, should be detectable through their gravitational interaction with their immediate cosmological neighborhood. It is now accepted by a clear majority of the cosmological and astronomical community that there exists certain type(s) of matter in our Universe which are both non-luminous and non-absorbing, i.e; dark matter. The exact identity of this dark matter is one of the long-standing problems in cosmology, dating back to 1933, when Fritz Zwicky first noted irregularities in the radial velocity measurements of galaxies in the Coma cluster [13]. He found that the orbital velocities of the individual galaxies were almost a factor of ten higher than what he expected given the total amount of visible mass in the cluster. The conclusion that Zwicky reached was that, in order to hold the cluster together, it must contain a massive amount of invisible (or dark) matter. Zwicky’s discovery still holds true today in that, probably the most convincing evidence to date of dark matter is still the observation that various luminous objects move much faster than we should expect if they felt only the gravitational effects of their visible neighbors. Probably the most important and well known example of this type of behaviour is the measurement of galactic rotation curves.  In this section we will illustrate a sample of the now large body of evidence which implies the existence of dark matter. We will discuss the galactic rotation curves, x-ray emission in nearby clusters, large scale structure formation, and the dominence of cold, as opposed to hot, dark matter.  13  Galactic Rotation Curves The rotation curve of a galaxy is represented as a plot of the orbital velocity of the stars or gas in the galaxy against the distance from the center of the galaxy. In this section we describe the discrepancy be tween observed galactic rotation curves and the predictions of Newtonian dynamics when only considering the visible mass. This is generally considered to be the most convincing and direct evidence for dark matter on galactic scales [14].  Consider an object in a circular orbit about the center of a galaxy. If the object experiences a centripetal acceleration a  =  —  due solely to the overall gravitational attraction of the galaxy then the  rotational velocity is given by  where G is Newton’s gravitational constant,  /GM(r)  =  v2(r)  is the radius of the orbit, and MQr) is the total mass  r  contained within the orbit. The total mass is defined as  M(r) =  where dfZ  =  f  p(r) 2 drdQr  sin OdBdçb and, assuming it is spherically symmetric for simplicity, p(r) is the mass density  within the sphere of radius r. If the object’s orbit lies outside the visible part of the galaxy then, assuming that the lack of visible light implies the lack of mass, we would expect that p  n.j  -  so that  çR  M(r)r..i  I Jo  drriR  where R defines the boundary of the visible sector of the galaxy. This implies that we should expect the rotational velocity of the object in orbit to be of the form v(r)  n.’ —z.  Instead, measurements of the galactic  rotation curves, such as the one in Figure 2 below for M33, usually obtained by measuring the 21 cm emission line from neutral hydrogen gas, show that the rotational velocities become approximately constant out to the largest distances from which the rotation curve can be measured. This constant rotational velocity implies the existence of a spherically symmetric halo consisting of dark matter with a mass density  PDM  4. This  dark halo then extends out, past the visible sector of the galaxy, to engulf the orbit of the object so that  M(r)  dr’  :  r  v  constant  This observation suggests that the mass of the galaxy continues to grow significantly when there is no more luminous component to account for this increase.  14  At some point the mass density of the dark halo will have to decline so as to render the total mass of the galaxy finite but it is not known exactly where this drop-off occurs.  Figure 2: The observed rotation curve for the galaxy M33.  Although quite convincing, rotation curves do not constitute the only observational evidence for the existence of dark matter.  15  X-Ray Emission in Nearby Clusters In this section we will illustrate how the very existence of hot X-ray emitting gas in nearby galaxy clusters implies the presence of dark matter.  Detailed observations of hot X-ray emitting gas in nearby galaxy clusters has been reported by the Einstein X-ray orbiting observatory [15]. The presence of’ dark matter in these clusters is confirmed by the very fact that the hot X-ray emitting gas is still in place. That is, if there were no dark matter in the cluster the gas would have expanded out beyond the boundary of the cluster under its own pressure on time scales much shorter than the age of the Universe. The existence of a large amount of dark matter is needed to add enough gravitational attraction to counteract this expansion and keep the gas in place. In this case, it seems as if the assumption of hydrostatic equilibrium is well justified and can be used to estimate the total mass of the cluster. The assumption of hydrostatic equilibrium simply states that the thermal pressure from the gas is balanced by the gravitational attraction of the galaxy dP dr  GM(r)p(r)  —  where P is the pressure of the gas, p is the mass density of the gas, and M(r) is the total mass within the spherical boundary with radius r. The pressure of the gas can be approximated by the perfect gas law so that  =  where T is the temperature of the gas and p is its mass in units of the proton mass mp. Assuming a homogeneous chemical composition, i.e; p is independent of the radius, the mass can be determined as  M(r)  =  —rT(r)  kB Gpm  dlnp dlnT(r) + dlnr dlnr  where we have also made the simplifying assumption that all contributions to the pressure are thermal. It is possible that there could be non-thermal components of the pressure, P, consisting of high magnetic fields or cosmic rays which would constitute an extra term added on to the mass relation of the form  M(r)  =  —rT(r)  kB  P?-t dlnP din r  Gpm Tota1  The thermal contribution to the pressure should far outweigh the non-thermal contribution in most circumstances though, so we choose to ignore it here. In general, the amount of dark matter needed to counteract the pressure turns out to be quite high.  16  For instance, the mass of the Coma Cluster is estimated to be  n-i  2 x 1O’ 0 within a radius of 3.6Mpc of M 5  the center. The contribution to this mass from the star content and intracluster X-ray emitting gas, though, is calculated to be  2.3 x 10 0 [15]. This means that in order to stabilize the hot intracluster gas almost M 14  90% of the mass density of the Coma cluster must be composed of dark matter! The evidence for dark matter presents itself on even the largest cosmological scales as well, in the theory of structure formation.  17  Large Scale Structure Formation and the Cosmic Microwave Background The theory of structure formation not only requires the existence of dark matter for agreement with the cosmic microwave background (0MB) results but places constraints on the properties dark matter must have. In this section we briefly describe the theory of large scale structure formation, the CMB, and how the 0MB results constitute the most definitive evidence for non-baryonic dark matter.  The current theory of structure formation proposes that large scale structures in the Universe are formed in a bottom-to-top type scenario where matter gravitationally clusters around initially small density fluctuations over long periods of time [16]. Matter falls towards regions of higher density while falling away from regions of lower density. In this way, initially higher density regions become even denser over time while initially lower density regions become ever more sparse. This type of behaviour, if allowed to occur for long periods of time, could then reproduce the type of structure we see today, namely the way in which galaxies are connected in filaments that stretch out very long distances to surround immense amounts of space almost completely devoid of galaxies. The evolution of large scale structure is described by the equations which dictate how small fluctuations in mass density, 6(f, t)  <<  1, grow in amplitude under the influence of gravity  in an expanding Universe. Suppose that a small amount of matter is added or removed within a spherical region of radius R so that the density within the region is p  =  ( 1 + 6). As the matter within the spherical  region becomes compressed by its own gravity, an internal pressure will begin to build which will fight against the collapse. If the pressure builds fast enough then the spherical region will relax into hydrostatic equilibrium, preventing the matter from gravitationally collapsing into the familiar structures we observe today. The condition that must be met in order for gravitational collapse to occur is that the radius of the region containing the density perturbation must be larger than the Jeans length which, in a flat, expanding Universe, is given by  where c is the local speed of sound. Using this, we can also define the Jeans mass  M=p(fA)  Our previous condition can then be translated into one concerning the mass, i.e; for gravitational collapse to occur, the total mass within the spherical region must be greater than the Jeans mass.  18  The existence of a cosmic background radiation stemming from photons propagating from the early Universe was predicted by Gamow, Aipher, and Herman [17] and unintentionally discovered by Penzias and Wilson [18] [19]. After an enormous experimental effort the CMB is known to be isotropic at the  io—  level [20] and correspond to a black body spectrum with temperature T  =  2.75° K. The analysis of  CMB anisotropies is widely used as a tool for the accurate and precise testing of cosmological models by putting stringent constraints on cosmological parameters. Using the Wilkinson microwave anisotropy probe (WMAP) data while taking into account various other CMB experiments, [21] [221 [23] [24], the contribution from the baryons and the total amount of matter in the Universe to the density parameter was shown to be [25]  0.0224 + 0.0009  =  2 flMh  =  In the early Universe, before the creation of the CMB, the photons, leptons, and baryons were all tightly coupled together through their various interactions and essentially comprised a single fluid. Considering the baryons as a negligible contaminant in the gas just before decoupling, we can estimate that Mj  n.j  M®. 5 7 x 10’  Shortly after this, as the Universe cooled enough to allow the baryons to combine with the charged electrons to form electrically neutral atoms (which we still call baryons), the baryons and photons decoupled from each other, evolving into two separate gases. The speed of sound in the resultant baryon fluid turns out to be quite a bit smaller than in the original composite gas, leading to a drastically reduced Jeans mass M  nJ  1 x 10 . These results indicate that, before the decoupling era, only the most massive objects that 0 M 5  we see today could have undergone gravitational collapse. Any of the smaller objects, including our own Galaxy, must have begun their gravitational collapse after the decoupling era. As it turns out, requiring that the density perturbations in a baryonic universe with no dark matter are large at current times, ö> 1, necessarily implies that 6 >  io  just before the decoupling time. This thoroughly contradicts current  observations of the cosmic microwave background [15]. However, if we introduce a dark matter component into the photon baryon fluid and assume that the dark matter is not baryonic then, due to the lack of any significant coupling between the dark matter and the photon gas, it can decouple from the fluid much earlier than the photon-baryon decoupling. Since the dark matter decouples much earlier, as early as the epoch of matter-radiation equality, its density perturbations have more time to grow before the photon-baryon decoupling. The baryons, once decoupled, then simply fall into the pre-existing gravitational wells produced by the dark matter density perturbations. The existence of non-baryonic dark matter therefore kick-starts the process of structure formation much earlier and is therefore absolutely necessary to explain the present large scale structure of the Universe.  19  Hot vs. Cold Dark Matter We can make one further distinction about the necessary character of the dark matter based on whether it was relativistic or not when it decoupled from the photon-baryon composite gas. In this section we inves tigate the consequences for each case and conclude that the dominant component of dark matter must have been non-relativistic when it decoupled.  The present day character of the dark matter is highly dependent on how relativistic it was at the time of decoupling from the photon-baryon composite gas. We refer to dark matter that was non-relativistic at the time of its decoupling as cold dark matter while, if it was relativistic, we refer to it as hot dark matter. Dark matter particles which were originally relativistic when they decoupled should have exhibited a type of motion called free streaming where they moved freely in random directions at close to the speed of light. The free streaming of the non-baryonic dark matter particles introduces a length scale called the free streaming distance dfS, which is the mean distance travelled before it becomes non-relativistic. The free streaming motion of the non-baryonic dark matter particles acts to smooth out any fluctuations on scales smaller than df . The neutrino is a good example of a hot dark matter candidate. Assuming the neutrinos 5 mass is  1eV, the free streaming distance is  immense structures with mass  ‘-.  41  (I-Y”) Mpc which suggests a top-down scenario where  m,, / 0 are the first to form [15]. The smaller structures are then formed M 5 lO’  by the eventual fragmentation of these larger initial structures. This result, however, is in direct conflict with current observations which show that the largest objects that we can see today such as superclusters are in fact just collapsing recently while galaxies are seen as far back as z  6. This implies that the dark  matter must have been non-relativistic when it decoupled from the photon-baryon composite gas so that any free streaming would be negligible.  The above arguments based on galactic rotation curves, X-ray emission in nearby galaxy clusters, large scale structure formation and the CMB, and the free streaming length suggest that there exists massive amounts of a new type of matter which avoids direct detection due to a seemingly vanishingly small inter action strength with standard model particles. Its present effects become truly prominent only on scales where its gravitational attraction is observable. Its discernable properties, so far, are that its dominant component must be non-baryonic and must have been non-relativistic from the time when it decoupled from the photon-baryon composite gas onward. In the following thesis, we will formulate a particle theory of dark matter consisting of many different but highly related particle species, each of which have all of the above mentioned attributes.  20  Kaluza Klein Dark Matter in the Universal Extra Dimension Sce nario The Universal Extra Dimensional (UED) model has been the subject of intense investigation recently due to its ability to naturally produce a dark matter candidate [26] [27] [28]. This model was originally studied by Appelquist, Cheng, and Dobrescu and gets its name from the fact that all particles are allowed to propagate out into the bulk, i.e; all particles have universal access to all existing compact dimensions [29]. This is in contrast to both the ADD type models where the entire standard model particle content is restricted to a brane and the, previously unmentioned, intermediate models in which only the standard model gauge bosons are allowed access to the bulk while the fermions live on the brane(s) [30] [31]. In this section we will outline exactly how a dark matter candidate naturally arises in the context of the UED theory. For simplicity, we will also restrict ourselves to simplest UED model, that is, only one extra dimension.  The most important feature of the UED model is that, since there is no brane, the translation invari ance of the extra dimension is preserved which leads to the conservation of momentum in the fifth direction. This, in turn, translates into the conservation of the Fourier mode number, called the KK number. In order to induce chirality in the fermions which propagate out into the bulk the extra dimension is orbifolded which allows us to project out the unwanted fermionic degrees of freedom. The process of orbifolding also has another effect, which is to break the conservation of KK number down to the conservation of KK parity. This affects the interactions of the odd numbered Fourier modes by projecting out all interactions except those involving even numbers of the odd numbered modes. The conservation of KK parity implies that the least massive of the first excited KK states cannot decay into any of the standard model zero modes and will therefore be stable.  In order to see all of this more quantitatively, we will take the simple example of a real free scalar field and allow it universal access to the bulk. The action for this is  S  =  f  xdy 4 d  [GAB8(x,y)8(x,y)  —  (x,y)] 2 mg  where xIL are the coordinates of the 4 dimensional spacetime, y is the coordinate for the single extra dimension, and m 0 is a bulk mass.  21  The spacetime indices, A and B, run from 0 to 4 and, assuming the spacetime is factorizable, the metric GAB  is given by  0 GAB(  0 where  i  —1  is the 4 dimensional Minkowski metric and the —1 is the metric for the circular extra dimension.  Due to the circular symmetry of the extra dimension the fields are periodic in y as  (x, y + 2irR)  (x, y)  where R is the length scale of the extra dimension. As before, we can immediately exploit this periodicity in order to decompose the fields into their 4 dimensional Fourier modes, (x), as  (x,y)  where the prefactor (2irR)’J 2 is a normalization factor. Substituting this into the original free field action and integrating over the entire extent of the extra dimension gives  S  =  [  fd4xdy  >  O(x,y)8”(x,y)  n=—oo m=—oo  f  —  2 (8(x,y))  —  m2(x,y)]  irR 2 j  dx  dy  [cbn8bm  +  —  Since any two Fourier modes are orthogonal, we find t2irR  J  dyeihh/’  =  m 2irR6  0  so that  S  f  x 4 d  —  22  (m +  )  mqci]  etm)/R  The effective 4 dimensional theory then contains an infinite set of real scalar degrees of freedom whose masses form a doubly degenerate equi-spaced hierarchy pattern  where the integer n is called the KK number. By generalizing the above method, every standard model particle is then associated with an equi-spaced set of particles with identical quantum numbers. Any set of equi-spaced degrees of freedom that arises in this way is called a KK tower. Just like the particle in a box scenario in elementary quantum mechanics, the momentum along the extra dimension, p, is discrete and equal to  .  As all particles can equally access the bulk, momentum in the fifth direction, and hence the KK  number, is conserved.  In order for the entire standard model content to propagate in the bulk we need to find a way to generalize the Dirac action to 5 dimensions. The Dirac action in 4 dimensions is given by  S =  where algebra  [7U,  is the spin 1/2 fermion field, 711]  =  ij1, 2  f  x (i 4 d 8 7  —  m)  and the set of matrices  =  “,  which obey the Clifford  are the Dirac matrices [32]. In order to generalize this action to 5 dimensions we  must find a way to extend this set of matrices such that the extended Clifford algebra, obeyed. The standard choice is  A 7  =  {eybt, i 5 7  }  where  75  matrices are used to form the chiral projection operators  =  A, 7 7 [ B]  AB, 2  is  3 [33]. In the standard model, the 75 2 1 0 7 i  L,R  =  (1 + 75) which project out the left and  right handed fermions. The ability to project out the different chiral fermions is absolutely necessary due to the fact that fermions of different chirality each interact differently with the weak gauge bosons, i.e; we must be able to project out the left and right handed fermions in a 5 dimensional model as well. This, however, is complicated by the fact that the counterpart to the  75  matrix in 5 dimensions is given by i( 2 ) 5 7  =  i, which  does not project out the chiral fermions. In order to regain the chiral fermions in the 5 dimensional model we then use a technique called orbifolding to change the topology of the extra dimension. This allows us to use the geometry of the extra dimension as an effective projection operator.  In order to have created the circular extra dimension in the first place we would have started with an infinite line segment, the same as the familiar 3 spatial dimensions, and impose the periodic identification y  ‘-  I,, + 2irR. Note that this requires no 2 dimensional embedding space. That is, there is no point in the  space whatsoever which acts as the center of the circle, i.e; which is equi-distant from all points.  23  In this case, the radius is just the quantity which, when multiplied by 2x, gives the total length of the extra dimension. In order to obtain an orbifold, we impose the additional discrete identification: y  n.’  —p. If  we view this identification as a transformation on the coordinate then, by applying it twice, we arrive back at the same point. Any discrete symmetry which exhibits this type of behaviour is called a  .Z2  symmetry.  If we had to apply the transformation 5 times to arrive back at the same point then this would be called a £5  symmetry and so on. The resulting space then no longer consists of all points in the range 0  since the combination of both identifications gives p are now identified with points p that 0  p  r’.  2irR  —  p  <  2xR  p. This implies that all points with y> yR  yR so that the resulting space then only consists of all points such  irR. Note that the end points of this space are not identified with each other through any  combination of either of the two existing identifications. In fact, they are fixed points, i.e; points that are related to themselves via the existing identifications  (p (p  =  xR)  p  =  0)  -y  .  (y  p =  —y  =4.  --irR)  (p p  =  n’  0)  p + 2irR  (y  =  irR)  The existing space is then the result of starting with the circular space, S , and “modding out” the 1 discrete symmetry  £2.  We label this modded space, generically called an orbifold, as S’ /Z . An orbifold 2  is, in general, any space which is obtained by imposing identifications which result in the existence of fixed points. In terms of topology, this process has the effect of folding the circle in half onto itself in order to produce a finite line segment with fixed end points.  y=O y=xR  y=O  y=xR  Identify y with -y  Figure 3: By identifying y with -y we can change the topology from a circle to a line segment.  We can now ask what effect does the orbifolding process have on the free scalar field theory that we saw earlier.  24  We can rewrite the Fourier expansion in an eigenbasis which separates the Fourier components which are even and odd under the y  (x,y) where  4  n  =  —y identification as  .çL [t(x) + E  t(x) cos  is the zero mode which is, by default, even,  ()  +  Z k(x) sin  are the even KK modes, and ç are the odd  KK modes. If we now impose that the 5 dimensional action be invariant to the .Z 2 orbifold symmetry then we must choose F to have some definite parity. In other words, it should be either even,  or odd,  ,  where  x,y)  [=ixi +Et(x)cos  =  r(x,y)  =  Zcx)sin()  Whichever choice we make we are effectively eliminating half of the 4 dimensional scalar degrees of free dom. In other words, any field propagating in a space with an imposed orbifold symmetry on it naturally has half of its 4 dimensional degrees of freedom projected out. Since the choice of whether the field in question is even or odd under the orbifold symmetry is completely arbitrary, we can choose the definite parity of each individual field based on phenomenological motivations, such as chirality. In the UED model, the Higgs field, the first 4 components of the gauge fields, the right handed chiral SIlL (2) singlet fermions, and the left handed chiral SUL (2) doublet fermions are all even under the orbifold symmetry while the fifth component of the gauge fields, the left handed chiral  L 511  (2) singlet fermions, and the right handed chiral SUL (2) doublet  fermions are all odd. The even fields all have zero modes which are identified with standard model particles while the fields with odd definite parity have no zero mode and have yet to be observed.  The act of £2 orbifolding the extra dimension lead directly to the introduction of two fixed points and therefore destroyed the translation invariance in the extra dimension and the momentum conservation with it. There is, however, still a discrete residual symmetry left over from the original translation invariance, represented by y  —÷  y + xR, which is different from the  25  £2  symmetry involved in the orbifolding.  Letting p  —÷  p + 7rR, the fields in the orbifold space behave as  (x,y)  +E(—lr(x)cos  =  C(x,y)  =  E(_1y’4c(x) sin  (!)  Since performing the same transformation twice returns the original fields, this corresponds to another £2  symmetry. Imposing that the action be invariant to this symmetry means that all n  be produced in pairs while all n lightest n  =  =  =  1 KK modes must  2 KK modes may be singly produced and, most important of all, the  1 KK mode is stabilized against decay. Therefore, the conservation of KK number then breaks  down naturally into a new symmetry, called KK parity, due to the orbifolding process. After taking into account all the various radiative corrections it has been determined that the first excited state of the photon is the lightest KK state. It is exactly this particle which constitutes the dark matter candidate within the URD model.  One of the main motivations for the current thesis arose from an attempt to answer the question: is it possible to produce a viable dark matter candidate within the context of an extra dimensional theory without resorting to orbifolds to produce the necessary symmetries to stabilize against decay? It turns out that it is indeed possible and outlining the exact methods which accomplish this is the subject of the next chapter.  26  The Scalar Dark Matter P henomenology There were two principle motivations for the present work. The first was to construct a particle theory which produces dark matter without resorting to the standard method of identifying any existing parities to stabilize potential candidates. The second was to determine and investigate the Boltzmann equation needed to simultaneously determine the relic densities of a multi-component interacting theory of dark matter. In this chapter we will describe in detail a new theory which contains multiple dark matter candidates arising from the propagation of a complex scalar field in a six dimensional compactified spacetime. The candidates are stabilized against decay via a globally imposed U(1) symmetry and a subset of the continuous symmetries of the extra dimensional geometry. In the higher dimensional bulk, the Fourier decomposition of the scalar field is more complex and results in the existence of many different KK towers associated with the single scalar in the bulk. It is found that the set of stabilized dark matter candidates is the set of the lightest KK modes belonging to each KK tower. In this way, each complex scalar in the bulk has, in principle, an infinite number of stabilized dark matter candidates associated with it.  In this section we will first solve the equations of motion in order to determine the correct harmonic expansion for the higher dimensional spacetime. The action will then be dimensionally reduced, revealing the 4 dimensional effective field theory, from which the Feynman rules for the effective theory can be properly read. All existing decay modes will be analysed in order to determine exactly which states are stable and therefore constitute the dark matter candidates. We will then end this section with a more detailed expla nation of the choice of action terms and a discussion on the gravitational stability of the extra dimensional geometry.  27  The Free Field Equations of Motion In this section the six dimensional free field equations of motion for a complex scalar field in the bulk is solved in order to determine the correct Fourier decomposition of the field. The six dimensional free field action is then dimensionally reduced to reveal the resulting four dimensional effective action.  We begin by introducing a six dimensional complex scalar field x(xM, 9,  )  which is allowed to propa  gate freely in a factorized bulk spacetime M 4 x S . The Fourier expansion for the complex bulk scalar field 2 is determined by solving for the equations of motion from the six dimensional free field action  =f  x 6 d  [AB aAx* aBx  where the indices A and B now run from 0 to 5, d x 6  =  —  xd9dq5, and m is a bulk mass associated with 4 d  the scalar x Our ansatz that the gravitational ground state is the factorized spacetime M 4  X  S leads to  the six dimensional metric  gAB  where  ij,  = (. I  0  0  crj 8  is the standard Minkowski metric for our regular 4 dimensional spacetime with signature  =  (1, —1,—i, —1) and  ,R 2 diag(R 2 9) is the metric for 2 sin  2  with constant radius R. As before, the  determinant of the six dimensional metric factorizes to give  =  —det  det (Safl)  (,j,)  =  2 sin9 R  The free field equations then take the form  86  =  =f  fd4xR2sin9d9d  xR sin 9d9dq 4 d 2  (  px* aax* 8  [a,x*8#x  —  )(  0 0  (  x 8  —  8afl)8x)  —  Assuming that the fields vanish rapidly enough as the four dimensional coordinates tend to infinity in either direction, x’  —*  +, we can integrate by parts to obtain  28  f  2 sin 6d8d dxR  [  —  mX*X] +  J  xdOd x 4 d 2 sin 8S8) 8 (R —  f  2 sin 9dOd SX*8X dxR BC  where the boundary term on the right hand side must vanish due to the fact that the sphere has no boundary. The equations of motion are obtained by varying the action with respect to the scalar field x  6 6S  f  xdOdb [R 4 d 2 sinO (—8&  2 sin 8S8’)] 6 m) + aa (R x  —  We now assume that x can be factored into eigenfunctions as x(x, ,) 8  =  0  (x)Y(8,). The functions  (x) and Y(9, q) represent any fluctuations in the four dimensional spacetime and the angular directions of the extra dimensions respectively. The equations of motion then become  (OO + m)  (sin 0 saY)]  =  Since the left hand side of this equation is a function of 4 dimensional spacetime only, the right hand side must be also. This means that  Y sin 9  ô (sin 9 S’8Y)  C  where C is a constant. This operator turns out to be just the Laplacian defined on S 2 which reduces to 1  where we have chosen C  —  ö  1  (sino.-) + .  =  Y 2 a =  -+ flY  With this choice for C, the solutions are then quite familiar and  are easily recognized as the spherical harmonics Y  Y (8, ql). For simplicity we will choose the orthogonal  =  normalization for the spherical harmonics which is given by  (9,) m Ye  1)  rnprn(9)im  where Pj (cosO) are the associated Legendre polynomials [34]. This then leads to the equations of motion  /  £(+1) 2’+mB}X=O 2 R  Since the most general solution to the six dimensional equations of motion is, up to a normalization, a linear combination of the eigenfunctions we can write  29  £=O  m—t  where C 2 is a constant to be determined by normalization. Integrating over the 6 dimensional subspace for some  , 1 4, m  £2,  , the normalization constant C 2 m 2 is given by  f —  =  dxf Rsin6d9j R 2 C 2 2 C R  f f  x12S6 4 d  x’’ 4 d  If we assume that the 4 dimensional fields are normalized so that  f then C 2  =  x’’ = 4 d  1  1/R so that the full Fourier expansion of our six dimensional complex scalar field is given by  £=O  m=—e  The four dimensional effective action which describes a complex scalar particle propagating freely through the six dimensional bulk is then given by substituting this expansion back into the action and integrating over the extra dimensions. This leads to the following free field action  s= £=O  m=—t  which dictates the equations of motion for each of the 4-dimensional harmonic modes or “Kaluza Klein” states.  30  The Action In this section we construct the full interacting six dimensional action for the theory and subsequently dimensionally reduce it to reveal the resulting four dimensional effective theory.  The action for the interacting theory of the scalar field x is chosen based on the idea that the entire standard model field content is restricted to the brane, forbidding it to propagate out into the bulk. Since the choice of any particular point in the extra dimensions is completely arbitrary due to the spherical sym metry we can, for convenience, choose our coordinates such that the point at which we choose to localize the standard model fields is at the north pole. Although it is possible that at some higher scale new physics could emerge which could break this spherical symmetry and thereby distinguish one point from any other, we assume that there exists a scale at which the symmetry is at least approximately spherical which is sufficient to move forward with the current line of reasoning. A complex scalar field, y, is then introduced and allowed to propagate in the bulk. The scalar field is assumed to be a singlet under the standard model gauge group but does have a global U(1) symmetry associated with it both on and off the brane. If we demand that the effective field theory be renormalizable then, due to its gauge representation, the scalar is only allowed to interact with the standard model via a quadratic coupling to the Higgs on the brane [35J. Since this interaction involves two spin zero bosons it will necessarily modify the Higgs potential. However, as we show below, it is possible to choose the interaction term(s) specifically so that the vacuum structure is left unchanged.  The entire 6 dimensional action can be written as  S  =  Bulk + 5 5 Brane  where the entire field content on the brane and its interactions are localized to the north pole of the spherical geometry of the extra dimensions by way of a delta function. The  Bulk 5  action, which describes  how the x field propagates in the bulk under its own self-interactions, and the  Brane 5  action, which describes  the interaction between the x field and the Higgs doublet on the brane, are, respectively, chosen to be  Buik 5 =  Brane 5  =  f  J  xR 4 d d 2 fl  xR 4 d d 2 fl [ãAB OAx*  [esM  + DI 2  aBx —  +  —  31  —  —  ] 2 0x12)  g2II2ix2]  6 (cos8  —  1)  where dQ is the measure for the 2-sphere  g2,  m is the bulk mass, 4 is the standard model Riggs  doublet, and ESM is the standard model Lagrangian density not including the Riggs kinetic and potential terms. Note that, due to the need to integrate over two extra spatial dimensions, the fields and their couplings no longer have the mass dimensions which are familiar from four dimensional theories. As we integrate over the extra dimensions to obtain the 4 dimensional effective theory we will be forced to redefine some of these quantities along the way in order to make sure they all have the proper mass dimensions. Before dimensionally reducing these actions we need to ensure that we are expanding all the fields about the proper vacuum state. Rowever, since we have specifically designed the interaction between the field  x  and  the Riggs doublet with the invariance of the vacuum state in mind, we are able to simply expand about the standard model vacuum state to obtain the true brane action  Brane 8 =  f  dQ 2 cPxR  [ESM  h8”h 4 + O,  —  h2  —  —  —  h2x2  6 (cos o  —  —  where the field h is the familiar real neutral scalar Riggs field and V is the Riggs vacuum expectation /22  value (vev) which is given in terms of the original Riggs parameters as V =  V 1—  Note that, due to  the presence of the delta function in the total action, both the bulk and brane potentials must minimize independently of each other. This allows us to exclude the four point function in the bulk when solving for the classical vacuum solution on the brane. It is clear that the value of potential is simply  2 1x1  =  0,  2 1x1  which minimizes the brane  which leaves us with the standard Riggs potential  VBrane  II 2  +  This then leads to the familiar form of Electro-Weak symmetry breaking and the brane action written above.  We can now begin to reduce our 6 dimensional theory down to its 4 dimensional effective theory from which we will extract information about any observable 4 dimensional effects. To save time and space, we adopt a non-standard summation convention such that, for each pair of indices (mi, £) appearing in the 4 dimensional effective theory, we assume the presence of the corresponding summation operator 4=0  =—4 1 ,n  unless stated otherwise. As an example, if we encounter a term in the effective Lagrangian density of the form  *m1m2]  fd4x[E 4=0  ,n,=—4  £20  32  =—t m 2  —  1)  then we will simply write this as  f  x 4 d  We are now in a position to dimensionally reduce the bulk action one term at a time by integrating over the extra dimensions. We have already determined the effective theory for the free field term in the bulk due to considerations in a previous section. The result is given by  f  x 4 d  [fm  (ao + L(L +1) + m)  ]  The integration over the bulk four point function is more involved, we start by expanding the  x field into  its harmonic expansion.  -  f  d2 2 dxR  2R  f  x 4 d  [±*rn1m2*rn3m4Y*m1Yrn2Y*m3Yrn4] 12034  f  Y*rn 2 dcrnl 4 Y 3  Since the spherical harmonics form a closed set, any product of two spherical harmonics can be written as a linear combination of other spherical harmonics. This statement is written mathematically as  y*rn3yrn4  =  +1)(2L+1)(2L+1) 3 (2L  ( )( 0  £=O m=—  L  £  L  £4  £  0  0  3 —m  m4  m  )  where the array-like objects above are the Wigner 3-j symbols’. For simplicity of notation, we define the symbol B7 as  12 B  ../(2L, + 1)(2L 2 + 1)(2L 3 + 1)  (L  £2  0  0  £  £3’\(  I I1 0) m  £2  £3  2 m  3 m  which we will from here on call B symbols. Then 3 m 1 4 01 3y T0 y*  meB  :  m  ‘For the definition of the Wigner 3-j symbols and a list of some of its properties refer to Appendix B  33  Inserting this back into the integral yields  I  co  (  d  £  7 fdQym1y2Y;m  The remaining integral over the three spherical harmonics is well known and is called the Gaunt integral. Its solution is known to be  f  ()mi+m  dQ  tm Y 12  =  (_1)ml+mf  B7  =  Recalling our new summation convention we can write the effective four point function as  _t(_1)m1+m3mB_7  7  7B7  7 7f  x;12n374 4 d  where we have redefined the four point coupling constant,  =  —a-—,  2 4-irR  so that it now has zero mass  dimension in the four dimensional effective theory.  The total effective bulk action is then written as  Bulk 8 4 =  f  x 4 d  [  ;m  + 1) + m)  +  “  7 B7  r  *rn1m2*rn3m4]  “:  We can now turn to reducing the brane action. Concentrating only on the pure Riggs sector for now, we have  J  xQ 4 d d 2 R [O,h8hth_  -h2  —  —  h4] 6(cos6— 1)  The presence of the delta function and the fact that all the fields are functions of 4 dimensional spacetime coordinates only makes the integration trivial. The effective terms are  2 2irR  —  j-h2  —  —  h4]  In order to correct the mass dimension of the Riggs field and its 4 point coupling we scale them as H  =  /iRh and A  =  A respectively. 2 2irR  34  This leads to  J  ctx  —  !LH2  —  —  H4]  =  where v  The interaction terms between the Higgs and the  f  f  dQ [h 2 dxR 2 + 2Vh + V ] 2  The spherical harmonics can be factored as before im  =  x field are  dn}:f1y72o (cos U  t ) N7’Py’ (cos 6  —  1)  where NI is a normalization  factor to ensure orthogonality of the inner product between any two spherical harmonics, Pt(cos 6) are the associated Legendre polynomials which have been defined in a previous section, and the exponential encodes all the dependence on the azimuthal angle  J  dQö (cosU  —  1)  =  The above integral involving the delta function then leads to  .  d(cos6)PI’ (cosU) Pr 2 (cos6) 6 (cosU  2 NI’N1  —  1)  In order to perform this integral we must first determine what is the value of the associated Legendre Polynomials on the brane, Pj” (1). The associated Legendre Polynomials are given by [34]  P1(x)  (  1)”’  dt+m  (1  —  =  2 )j÷m x d 2 m/ (x  —  We can rewrite this as  P1(x)  (1— 2 x )m/’2  T d d  2 (x  —  =  Then, through induction, we can calculate that ct dxt  —  1)  =  (2x)t!  so that  P1(x)  =  (_1)m(l  —  In order to calculate the factor __xt we must consider three separate cases m < 0 m tm dx  35  =  0 and m> 0.  Again, using induction, the Legendre Polynomials take on the form (_1)mxe+ImI P°(x)  (_1)m(1  =  —  if m <0  2)m/2 X  if m —  1)(f  —  2)  (  —  Im + 1)xe—ImI  0  if m > 0  Once viewed in this form, it is clear that the value of P° (1) is simply 6. Substituting this back into the above integral gives  f  ’17° (cos m dfW7 6 2  —  1)  =  2NN  1 + 1)(22 + 1) /(2e  The effective coupling term on the brane is then  f  2+ x [H 4 d 2 + 2vH + v ] \/(2 + 1)(2e 2  where the Higgs field and all the parameters have been rescaled and we have also defined the dimensionless effective brane coupling as  4 SBrane  f  The entire effective brane action is then  =  X [a,Hah1H 4 d  2 + 2vH + v (H ) /(2 + 1)(2t 2 2 + 1)XX ] 2  —  —  —  —  Combining the two effective actions we obtain the total 4 dimensional effective theory  4 S =  gl( —  2  f  dx  [  sm  (oo + ee + 1) + m)  1 —m 3 —m 4 2 —m B_m3 m4 m —*m 2 —*m l\ml+m3+mB_ml m 2 Xe Xe 3 Xe 4 £, £2 £ £3 £4 £ Xe  +8Hã’H  —  !H2  —  —  2 + 2vH + v (H ) /(2 + 1)(22 + 1)XX 2 ] 2  —  Note that the total angular momentum 2 is not a conserved quantity on the brane, whereas the angular momentum in the azimuthal direction, although it is certainly constrained, is still conserved. This makes sense since the presence of the brane breaks the over all spherical symmetry but not the rotational symmetry about the polar axis. This complicates the theory by generating an infinitely large mass mixing matrix for the Kaluza Klein states in the non-magnetic tower,  ,  which, in principle, must be diagonalized. The  offending term which gives the mass corrections is small enough to be considered a perturbation on the bulk mass relation,  =  . 2 0.032 TeV  Note that the “total angular momentum” here refers to the total angular momentum in the extra dimensions only. 2  36  We can estimate the mass corrections by investigating the mass mixing matrix more thoroughly  Smm =  f  dx  + 1))  [° (m +  +  + 1)(2t 2+  1)2]  To make the form of the mass mixing matrix more clear, we redefine the KK states, and mass term arising from the bulk interactions, in?  =  (2t + i’  +  +  =  /2t + 1,  n). The action is then  written as  Smrn =  where we have also defined C  =  2?—  f  cøx [m° + G? ] 2  for brevity of notation. The mass mixing matrix then takes the  form  m+G  C  G  G  m?+G  G  G  C  n4H-C...  The diagonalized masses of the physical KK states are the eigenvalues, which are determined by finding the roots of the characteristic equation given by  det  0 M  C  C  C  1 M  C  C  where M 1  =  m? + C  —  =0  CA..  . By calculating the determinants of successive truncations of this matrix we 1 A  see the consistent appearance of a term of the form  1 flL M  in the characteristic equation whereas all other  terms are suppressed by various powers of C. The approximate eigenvalues of the matrix are then simply given by the approximate characteristic equation  1 0M flL  0 which leads to A 1  m? + G. Standard  Schrodinger perturbation theory yields the corrections to the harmonic eigenstates to be  Y’°Y°-Cv2E+1  £(t+1)  k(k+l)  Since the first order corrections are suppressed by a factor of R C they are deemed to be negligible and 2 we can simply set  yO  =  37  We can therefore rewrite, to leading order in G, the mass mixing sector of the action as  Smm  f  dx [(my + G)°] =  f  x 4 d  [(e(e +1) + m  +  (2 + 1)) ;o]  Therefore, written in terms of the physical masses and the perturbed eigenstates, the final form of the fully dimensionally reduced four dimensional effective action is given by  4 S  =  I  d4x[sM  —  *m  (aa +  _(_1)m1+m3+mB  —  —  1)+m +2(2+ 1)) =4 Tn  7 —  *rn1 m2 *Tn3  fl4 —  2 + 2vH) /(2 + 1)(22 + 1)X (H ] 2  where the entire field content contained in the standard model Lagrangian density is assumed to have been appropriately rescaled.  38  The Feynman Rules In this section we derive all the Feynman rules from the dimensionally reduced four dimensional effective theory. Most of the Feynman rules can be easily read from the action as  H  H  p 2 —p 2  =  Figure 4: The Higgs Propagator.  —2  — 2 p Figure 5: The Effective Scalar Propagator.  H H  =  —i3Av  H  Figure 6: The Higgs Three Point Self Interaction.  H  H —  =  —z3A  ——  H H Figure 7: The Higgs Four Point Self Interaction.  ‘-0  H  =  —iv(2i+1)(2+1)  —*0  Figure 8: The Effective Scalar-Higgs Three Point Brane Interaction.  39  H .4.  S..  S..  H Figure 9: The Effective Scalar-Higgs Four Point Brane Interaction.  1 x  The Feynman rule for the four point bulk interaction is more complicated and deserves a more detailed explanation. We start in position space where the operator for the four point function is  t 4 S  = —  f  dx  [(  l)ml+m3+mB_  3 m £  m2  £  —  .*m 2 —m 4 m m B_m3 m 4 m *mj 3 Xe e 2 £3 £4 £ 3 e 4  ]  The momentum space Lagrangian is obtained by inserting the Fourier transformations of the individual fields  (x)  fd4 (p) e  The spatial integral then translates the exponential dependence into a delta function which serves to ensure conservation of 4 momentum  )ml÷m 1 1( + 3 m_ m 1 £  m2 —m £2  £  B_m3 £3  4 m m £4  £  f  3 m42x) *m1 -m2 -*m ö 4 pid P4Xt 3 d 2 p 4 (P1 + 3 1 Xe 2 Xe 3 Xe 4  — P2 — P4)  Since the functional derivative acts on these Fourier transforms as  (k)  fd4p(P) =fd4p4p_k)  =o  the amplitude is then obtained by taking successive functional derivatives of the momentum space opera tor. Letting Z  (  =  l)ml+m3+mB_ml  £j  m2 —m £2  £  m4 m 3 B ö (pi + p3 4 e and omitting the factor of (2ir) 4 £3  for brevity, these successive functional derivatives are given by  pt 4 6s  —  —  e cmi m3 *m3 -m4 1 Z 3 2 [fdP d P4 ° n 2 0 1 Xe Xe 3 Xe 4 +  40  f  pid d 2 p 4  1 -m 1 4 2 m 3 *m m i  1 ‘i.e  2 Xe Xe 4  j  —  P2  —  p4)  6 (k )6f 711 (ki)  =  Z  (o &‘ o ö + S 6 6 oml)  84pt 4 6 — —  =  2 p 4 d  12  ts m m 5 i s i z loEi mi4ms +6 ni Es Es El  7B  IH1)nl+ns+m  [f  fl  7+ B  ‘  7B  713)  5  +  f  dp 2 dp  t4 t ( 2 5 4 m 6 2 6 71 714  Es  7+ B 714 713 +B E3 E4  f4  m2  46 4 5 2 e 2 3 +m f E 714  7B —m £  ’ 7 B 1Ei  7  n  E2 £  Due to the fact that the B symbols are totally symmetric with respect to permutations in the columns of indices 3 we can simplify this result down to the following amplitude  fl  424  XE.  4 XE  =  1 XE  _i(_1r1+ns+m[B2  : 7 fl 7 +B 7 Bfl7]  5 XE Figure 10: The Four Point Effective Scalar Bulk Interaction.  where we have renamed our indices, (mi, £)  —*  , 1 (n  j),  in order to represent the fact that specific integer  values have now been assigned, i.e; the only indices for which the above mentioned summation convention still applies are (m,t).  Now that we have determined all of the Feynman rules we can inquire about the various observables of the theory. In particular, we can investigate the various decay modes for all of the 4 dimensional KK states in order to determine which, if any, are stable and might constitute dark matter candidates.  See Appendix B 3  41  Decay Modes The most important property which all dark matter candidates, regardless of theoretical context, must possess is stability, at least on cosmological time scales. In laymen terms this simply means that the candi date must live long enough and exist in great enough number to exert the observed gravitational effects that are the signatures of dark matter. There are two ways in which any particle may satisfy this constraint: i) The particle either does not interact or interacts very weakly with the standard model particle content and has an extremely small decay rate. The particle is then essentially stable on cosmological scales. ii) The particle is coupled to the standard model particle content. This would then present many opportunities for decay, especially since most dark matter candidates are massive. It must then have some sort of symmetry associated with it which will preserve it against decay. In this section, we will show that it is the the com bination of the imposed parity and the continuous rotational symmetry about the polar axis of the extra dimensional geometry which stabilizes the dark matter candidates in this theory.  Recalling the Feynman rules of the previous section, there are three possible decay modes for the excited KK states. The only one that the excited KK states of the magnetic towers (all towers for which can participate in is the  n  $  0)  decay mode involving the four point function in the bulk. The  —*  other two decay modes are associated only with the excited KK states from the non-magnetic tower (the tower associated with the  n =  0 states). These are the  HH modes involving  H and  —*  both the three point and four point interactions on the brane. The amplitudes for both of the decay modes on the brane are very simple and imply that the decay rate depends directly on the level of excitation of the interacting KK states. In other words, the higher the excitation of the state, the more likely it is to decay. There are no symmetry constraints on the amplitudes so they are allowed as long as the relevant kinematics are satisfied. Combining the two and three body phase spaces with the relevant amplitudes gives the total 4 rate  1’  — —  —*  =  2(2  where m and  mj.j  H) + F( 1  + 1)(2j + 1) l6irm 1  —  HH)  (v2./(m  are the masses of the  +  m — m) 2  m?  —  4mm  +  (mi  —  KK state and Higgs respectively.  See Appendix A 4  42  m2 —  2 16ir  2 2mH)  Since the imposed global U(1) symmetry prevents us from writing down a three point function on the brane involving a KK state and two Higgs, the lightest KK state in the non-magnetic tower,  ,  is stable  and thus constitutes our first dark matter candidate.  The decay mode involving the four point function in the bulk is more complicated and deserves more attention. Since the amplitude for this process is  M  =  _(_1)fh+Th3+m  [B7  fl  7 + B7 7 Bfl7]  the possible interactions and decay modes are highly constrained by the symmetries of the 3-j symbols. The most relevant of these symmetries in this context is the triangle inequality. For a given B symbol, the triangle inequality imposes a selection rule which states that the B symbol vanishes unless i  £  —  + 2. Since the amplitude contains products of B symbols which share the £ quantum number, each term  in the amplitude must vanish unless two separate selection rules are simultaneously met. For the first term, the selection rules are  2 + 1 t  4 + 3 34I t  while, for the second term, the selection rules are  4 + 1 te  e3—2I  +e 3 t 2  Clearly then, the first term in the amplitude will vanish unless £ is simultaneously less than or equal to and greater than or equal to  +  upper limit, we can then write  —  Since the lower limit must always be less than or equal to the  —  +  The phase space for the decay will vanish unless the mass  of the initial state particle is greater than or equal to the masses of the final state masses so we can be sure that  The selection rules imposed by the symmetries of the 3-j symbols then impose the upper limit  > e2  +  limit on  3  i  +  .  The exact same procedure applied to the second term in the amplitude returns the same  which means that this limit applies to the total amplitude for the decay of an excited KK state  through the four point function in the bulk. The kinematics of the problem impose the condition  +  +n4 +  + + 1) + 1) +  + 1) V-(2a 2 63  + 1) +  43  + +  +1) +  ofl22V(2  + 1) +  +  1) + 1)  Since the augmentations to the mass relation from the Electro-Weak symmetry breaking on the brane are on the order of the weak scale while the bulk mass and the compactification scale are generally  (9(TeV),  their addition to the above inequality is superfluous and can be neglected without loss of generality. The condition can then be written as  iJ?4R2 + ‘(‘ + 1)  Inserting the largest value of  &a +  +  +  n4R2 +  + 1)  allowed under the 3-j selection rules leads to  2 + (mR2 + 2mR  + n)  + (s + 1) + n4R2 +  + 1) +  (mR2  + 1))  (mR2  +  + 1)) + (mR2 +  +  + 1))  + 1))  (mR2  +  + 1))  Since the inequality  &  +  i(&  + n) (mR2 + ei(ei +  leads to  0  4 + mR n4R 2  +1) + i(ei +  +  +  + ei  which is a clear contradiction, we see that there is no way to provide enough phase space for the decay at tree level even when using an initial state with the largest mass allowed. This implies that no decays of the excited KK states can occur via the four point function in the bulk. This essentially stabilizes every KK state from all the magnetic towers. However, by considering i-loop corrections to the decay modes we can drastically reduce the number of stable KK states.  44  Since the magnetic and non-magnetic towers interact through the four point function in the bulk we can introduce the following i-loop processes  , 1 X  11  , 1 X  I *0  3 X ,  •  •..-a  .  •  :  Xe,  .  *0  .  ’ 1 Xe  3 *Xe  ..  4  4  ,  H  4 Xe  H  H Figure ii: The i-ioop Induced Decay Modes.  These are clearly kinematically allowed if the Higgs mass is on the order of the weak scale and the corn pactification is O(TeV). Due to the fact that the magnetic quantum numbers of the initial and final KK states have to be the same, this allows any excited KK state to decay down to a final state containing the lightest mass state in its respective tower and one or two Higgs states. The equality of the magnetic quantum numbers is directly related to the azimuthal symmetry, or the rotational symmetry about the polar axis. Since this prevents any decay to lighter states in other towers the rotational symmetry therefore stabilizes the lightest mass state in each KK tower, leaving us with many different dark matter candidates. In practise, of course, there will be some cut-off which will allow us to consider only a finite number of these candidates.  The amplitude for the first i-loop decay diagram with one final state Higgs is given by summing over all possible KK states which can participate in the loop.  M  + (-i)’B  [-i-’r’ (B’  =  B  Es=0 f*=0  k 4 IIcl  if  2x)  i  -m 2 k  [  i  H  Lk+p2_mi1 [-iv2e+i2+i)]  where p is the 4-momentum transferred into the loop by the decaying KK state and max is the maximum allowed value of the orbital quantum number.  45  This maximum value corresponds directly to the cut-off discussed above. Defining A 34 as  flj /(23 + 1)(24 + 1)(_1)fh [B—ni i mu  £  ° ° £  B°  ’B Th + (—1) 1  fli  £  _ni] B°3 ‘ nil £  and using the method of Feynman parameters, we can rewrite the amplitude as  mox ma,,  M  =  -v 3=O 4=O  3 A 3=O 4=O  f  Renaming the internal momentum as k x(1  —  2 x)p  —  space, where d t 4  m  —  x(m  —  d’k (2ir)’ [k 2  f  A  dxf  =  £  —  (1  )4  —  —  1 mj [(k + p) 2  2 [x(k  —  m]  m) + (1- x) ((k + p) 2  -  -  m)]  x)p we can rewrite the denominator as  2  £2  +  where  m). The integral can then be calculated by going to 4 dimensional Euclidean  d/3 sin 3 /3 2O d sin 2 1 d d çb. 6 This leads to 1 4 d 1 dxf_ M=-jEv f 34 2 A 4O  Since there is no angular dependence in the integrand this is evaluated simply to be  2  z  ) 2 dz(  M  =  32•  Defining A  =  , B 2 p  =  2+m p  —  m, and C  9V  16ir  3O 4O  where z  —  =  34 A 3O 4O  dx in  m we see that the final integral over the  Feynman parameter results in  / / B A—B\ 2 A “ fdx1n( A 2 ABC)=2+ln1+ )+lnAB+C) C 2 + i(B  —  46  4AC)ln  2 A  Jo () I  1 2 A(2C 2A(2C  -  —  B) i2A/4AC B) + i2A4AC -  -  —  B\ 2 B2)  It is clear that when we take the absolute square of the amplitude we will obtain a result which will be proportional to  [2+ln(1+  2  /  A—B\  )+lnA  A 2 —4AG) 2 2 +(B 1n 2A(2C_B)+i2AV4AC_B2) B+G)]  which, in the limit as A becomes large, becomes  2 A 1n2(AB+Q  J 3 \m  The absolute square of the amplitude is then given as  2= A’!  ggfr 1rnaxErnax 2560  IE  ln  E4=O  2 A  12  ()j  Combining this result with the three body phase space we obtain the decay rate associated with the first radiative process.  remax  -*  =  40960m?  (m? + m  -  4)2  -  4m2m2  I  E t Erna  ln  12  2 A  [e3=oe4=o  The square of the amplitude for the second radiative decay diagram is the same as for the first but without the factor of v . Combining this with the four body phase space leads to the result 2 2 Y1Y2  =  1 m 7 65536x  (mi  —  m2 —  ln  2 2mH)  A2\l  e3=O C4=Q  The total decay rate of an arbitrarily excited KK state in an arbitrary magnetic KK tower is then the sum of the rates for each of the individual decay modes F  = F( —÷  + F(  —+  HH). 1  We can make one further simplification to this result by choosing the appropriate form for the cut-off. Since the theory breaks down when the compton wavelengths are on the order of the size of the extra + 1) 2 = dimensions, a natural choice for the cut-off is A  47  For this choice, the total decay rate becomes  (v2i/(m  4 l6irm (4ir) 1 x  E  t0t40  +  n4  —  2 i4)  —  4m?m  +  (ini  —  2 in  in?  in  (  —  2 2mH)  2 16ir 2  max(max±1)  (a+i)+Y2 3 \mR2+e  2  (23+1)  The general conclusion, then, is that all excited KK states will decay, either through i-loop corrections or tree level processes, down to the lowest mass state in their respective KK towers. Since these lowest mass states are then stabilized by either the globally imposed U(i) symmetry or the continuous rotational U(i) symmetry about the polar axis, they constitute the set of possible dark matter candidates contained within the theory. While we have had to resort to a globally imposed symmetry to stabilize one of the dark matter candidates, we have succeeded, barring the introduction of gravitational fluctuations, in stabilizing all of them without the use of any discrete symmetries at all. Each one of the dark matter candidates contained within the theory is the lightest Kaluzä Klein particle in its resepctive tower but since they cannot all be the “lightest” Kaluza Klein particle we will henceforth refer to them as the stable Kaluza Klein particles (SKPs).  As a cautionary note, since all extra dimensional theories are inherently non-renormalizable, at some higher energy this effective description of the theory will break down. In particular, the eventual introduction of propagating graviton modes should, in principle, allow for the decay of all of our SKPs. Since the graviton modes are geometrically realized as fluctuations in the extra dimensional geometry, the problem of insuring stability for the SKPs is translated into the problem of stabilizing the extra dimensional geometry. Although, in the present work, we simply assume geometrical stability, there has been considerable interest in the literature on this particular subject. For further details on some of the various methods in the literature see, for example, [36] [37] [38] [39].  48  The Scalar Dark Matter Relic Abundance The recent precise determination of cosmological parameters by the Wilkinson microwave anisotropy probe (WMAP) allows us to constrain the parameter space of our theory. The particular experimental result which we will use for comparison is the indirect measurement of the cosmological relic abundance of cold dark matter given by  2 Oyh  =  0.1099 ± 0.0062 [40], where Ii  0.72 [41] is the Hubble normalization  parameter. In this section we will determine the combined relic abundance of all of our dark matter candi dates (the stable SKP5) by simultaneously solving the first N Boltzmann equations, where N indicates the number of SKPs whose contribution to  cDM 11  is significant to  ion.  In other words, we will calculate, from  first principles, the contributions to the cosmological relic abundance from the SKPs and determine exactly which regions of parameter space can satisfy the physical constraint  cDM 12  0.212.  Stable weakly interacting massive particles (WIMPs) with weak scale masses,  n-i  TeV, decouple from  the primordial plasma when they are non-relativistic and therefore constitute cold relic particles [42]. In the standard model of cosmology, this decoupling proceeds after inflation in the radiation dominated era [43]. On the other hand, if the stable particles are relativistic when the decoupling occurs, making them hot relic particles, then the relic abundance is simply given by the value of the abundance at the time of decoupling, i.e; the abundance is insensitive to the details of the decoupling process [44]. In the non-relativistic regime, approximate expressions have been derived for the relic abundance [41] [44] [45] while, in the relativistic regime, the relic abundance can be calculated analytically. Since we have multiple dark matter candidates and don’t know, a priori, whether any of them decouple in the relativistic or non-relativistic regime, we must take on a more democratic attitude towards the solving of the set of Boltzmann equations. In fact, even if we could, for some reason, assume that a certain set of SKPs do decouple in one of the specific regimes, it would still be possible for other SKPs not belonging to this set to then decouple in the other regime. This simply means that we cannot a priori afford the luxury of any of the simplifications that come with the standard approximations and must therefore rely more heavily on numerical methods to provide us with information of the various decoupling processes. We will see all of these ideas developed more quantitatively later.  49  The Boltzmann Equation As mentioned before, in the early Universe the SKPs should have been thermally produced in large numbers and experienced high enough interaction rates to keep them in thermal equilibrium with the rest of the thermal bath. The SKPs are, however, allowed to reduce their number density via annihilations and other 4-body scattering processes that can occur in the bulk and on the brane. As the Universe expands and cools and their number density decreases, it becomes consistently more difficult to find other particles to annihilate with. At the point when the rate of interaction slips below the rate of the expansion of the Universe the co-moving number density becomes constant and the particle species “freezes out”. The result is a relic number density, or just relic density, which is much greater than one would expect from thermodynamic arguments. In order to calculate the relic density we are then immediately concerned with how to track the evolution of the number density of a particle species which is undergoing a specific set of interactions within the context of an expanding Universe. The Boltzmann equation is exactly suited to this task, it assumes that the evolution of the number density of a particular particle species is given by the difference between the rates for producing and eliminating that particular species. In the bulk, we will deal entirely with 4-body interactions since these are the only type available to the SKPs. On the brane, we will be concerned with both the four point interaction and the three point interaction which will provide a way for the  SKP to annihilate into various different standard model particles via the Higgs. Since all of the  above interactions are four point interactions, the Boltzmann equation will be introduced in the context of these types of interactions. This introduction is based on the analysis of [41].  50  The Standard Case: One Particle With One Interaction In this section we will derive the Boltzmann equation which will track the number density of a single particle undergoing a single type of interaction.  Consider a general single four point interaction where the four interacting particles are labeled as 1, 2, 3, and 4. The interaction in question can be written schematically as 1 + 2  3 + 4, where particles 1  and 2 can annihilate into particles 3 and 4 or vice versa. In order to determine the evolution of the number density of particle i,  a  d(rtia 3 _ ) dt  , 1 n  —  —  we must solve the Boltzmann equation which is given by  [ j  1 p 3 d 1 2 3 (2ir) E  2 p 3 [ d f 2E j 3 (271) I 2  ± fi)(1 ±  f2)  -  p 3 d 2E 3 (2ir)  hf2(1 +  j  4 (27r)2E  2 t)  (P1  +732  P3  p H 4 A4I 2  f)(1 + f4))  where a(t) is the scale factor of the Universe in which the four point interaction takes place. distribution functions,  f,  The  of the i’” particles are defined as  L  1 e(Eri)/T  =  ±1  where p is the chemical potential and T is the temperature. The sign in the denominator determines whether the particle in question obeys Bose-Einstein type (1 +  f), (+)  for Bose-Einstein statistics and  (—)  (—)  or Fermi-Dirac  (+) statistics. Factors of the  for Fermi-Dirac statistics, serve to incorporate Bose  enhancement and Pauli blocking into the calculation. These simply refer to the fact that, if particles of type 1 are present when an interaction which produces more particles of type 1 occurs, the interaction is more likely if the particles of type 1 are bosoms and less likely if they are fermions. Since we are typically interested in temperatures much lower than E  —  ji,  we can safely assume that the exponential is generally  large compared to the ±1 in the denominator. We can then, in this limit, ignore the problem of quantum statistics and assume that all particles obey Maxwell-Boltzmann statistics, i.e;  f  —÷  ’ 7 e’  In this limit, the factor containing all of the distribution functions simplifies to  (f3f4(1 ± f)(1 ±  f2)  —  flf2(1  + f)(1 + f4))  where we have used energy conservation E 1 +E 2  =  51  e_12)/T (e(P34)/T  —  4 to obtain this result. 3+E E  e12VT)  We can rewrite this last result again in terms of the relevant number densities in order to eliminate all dependence on the chemical potentials. The number density, assuming Maxwell-Boltzmann statistics, is defined as  p/T ,—g, 6  [_Le_Ei/T  —  j (2ir)  where gj is the degeneracy of the species. When a particle is in equilibrium it has no chemical potential, the equilibrium number density is therefore simply given by 3/2  g(—J \27tJ  ‘-‘3  —  I  t±t/T  3 j (2ir) t  ” m e  ifm>>T  —  —  if m 1 <<T We can then rewrite any exponential factors of the chemical potential in terms of the number densities of the particle species as  =  etui/T  ni  Since the number densities are independent of momentum, we can extract any factor involving them from the integrations within the context of the Boltzmann equation.  Letting  2 be the absolute square of the reduced matrix element for the interaction, suitably MI  summed over all final spins and averaged over all initial spins, we can integrate it over the Lorentz invariant phase space (dLIPS) for the four point interaction as  f  2 dLIPSM =  I  2E (2x) 3  I  260 4 (2ir)2E ”  +732  P3  )1M1 4 —p 2  Since dLIPS is Lorentz invariant it depends on the momenta only through the Mandelstam variable s  =  + 732)2. We can then, for convenience, define  w(s)  =  JdLIPSIMI2  For fixed incoming momenta g and 732 we define the product of the cross section with the relative velocity to be [32]  w(s) =  2 E 1 4E  52  The thermal average of this quantity, denoted as  (cv)  — — —-— 1 4_?L_?1 f2 cv 3 3 (2x) — ri?n j (2ir) —  (cv),  n?n  is then defined as [45]  pi 3 2 e_1+E2Tw p 3 d [ d 1 2 3 E (2ir) 2 2 3 E j (2ir)  a  These definitions allow us to rewrite the Boltzmann equation as  a  1 d _ ) 3 a (n /‘fl3fl4 l2 =n ( 2 n 1 cv)i———-—— dt \rign  In order to see how information about the expansion of the Universe is contained within this equation we concentrate on the left hand side for a moment. Using the product rule, we are lead to 1 dn  o o  4 n 3 (n  ——  = n (cv) 2 1  ’\ 2 1 n —  fl 3 \Th 4  where H  =  1  ——; / 12 fl  —  H 1 3n  is the Hubble rate. In this form, we can see the correct way to interpret the Boltzmann  equation: the change in the number density is decided by a competition of the rates of the various interac tions associated with the particle species of interest. The effect of the expansion of the Universe enters the equation as simply another rate which competes for dominance against the various interaction rates. The effect of the annihilation of particles 3 and 4 into 1 and 2 clearly adds to the number density,  , 1 n  while the  effects of both the expansion of the Universe and the annihilation of 1 and 2 into 3 and 4 clearly depletes it. If the particle of interest had no interactions then its number density would eventually deplete completely as there would be no competing rate to slow down the effect of the expanding Universe.  In order to put the Boltzmann equation in a form that we may use more readily, we instead use the first derived form. Recalling that the temperature scales as a 1 we can multiply and divide the product 3 a 1 n  within the derivative by T . Since T 3 a is constant in time, we can remove it from the derivative and 3  define a new variable  ,2 Pt  —T 3  so that the Boltzmann equation becomes  3i —T (cv) 2 1 3 —  53  (/33/34  —  /31/32  This equation, at the moment, is made slightly awkward by the fact that we have the explicit temperature dependence. In order to correct this we can define a new time variable as x  =  ,  where m 1 is the mass  of the particle of interest. To find the relationship between this variable and the original time variable we note that we expect our SKPs to have been produced early in the radiation domination era of the Universe where the solution of the Friedmann equation gives [46] /4ir3g*2 H—  V 45M  where g. is the number of relativistic degrees of freedom present in the cosmic plasma at that time and M is the Planck mass. Since the temperature scales as a 1 where the proportionality constant is the present  day temperature T , the above equation represents a differential equation given by 0  /43g  2  45 V MT0dt The solution to this differential equation is 1  ill Rewriting the Hubble rate in terms of the new time variable leads to ) 1 /4ir3gmH(m H—  V45M  2 x  Using this yields the relationship between the two time variables as  d  xcix H(mi)  Rewriting the Boltzmann equation in terms of the new time variable then gives  d/3  —  —  /45M$mi (P3/34 (u’0) 2 P 1 -fl g 3 ‘i7r  V  /31/32 —  So far, we have only written down a Boltzmann equation which tracks a single particle species undergoing a single general four point interaction. In order to generalize this equation so that it may be used to simultaneously track many different particle species, all undergoing many different four point interactions, we must first redefine some of the quantities above.  54  The Generalized Case: Many Particles With Many Interactions In this section we derive the set of Boltzmann equations which will track the number densities of a set of particles undergoing many different types of interactions.  To apply the Boltzmann equation to our particular theory, we must first identify the dependence of the above quantities on the various quantum numbers involved in the theory. Since both the equilibrium and non-equilibrium number densities depend on the energy of the particle and therefore its mass, the number density must then depend on the two quantum numbers which are needed to specify the mass. That is, each number density can be written as  =  where Ema  =  y2  +  and  geuhi/T  f  ?!e_Emiei/T  (2ir)  ( t + 1 l)+n4+6Wi%L(2+l). t As mentioned before,  =  we have no a priori way of knowing in what regime the SKPs will freeze out in so the strategy has always been to be as democratic as possible in our assumptions. One way which is immediately applicable regarding the number density is the exact form of the equilibrium number density. Previously, the equilibrium number density had two exact forms. Which one to use, though, depended on the regime in which we were interested in exploring, i.e;  mm€t  <<T or  mmj, >>  7’. In order to be democratic at this point, we will use the more  complicated, but more general, form given by [47]  2 K 7l Xm.t.  where  =  mrn,t,  and K 2  (Xmje)  (xmg.)  is a modified Bessel function of the second kind. This relation holds  for any temperature regime under the assumption of Maxwell-Boltzmann statistics. It is now clear that any dependence on the masses necessarily translates into a dependence on the quantum numbers. This includes the following  ma 3 I  (cv)  (cv):73m4  where the dependence of the thermalized cross sections on the quantum numbers comes from the quantum mechanical amplitude associated with the four point interaction in the bulk as well as from the masses. The generalization of the Boltzmann equation to track multiple particle species also introduces an ambiguity in the new time variable x. This was defined as x  =  ,  density we wish to track.  55  where  m  is the mass of the particle whose number  Since we would like to now track many different number densities simultaneously tbis introduces a severe element of ambiguity into the equation, i.e; which SKP mass do we choose to appear in the new time variable? In order to dispel this ambiguity, we choose the mass of the lightest SKP state, is then used to define the new universal time variable x  =  3, as a universal mass which  From here on, everything will be written in  terms of this universal time variable, including our general relation for the equilibrium number densities  m  £•  gim. m 2 K 0 x 2 2irx 0 \ Tn  and the generalized Boltzmann equation given by  —  clx —  /45M m o o g 11m22 3 4x  V  4 3 2 m 1 n  44 (i3mataflm  22 m 3 / 1 fln  23 (Uvtv —  The only particle species that we are interested in tracking are the SKPs, all of which are characterized by the lowest mass KK state in each of the various KK towers, i.e; the  states. The number densities of these  SKP species are written as n 1 and they therefore should appear in the above Boltzmann equation as However, since we only need one quantum number to specify which SKP we are considering, we abbreviate this simply as j3. It is also important to note that the index on  is not summed over but acts instead  as a vector index for the Boltzmann equation. We have also made use of the summation convention used in Chapter 2 in that there are implicit summations over all pairs of indices of the type (m , £). Summing over 1 all possible quantum numbers in this way has the effect of taking into account all possible interactions that the particular SKP associated with /3 can participate in. In this way, the original Boltzmann equation has been modified so that it now has the form of an, in principle, infinite number of coupled differential equations. The coupling terms occur each time the summations produce a term which implies a four point interaction in which multiple SKPs are participatory. The simultaneous solution of these coupled differential equations gives the set of relic densities of the SKPs. Thankfully, it will turn out that only the first two equations, 1 = ti  1 0 and n  =  1, will contribute significantly to the relic density results allowing us to reduce the problem  down to simply solving two coupled differential equations. The factor of 6 is inserted to avoid incorporating any non-number changing interactions since they do not contribute to any changes in the number density. In order to enforce this, the 6 is defined as  6=1  1 0  I+t+Im 3 ifm + 4 %— m I—t—2friiI 2  56  =0  So far, the right hand side of the Boltzmann equations describes all the interactions which enable the magnetic SKPs to remain in equilibrium but mentions nothing about the interactions between the nonmagnetic SKP,  3,  and the standard model particle content. These will contribute to keeping the non-  magnetic SKP and, through its interactions with other SKPs, the magnetic SKPs in equilibrium. The non-magnetic SKP interacts directly with the Higgs field only but can undergo scattering processes with other standard model particles via Riggs exchanges. All number changing interactions of this type involve either one or two non-magnetic SKPs and are of the form  SM, where £  —*  e {O, 1,2,.  ,  oo}. Based on  this information, we can write down the Boltzmann equation describing the evolution of the non-magnetic SKP number density based solely on the interactions with the standard model particle content is given by  d/3 0 dx  /45M 4xg  — —  V  s—’  m 2 ((/3sM) 0 0 o 3 Ld/ O (Ut)SM /  \C6M)2  where we assume a summation over all values of £. The sum  nOot 3 / —  is meant to represent a summation SM  over all possible allowed standard model processes, i.e; interactions of the following types —  ZZ,  —*  ww,  —,  HH. Note that we have not listed  —*  —*  ff,  H as a possible process,  this will be discussed in detail in the following sections. At the energies which the SKPs are likely to have frozen out it is a safe assumption that the standard model particle content is still in thermal equilibrium and we can therefore assume that  5M 3 /  5M. 3 /  =  0 d/3  This modifies the above Boltzmann equation as  /45Mm-. =  J)5M 2  V  00  (/3o/3ot  —  i3oi3o)  We can now combine our results to obtain the set of Boltzmann equations whose simultaneous solution leads to the SKP relic densities  ouvp4 t 2 [i  =  9*  Earlier we saw that, in the mm 1  flrn* 3 (/3rn*i 33 m  >>  44 m  -  /3n15rn2t2) ni  22 m  + ó’  -  !3o[3oe)]  SM  T regime, the equilibrium number density is exponentially sup  pressed, i.e; if the KK state is kept in equilibrium long enough the temperature will drop until this is a valid regime and its number density will drop. Since the excited KK states are thought to all decay down to the SKP in their respective tower, we don’t expect any of them to be around in the present. Since, in terms of the number density, the effect of keeping the excited KK states in equilibrium and allowing them all to decay is the same, we make the approximation that all of the excited KK states are kept in equilibrium.  57  In this way, we can associate only equilibrium number densities with all of the excited KK states so that, if m  $  £, then  fimjtj  making the only unknown  fi  —*  This simplifies the problem of solving the Boltzmann equations by  functions the exact /3 functions which are associated with the SKP number  densities. We also note, only in passing, that it is, of course, an insurmountable job to solve an infinite number of coupled differential equations. The more realistic strategy is to solve as many as is realistically feasible and determine from that how many of the particular relic densities contribute significantly to the total dark matter relic density. Only those which deliver a significant contribution to the total relic density, and therefore the dark matter density parameter  DM, 12  actually need be considered. This will be discussed  in much more detail in further sections.  Now that we have the correct set of Boltzmann equations, whose solutions give the desired relic densities, we must go about the task of calculating all the necessary thermalized cross sections needed in order to solve these coupled Boltzmann equations.  58  The Thermalized Cross Section Calculations In the following sections we present, in full detail, the calculations of all of the thermalized cross sections which are relevant to the Boltzmann equations. The first section will discuss the set of thermalized cross sections which describe the four point interactions occurring in the bulk. Although the non-magnetic SKP undergoes both four point interactions in the bulk as well as a number of brane interactions which act against the expansion of the Universe to keep it in equilibrium, these bulk four point interactions are the only interactions which sustain the thermal equilibrium of the magnetic SKPs. There are, in principle, an infinite number of possible four point bulk interactions but we will find that the associated thermalized cross sections all take on the same form so that the only difference between any two cross sections is simply a reshuffling of the quantum numbers. The second section will discuss the set of thermalized cross sections associated with the brane interactions of the non-magnetic SKP. These are slightly more involved due to its interaction with the standard model particle content via the Higgs field. There are 4 four point interactions and, in principle, one three point interaction. We will, however, find that the three point interaction is only allowed if the mass of the non-magnetic SKP is assumed to be less than the Higgs mass.  59  The Magnetic Thermalized Cross Sections In this section we calculate the general form for the thermalized cross section associated with the four point bulk interactions.  In order to simplify our notation for this section, we suppress the use of the quantum numbers as a label for the various quantities that we will need. For example,  nj, Em  —,  , etc. In the previous 1 E  section we defined the thermalized cross section to be  (cv)  2E 2 3 (2ir) 1 2E 3 3 (27r) 2E (2 (2ir)  =  1 (2ir)S(p  2E:  +P2  )MI 4 —p —p3 2  In order to calculate the thermalized cross section for the four point bulk interaction we simply insert the correct amplitude while assuming that  In’ I. The amplitude for the four point bulk interaction  has  been calculated previously and, modifying it so that it complies with the assumption that at least one SKP is involved, is given by  M  =  _i(_lr1+m3+m[B:im:7  [7 1 B2 7+B  where there exists an implicit summation over the  (m, £)  pair of indices. Using the symmetries of the  3-j symbols, in particular the symmetry that enforces the vanishing of the sum of the magnetic quantum numbers, the absolute square of the amplitude is  MI’  =  g—’  [  2  1 ((_1r2B  (nj—na)  B  (n2fl1)  + (—1)B 1  7Z4  (ni—ru)  B7  (n4;ni))]  where we have made the final implicit sum explicit so as to emphasize the fact that the absolute square of the amplitude is the square of the sum and not the other way around. This amplitude is a constant and is dependent on all of the quantum numbers involved in the interaction. Due to this, we can extract the absolute square of the amplitude from the integrals in the thermalized cross section to arrive at  (cv) =  l6(2ir),4i4  1  4 E 3 E,E,E  where we have assumed the center of mass frame so that s calculate the result of integrating ji and  eT S (/—p, —pa) 4 =  over the delta function.  60  (p + p,)’  =  (E, + E,)’. We will first  In order to do this we will use the fact that  f  dp6(p  —  m)  =  f  6(E 4 f dE 4 d3p  —  4I2 —m)  =  f  2 _— Vi14 6 4 dE  +m)  2/I2+m  4 J2E  so that  f  3 E  ?ó4v  4 E  —  p3  —  p4)  =  2f --dp ö(p 4  —  (v 4 m)ö  —  p3  —  Since there is no angular dependence the rest of the integration is somewhat trivial  f  4ir  Pao(22 3 Id r E) 2 2  f  2ir _im_m)2_4sm S  Inserting this back into the formula for the thermalized cross section gives  (0V)  2 IM l6(2ir)7n4  I d 1 d p 3 p ./(s + 3  I  m  —  2 m)  —  4sm  s  2 EE  e  In order to write this integral in terms of the new time variable x, we must first recall our relation for the equilibrium number densities  =  mmo. /m \ x 2 27r \m J  Substituting this in gives  (ov)  =  M1 x 2 /m 1 2 ( —x ) 64(2)m?mmK 0 j \m  2 K  m 2 \mo  I I  61  31  2 /(s + m p 3 d  —  2 m)  —  4sm  0 e m  Since there is no angular dependence in the integrand we can, again, perform the angular integrals to arrive at  x 2 M1  (uv)  (m ‘\ 1 2 (m 2 2 2 32-irm K 0 2 m 1 _x) 2 K 0 m 0 m  J(E 21 fdEidE  —  m)(E  —  /(s + m  m)  —  m)2  —  4sm  s  ‘\  At this point we make a change of variables from the energies, E, to the variables, y, where the relation between the two is given by  E  —  m (i + mjx \  In order to understand the physical meaning of the new variables we expand the energies in the regime where  m  <<1  I  2  m  V  /  —2\  /  \  2m,,j  \  K m  where K is the non-relativistic kinetic energy of the particle. By letting  i =  -  we can then obtain  the above transformation. The new variables then represent a sort of thermally normalized non-relativistic kinetic energy. Using this relation we can also determine the transformation for the momenta  iJE  -  m  +  and the Mandelstam variable  \  x 1 m  x 2 m  Writing the thermalized cross section in terms of these new variables gives  1 (m  =  +  m2)  167r(mlm2)3/2x K 2 (i) K 2  0 m  () f  /(Yi 2 dyidY +  0 m  )  + (Y2  where II  )+m _m) 2 / (s(yi,y f(yl,y2)  s(Yi, Y2)  62  —4s(yl,y2)m  mo2) f(yl,y2)  e12  where the limits of integration are over the full domain [0, cc) for both the averaging integrals. This is the simplest analytic form of the thermalized cross sections that we are able to obtain. Unforti.mately, as with many thermalized cross sections, it is not possible to perform the thermal averaging integrals analytically. Although there do exist approximate methods, we will rely more on numerical methods to advance any further in terms of calculating the cross sections.  63  The Non-Magnetic Thermalized Cross Sections In this section we calculate the various thermalized cross sections associated with the brane interactions between the non-magnetic SKP and the standard model particle content.  The non-magnetic SKP participates in many different interactions with the standard model particle content via Riggs exchange. The first step in determining the relevant thermalized cross sections associated with these interactions is to determine the relevant effective field theory. We have included a brief introduc tion to the standard model in Appendix C in which we specify the particular sectors of the standard model Lagrangian which are of most interest to us here. Localizing these interactions to the brane leads to the action  S  =  xIZ 4 d d R —f2  Jfk +  where we have summed over all the fermions  WW”h + 4JZZIth] 6 (cos  f,  — 1)  W, and Z are the weak gauge bosons, and, as before,  h is the Riggs field. Integrating over the extra dimensions leads to a rescaling of the fields and Riggs vev as v Vr  H h=__  F  w+__vv±  z—  -  -  so that section of the brane action which plays a role in the cosmology of the non-magnetic SKP is  Brane 5  f  x [oH&LH 4 cl  -  —,  jH2  -  -  -FH  -  2 + 2vH) ‘12t2 + (H  where we have assumed the existence of at least one non-magnetic SKP since we are not concerned with non-numb erchanging interactions.  64  The new Feynman rules can now be read off of the action as  H=  -i  Figure 12: The Higgs-Fermion Three Point Interaction.  >  >  =  Figure 13: The Higgs-Gauge Three Point Interactions.  The four brane interactions that fight against the expansion of the Universe to keep the non-magnetic SKP in equilibrium are then given by  •  —  :<  A  / F..  F..  Figure 14: Effective Scalar Annihilations into standard model End States.  65  Note that we have not included the three point interaction between the non-magnetic SKP and the Riggs field. The amplitude for this interaction is constant and the thermalized cross section for this process is given by  f  (uv)g;oH  2 (2ir)2Ei 2 2E ( 3 (2ir)  (2ir)6(pi +P2  e 3 E 2  In the center of mass frame we can factor the delta function as Integrating over  p3  (p 4 6  ) 3 +P2 —p  P3)  1+E 6(E 2  =  —  ) E ( 3 r53). S  yields  (ov)g,;oH =  f  mHn?n 5 8(27r)  P d P 3 ) 2 /To(E 2_(El+E ld + E 2  —  mH)  Since there is no angular dependence in the integrand we can write this as  (UV)O*OH =  m n°n° 4 4(2) e_mH/T 2 M mHn?n 4 4(2)  e_mh/T 2 M mHnn 3 2(2)  where the upper limit, E 1  mjq  —  , 2 m  f f  \/E 2 !dE  3 d  /(m  —  —  me_  2 ) 1 E  —  1 1+E2Tô(E  2 +E  —  mH)  m  mH—m2  E? 1 dE  —  mmH  2 ) 1 E  —  m  fmi  is necessary for the integrand to be real. The integral is finite  and, at first sight, there doesn’t seem to be any problems but because the energy, E , is positive definite 1 this means that  0 <  1 E  mH  —  2 m  .  2 mH > m  The thermalized cross section is then physical only if the Higgs is more massive than the KK state Unitarity constraints give an upper limit for the Riggs mass as  185 GeV. Due to the fact that no scalar  particles have yet been observed we reject this out of hand and conclude that this particular process is kinematically forbidden.  The thermalized cross section of the four point interaction is perfectly allowed and the simplest to calculate.  66  Since the amplitude for this process is constant, the thermalized cross section takes on the same form as with the magnetic SKP four point interactions with m 1  -  +1)e 2 (2 (UV)xoxoHH=  (  =  0 and m m 3  =  4 m  =  mH given by  m2”\  1  +—Ix  j 0 m  , 1 f(y ) 2 e”’ y  2 xK 167r,’m 2 (x) K 2  2xj  —x) 0 ) \m  \  m2.2x)  where  (y /s , 1 ) 2 y f(yl,y2)  —  ,m 1 4s(y ) 2 y  s(yl,y2)  and  2 x) m  Again, there is no analytic solution to the thermal averaging integrals so we have to calculate the result numerically. This will be the running theme with the rest of our thermalized cross sections as well.  In the following calculations we will always use the following naming convention for the various momenta  The amplitude for the scattering process involving the non-magnetic KK states and the fermions is given by  —iM  ) 3 = Il(p  i mFl vi/2 + i] 2 [_i v(p4) IL —i——i VJ s—rn 2  which leads to  M  —  —  m/2 +1 2 s—m  Taking the absolute square and summing over all final spins yields 67  > Since s  2 + 1) m(2e  —  -  (s  (pj + p4) 2  =  =  -  [(p3  m)2  + m)(4  -  4m(22 +1)  -  ms)]  (s_m)2  —  (P3 P4- m)  2m + 2p3 p4, we can rewrite this as  2 1M1  —  2 + 1) 2m(2e  s  (s_m)2  -  —  4m)  Inserting this into the definition of the thermalized cross section then gives  2 + 1)f d 2m3(2t 1 d p 3 2 d p 3 p d 3 4 p 3  (v)o*o  16(21r)8n?n  =  4 3 2 1 E  eT  (s (s  4m)  —  m)2  —  —  —  )  Concentrating on only the last two integrals, we find that  I  pd 3 d 4 (s p 3  —2 —  —  4m)  (s—4m)  [d p 3 a (s_4m)( (s—m 3 E  2fE)  J  =  2ir  2  (E 5 2(84mF fdE2 —mF( E 3 2)2  \  (s—4m-) /4m  Substituting this back in and performing the trivial angular integrations gives  (uv) 2 X 0 X F F e  =  m(22 + 1) d 1 fdE ( 2 E? E nn 5 2(2)  —  m)(E  —  m)eTXF(s)  where  (s—4m)  /_ s  (s_m)2V  Finally, we can rewrite everything in terms of the universal time variable variables from (E ,E 1 ) 2  —  x  and perform the change of  (yi, Y2) as before. This leads to  \ 2 m  1+—x mo)  2 + 1) e m3(2e °  £2  (x)K I 8ir/momxK 2  —x)  0 J \m  f  iJ( y + 2 dYidY  y  —) 2xj  / (Y2  \  +  m y” 0 2xj 2 m  —  which is the simplest form which we can massage the thermalized cross section into.  68  (s(y, Y2))  e12  The amplitude for the scattering process involving the non-magnetic KK states and the W± bosons is given by  —iM  v + i] 2 [_i  =  2M [—i----_w  s—m  ]  (*(p3)f*V(p4)  which leads to  M  M2 + l**l() 2 s-m  Since we have no interest, at this point, in measuring the polarizations of the W bosons, we can sum the absolute square of the amplitude over them which gives  — 2 MI —  +1) 2 4M(2  /‘ g2  (s_m)2  IVIW  —  1  IL 1  +— ‘wJ  I \  +1) 2 4M(2 (s_m)2  32  (2+ ‘  Mw  I  As before, we can rewrite this purely in terms of the Mandelstam variable s by using the fact that p3• p4  —  M. This yields (_M)2  +1) 2 4M(2e (s_m)2  (2+  2  I  4Wj M  The thermalized cross section is then  (8  =  fd d 1 p 4 e T 2 p 3 +1) 3 2 M(2 (2+ 4(2ir)8nn 4 (s — m)2 3 E 2 EiE  2  2  M) 4 M )  As before, we concentrate on only the last two integrals which, again, give  I  (,, P d 4 P 3 64 ad 4 3 E  —P4)  =  27r\/1 —  2 4M S  Letting 2 4M  Ji———  S V Xw±(s) — — (s_m)2  the thermalized cross section reduces to  69  /  12  (5  +  M  )  f  =  x/(E 2 dEidE  -  -  m) eTXw±(s)  Rewriting this in terms of the universal time variable and performing the change of variables as before we arrive at the final form given by  =  (Yi 2 fdYidY (nz)  +  )  (Y2  +  x± (s(yl,y2))  e12  The only difference between the scattering process involving the Z bosons and the previous result is the statistics of the final state. Since the two final state bosons are identical, the range of the integration over the p and p4 integrals must differ by a factor of 2. This range need only be over 2ir steradians instead of 4ir because it is impossible to distinguish which Z boson went in which direction [48). The end result is that we don’t have to recalculate from first principles the thermalized cross section for the scattering process involving the Z bosons, from the above argument we can simply write 1 (uv)o,?;ozz  =  Mw —*Mz  All of the thermalized cross sections which appeared in this section make their appearance in the set of Boltzmann equations in the term  6 (uv)jvj (,3gI3g  —  /3oi3ot)  SM  Since the  variables only involve number densities associated with KK states they are not affected by  the sum. In other words, the sum is given by simply adding up all the thermalized cross sections which were calculated in this section  E SM  (cv)SM = (uv),?o,?*oiiH  + (Jv)l;op +  +  F  As mentioned before, this sum will have to be calculated numerically in the next section.  70  Solving The Boltzmann Equations In this section we describe our solution of the Boltzmann equations and discuss the results. Sample calculations of the relevant freeze out temperatures are also presented and the excited KK state decays are shown to occur in equilibrium.  The Boltzmann equations are in the form of an infinite set of coupled differential equations. The relic densities associated with the SKPs are simply related to the solutions of these coupled differential equations as  =  where this relation is evaluated at a sufficiently late time so that the SKP species has long  frozen out. Once the relic densities have been numerically determined we can view the corresponding freeze out temperatures directly by plotting the solutions as functions of the universal time variable x. As mentioned earlier, we do not intend to solve the complete infinite system of Boltzmann equations here. Instead, we must employ a cut-off of some sort in order to only consider the first N Boltzmann equations. The criteria for our cut-off is based on whether or not the parameter which is significant to  Nt  relic density provides a contribution to the over all density  After they freeze out, the SKP relic densities fall off as a 3 which  leads to a current value of their energy density given by m 1 aj times its relic density where m 1 is their mass and af is the value of the expansion parameter after freeze out. Since the number density at that time is given by  1 (x)Tj, fl,  the mass density at late times is given by [41] af3rr3f 3 0 / =m ( 1 xf)T 3 —1 — p, 0  m3 0 T 1 —  30  The present day density parameter is then given by the ratio of the total current SKP mass density and the critical density  (1.02785 x 1010 TeV’) Zmfl(xf)  Pa  where only the first N  functions which contribute significantly to the WMAP result will be kept.  However, before we can numerically calculate the  functions we must deal with yet another cut-off. The  current form of the Boltzmann equations allows for each SKP to participate in an, in principle, infinite number of interactions with both excited KK states and other SKPs. Since, in our case, a finite set of differential equations where each equation has an infinite number of terms is just as unreasonable as an infinitely large set of differential equations we must employ a new cut-off which filters out only the dominant terms in each Boltzmann equation. This cut-off is based on the relative mass between the primary SKP and the other participating particles in a given interaction . 5 We refer to the SKP whose number density is described by a given noltzmann equation as primary. 1  71  To motivate the criteria for this new cut-off, consider a SKP interacting with at least one excited KK state. If the primary SKP is currently freezing out then the temperature must be on the order of its mass. Recalling that the number densities of the excited KK states can be approximated as their corresponding equilibrium number densities, if the participating excited KK state is more massive than the primary SKP then its number density is approximately exponentially suppressed as  e_m/T.  We would then expect  that the majority of interactions between the primary SKP and excited KIC states would involve only those excited KK states whose mass is less than or equal to the primary SKP mass since there are many more of these states in the thermal bath. Besides excited KK states, any primary SKP can also interact with other SKPs. Consider the case where a primary SKP is currently undergoing freeze out and particpates in interactions involving at least one other secondary SKP. Since all of the SKPs experience freeze out, if the secondary SKP is more massive than the primary SKP it follows that the secondary SKP has already frozen out. If this is the case then it has already drastically reduced its interaction strength with those particles still in equilibrium with the thermal bath. Since the primary SKP is just in the process of freezing out we assume that it would then interact much more strongly with secondary SKPs which are just as or less massive than it. All of the above arguments can be summarized by imposing the following constraint on the implicit summations in the Boltzmann equations  £  In’  This insures that we have neglected any interactions between the primary SKP and any particles which are more massive than it as per the above discussion. This simplification allows us to easily write down the first two Boltzmann equations d5 0 dx  =  /45M;  /45Mmo / 02 2 3 —-E(uv)5MI(Øo) -(/3o) 4 x g.x SM  [(:  —  C8i)2)  () + 2 ) 0 (  (c13?)2  —  (/3)2)  (UV)2]  while the other Boltzmann equations have far too many interactions to write down. For instance, the third Boltzmann equation already has 25 interaction terms on the right hand side. Although the number of interaction terms is restricted by the cut-off and the 3-j symmetries, the escalating pattern occurs because as we increase the magnetic quantum number we inadvertently increase the SKP mass and therefore allow for more interactions. However, it is precisely these interactions which keep the SKPs in equilibrium. Hence, we should expect the departure from equilibrium of the heavier SKPs to be delayed due to their many interactions.  By delaying their freeze out and maintaining thermal equilibrium, however, their number  densities quickly become exponentially suppressed.  72  We should therefore expect that the relic abundances of the first two SKPs should dominate the con tribution to the density parameter while the contribution from each successively heavier SKP should be increasingly more negligible in comparison. If this is true then we can perhaps learn something by finding an approximate solution to only the first two Boltzmann equations. We will first define the following variables L45M  0 = A  1 A  /45M g 3 4ir  lrg*  0 In SM  /45M 2 mo(crv> g 3 4ir  2 A  mo(uv)i  V  The Boltzmann equations can then be written as d/3 02 ( A 0 —i (/30) dx 2 x  —  -(/30)  (i3)2)  +  2  (o  —  (/3)2)  Since the equilibrium number densities  xrnlK(ml)  /3’= 2x  2ir  0 m  0 m  are both proportional to the modified Bessel functions they rapidly become negligible in comparison to their corresponding SKP number densities as x increases. This is just a more quantitative restatement of the assertion that, at late times, the SKPs will not be able to annihilate fast enough to maintain equilibrium. Also, since the argument of the Bessel function in the second equilibrium number density is enhanced by the positive quantity  ml —,  the ratio  mo 2 m ( 2 K 1 —x  0 —  (x) 2 mK  should also rapidly become negligible. Thus, at late times, we should expect the Boltzmann equations to reduce to  d/3 0 dx  -  0 A 2 (/3o) 2 x  d/3 1 dx  where A = A . 2 1 +A  73  -  A x 2  -—(/31)  2  We can then integrate each of the equations analytically from the freeze out time  Xf  to late times x  oo  to obtain  1,2\  Xf  0)oo 1 U  where we have treated the variables A 0 and A as constants even though they are actually functions of x. Although this does lead to slight numerical changes in the following, it does not change the behaviour of the qualitative solutions [41]. Since the variables A 0 and A are each proportional to thermalized cross sections we now know that, at late times, we should expect the relic densities to be inversely proportional to the square of the effective coupling constants (fio)  r’.  2, (/31)IXD  2 T  i.e; the relative size of the couplings is  the deciding factor in which relic density dominates.  To summarize, the first two relic abundances should dominate over all the others due to the fact that they participate in, comparitively, very few interactions. Also, the dominating relic density should correspond to the weaker effective coupling. This turns out to be exactly the behaviour that we see in the numerical results. The following log plot demonstrates this by plotting the first 5 SKP  functions in blue  alongside their corresponding equilibrium number densities in red for the parameters, y  =  .5 TeV, and 1/R  =  .95,  1 TeV.  —  /31 50  1.31 x 10_6%  2.37 x l’ % 2  1.03 x lobG%  Figure 15: The natural logs of the first 5 butions to the total density parameter, contribute significantly.  1 (fi,) fi  133 I /34  functions are in blue (red) while the percentage contri for the last 3 implies that only the first two densities  74  It should be stressed that the total density parameter in this calculation is not equal to 0.212, as is needed to agree with experiment. This is simply a sample calculation to illustrate the above behaviour. We have specifically chosen the bulk coupling to be much smaller than the brane coupling so as to test just how many relic densities make contributions to the total density parameter significant to 10* In fact, the total density parameter for this sample calculation is  Q7  density to the overall density parameter are given by 1.2685 x b— , Q’j 5  9.69571,  fl/,J  =  =  =  9.70484. The contributions of each relic  1.77636 x  io’, Qjj  =  2.29647 x 1O,  9.13 x iV . Due to the very small relative bulk cou 3  pling the second relic density dominates over the first and only the first two densities contribute significantly. By looking at more realistic combinations of the coupling constants, i.e; lowering  , 2 g  we will only raise the  non-magnetic relic density higher, maintaining the dominance of the first two relic densities.  The natural question to ask now is: for what values of the parameters can the combination of the first two relic densities satisfy the WMAP measurement of  0.212 stated earlier? We present  cEM 12  a representative sample of the parameter space in the form of various two dimensional slices of the four dimensional parameter space. Each slice is obtained by fixing the values of the mass parameters m and 1/R and varying the values of the couplings  g and  2  to determine the curve in the g  —  2  space that  agrees with experiment. A 45° line can be drawn, starting from the origin, through each slice separating the two regions j > g and  2  >  th,  in which the non-magnetic and the first magnetic relic density dominates  respectively. Each two dimensional slice is organized so that all the curves represented are associated with a single value of mB. Each of the curves in a given slice correspond to different values of R and are color coded for simplicity.  I .0 —  —  —  J?=1TeV R=.75TeV 1 1? = .5 TcV  0.6  0.4  i3  0,2  02  0.4  <5o 0.6  Figure 16: The curves associated with mB  75  0  =  1 TeV.  IS  I.0 —  —  0.11  —  1? = 1 TeV’ J?= .75TeV 1 I? Tei .5 1  0.6  2 g 0,4  /3i  0.2  02  04  </3o  0.6  Figure 17: The curves associated with  Ui  mB  Ii)  .75 TeV.  —  —  —  II  =  .5 TeV”  2 g 0.4  i3o<i3 0.2  ,8i 02  </3o 0.4  0.6  Figure 18: The curves associated with  76  Ui  mB  .5 TeV.  If)  1.0 —  —  —  —  R=1TeV’  11= .75ToV’ R= .5TeV’ R=,25T6V  06  2 g 0.4  0.2  7 0.0  02  0.4  0,6  Figure 19: The curves associated with  01  mB  IX  .25 TeV.  It is assumed that any kinks or departures from smoothness in the curves are due to slight numerical instabilities and are not physical in nature. For each point on a given curve, a plot of the two relic densities was produced. Since all of these plots were very similar to one another we will only include a few representative examples here. The solid blue (red) curves are the numerically obtained relic densities for j3 (6) while the dashed curves are the corresponding approximatiOns, (/ o ) and (i3i ), derived above. Note that the validity of the approximation 3 is well demonstrated in the high x limit.  Lr8  —10  —20  —30  -40  —50  Figure 20: /3 and j3 for  mB  .75 TeV, 1/R  77  .5 TeV,  .75, and g 2  .555.  x 50  40  30  —10  —20  —30  —40  —50  Figure 21:  and j3 for  mB =  .25 TeV, 1/R  1 TeV,  =  .425, and  2  .12685.  .65, and  2  .505.  LiB  Figure 22:  and  for  mB  1 TeV, 1/R  1 TeV,  With the numerical solutions in hand we can now inquire as to when the SKPs froze out. As can be seen in the above plots, the  0 and  1 n  1 n  1 SKPs have completely frozen out at x  respectively. Their freeze out temperatures are then given by  T71=°= —  50  and  respectively.  78  1=I 7 T  25  50 and x  25  We can get a feel for when these freeze out temperatures occured by recalling that the last scattering surface is at redshift  z =  1100 [15] which corresponds to a expansion parameter a,  =  9.0826 x io—. This  then implies a temperature of  —p-  =  =  2.585 x io—’ TeV  For the above freeze out examples, the corresponding freeze out temperatures are listed below Table 1: Example Freeze Out Temperature Calculations mb .75 TeV 1.52 x 10_2 TeV 3.045 x 10—2 TeV .555 .25 TeV .12685 5.15 x 10 TeV 1.03 x 10—2 TeV 10_2 1 TeV 2.015 x TeV 4.031 x 10—2 ifl .505 Notice that all of the freeze out temperatures indicate that both of the relevant SKPs froze out long before the last scattering surface near redshift z  5.56 x io’. This is to be expected if they are to be  n.J  responsible for the initial density fluctuations which allow for the formation of large scale structure.  We can also now answer the question of whether or not the KK decay rates occur in equilibrium or not. The Hubble rate during the radiation domination era is given by  H—  V45M  x  where g is the number of relativistic degrees of freedom in the thermal bath at time TeV, all the species in the standard model are in thermal equilibrium, which leads to g. the three examples above, the at the time of the  1 = n  =  For T  0.3  106.75 [49]. For  1 SKP freezes out first. Therefore, when calculating the Hubble rate  1 n  1 SKP freeze out, we have to take into account the extra degrees of freedom in the  thermal bath due to the n out at the time of the  IEf.  1 n  =  0 SKP, i.e; g+  =  106.75 + 1  =  107.75. Since the  1 = n  1 SKP has already frozen  0 SKP freeze out, we use the standard model value of g at this time. Under  these assumptions we calculate  =  The decay rates for both the  1.647 x 10  1 = n  0 and  1 = n  Ha’’  =  6.61879 x  10_li  1 SKPs have been calculated in detail previously. The  criteria for thermal equilibrium is defined as I’> H(xf) [49]. Assuming a Higgs mass of .15 TeV and that only the first 10 excited KK states participate in the one-loop decay process for the n 1 rates for the three above examples are listed below.  79  =  1 SKP, the decay  Table 2: Example Decay Rate Calculations m  .75 TeV .25 TeV 1 TeV  1/R .5 TeV 1 TeV 1 TeV  1 Ff1  i  .75 .425 .65  .555 .12685 .505  2.887 x 10 TeV 5.5178 x 10 TeV 5.64546 x iO TeV  5.2107 x 10 TeV 1.4896 x 10 TeV 4.47671 x 10 TeV  Since all of these values are quite a bit larger than the Hubble rate, we can conclude that the decays mmt be occuring in equilibrium. Since our assumptions at the beginning of this section were dependent on the decay rates for the excited KK states occuring fast enough, we can now be more confident in our results.  80  General Conclusions We have demonstrated a new particle model which uses an ADD type braneworld scenario to produce a multi-component theory of dark matter. Compactification of the extra dimensions onto a sphere has lead to an infinite number of KK towers associated with each four dimensional Fourier mode of any bulk fields in the effective theory. The stable dark matter candidate states were shown to arise naturally as the lightest KK states within each tower. The multi-component dark matter theory was therefore a direct consequence of the compactification geometry. The Boltzmann equation was then generalized so as to provide the relic densities of the many different mutually interacting dark matter candidates. 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Hatfield, “Quantum Field Theory of Point Particles and Strings”, Addison-Wesley Publishing Com pany (1992)  84  Appendices Appendix A The calculation of the decay rates for the non-magnetic excited KK states require us to integrate the relevant constant amplitudes over the two and three body phase spaces. These calculations will be reviewed here for completeness.  The 2 Body Phase Space  The decay rate for the  Q (P1) 1  where m 1 v 2  =  —,  [  ( 5< ) 2 H(ps) p process is defined as  2 p 3 d p 3 I d 2 44 ( 3 2E r 7 1 )3 2 (2ir) 6 2E ( 2 2ir)j M  (‘ + 1) +  +  (p1—p2—p3)  2 is defined similarly, and the amplitude is given by M m  =  + 1) (22 + 1). Since the amplitude is constant we can bring it out of the integral and write the  rate in terms of the Mandelstam variable s  —  (2 + 1)(2e2 + 1) 2 gv 1 32x2mi  —  1 j  d [d 2 p 3 35 p 3 P2] E  (/—P2—Pa)  Using the fact that  f  6(p 3 p 4 d  —4)  1 1 =jdp 3 dE 31 J  J  =  f  6(E 3 d3psf dE  —  — 6(E + 3 i4) .fj3  +m3 2 2\/ps  85  2 4) Iflsl [d p 3 3 2E  =—  j  we can rewrite the rate again as  1d 2 p 3 c5(P — m)6 3 ——fd4P 4 (v—p2 16ir 1 m 2 v2(2 +1)(22+1) 2 P 3 fd . s[E) 2o(22 1 m 2 16ir (2 + 1)(22 + 1) 2 v 1  —  —  J  P3)  Since there is no angular dependence in the integrand, the angular integrals are trivial, giving  — —  —  (2 + 1)(22 + 1) 2 v 1 1 47rm (2 + 1)(22 + 1) 2 v 1  s 1 l6irm  —  Finally, if we evaluate the rate in the  —*  H) 2 =  f  E 2 dE  /(s + m  —  —  mc5(s + m  m)2  rest frame then s  =  —  —  m  —  ) 2 2/E  4sm  m, which leads to  v2(2i + 1)(22 + 1) J(m? + m 16irm  —  m.,)2  4m?m  —  The 3 Body Phase Space  The decay rate for the 1 (p)  —  ( 2 ) 3 ) 4 H(p p process is determined simply by extending the  phase space to accommodate one extra Higgs field in the final state 1  2m  f  I d p 3 d p 3 3 ( 2 2E 2ir) (2ir) 3 2E  I  f  4 3  (2ir) 4 2E 3  4 M(2ir)6  (P1  —  P2  As before, the amplitude is constant so we can bring it outside the integral.  —  P3)  Instead of using the  mandelstam variable, though, we will factor the delta function as  + 1)(22 + 1) 16(2ir) 1 m 5  2 p 3 d  p 3 d  f f f  It will be useful later to integrate over the  =  p4S3 (—p2 3 d  —  33  —  J34)  ö(mi  —  integral first, which yields  4 p 3 + 1)(22 + 1) 1 d 6(mi 16(2ir) 1 m 5 4 3 E 2  I  86  —  2 E  —  3 E  —  ) 4 E  2 E  —  3 E  —  ) 4 E  Since 2 dE E  p31 P4  =  d(cos 8) we can rewrite the integration measure as  f where clfl  =  4 p 3 d  =  JIfl2dIfiIdQ2dIflIdQ  sin OdOd. If we choose our coordinate system so that the z axis lines up with  p4  then we  can integrate over Q 4 to give  =  2 8x  f  dE 4 IflsIEsdEsd(cos9)sIE  =  8112 3 JE 2 d 4 E E  Substituting these results back into the decay rate we obtain  =  (2 + 1)(2e2 + 1) 64x in 1  In order to perform these last two integrals we choose a new variable defined as x  =  2 Pz 1 in  =  1 in  This leads to  —  —  + 1)(22 + 1)m 256ir  The minimum of both x 3 and x 4 are given by (xj)mjn  I4 dx 3 dx  j  =  To determine the values of  .  (xj)max  we investigate what occurs at the minimum of E . The minimum of E 2 2 is simply in , which occurs when 2 =  —p4,  or when E 3  =  . From this, it is clear that when E 4 E 2 is minimized, E 3 and E 4 both receive equal  amounts of the rest energy of the initial state and are therefore maximized. Using basic energy conservation  1 in  =  2+E E 3+E 4  we can determine (Ej)max by considering the point at which E 2 takes on its minimum value. This yields =  (E4)max  1 =  —  7n2) or  simply  (x3)max  =  87  (x4)max  =  1  2 in —  Applying these limits to the rate integral gives  —  HH) 2  + 1)(22 + 1)mi  2567r  fxm  dx 3 dx 4 Xm  (21+1)(22+1)ml (21+1)(22+1) 2 (m2+2mH) 3 2567r 1 m 1 m 3 2567r (1—  )  1 (m  —  2 m  —  mH) 2  2  —  The total decay rate is then the sum of the individual rates which gives the result quoted  —  rTotal —  —2(2  + 1)(2e2 + 1) 1 l6irm  (v2/(m  +  m  —  2 m)  m  88  —  4mm  +  1 (m  —  2 m  —  2 16ir  2mH)2’  )  Appendix B The Wigner 3-j symbols arise in the description of the coupling of two angular momenta in quantum mechanics and are related to the Clebsch-Gordon coefficients as [50]  ( I  £  £2  e  1 m  2 m  3 m  I  =  4t2;t3 —m3) 2 (tj4;mlm 12 + 1)”  )  where all of the parameters of the 3-j symbol are either integers or half integers. In quantum mechanics, £ is the eigenvalue of the angular momentum operator z-component of the angular momentum operator  II2m  =  I.  I  =  ‘  x  fl and  m is the eigenvalue of the of the  These are most familiar from the eigenvalue equations  £(t + 1)Y  m LY  =  Specifically, the symbols arise in the description of how two angular momentum states, couple together in order to form a third coupled state 13  =  1  I  and 12,  + 12. Due to the symmetry properties of  the Clebsch-Gordon coefficients, we can immediately conclude that the 3-j symbols satisfy the following selection rules  i.)  ii.)  3 1 m + 2 = m 0  iii.)  Vi  2 t t 3  2 <1 +  iv.)  where the third selection rule is known as the triangle rule and the fourth implies that the sum of all the total angular momentum quantum numbers must be an integer. If one of the above selection rules is not obeyed then the 3-j symbol vanishes. The 3-j symbols are symmetric about any even permutation of the columns  (  £  £2  £3  1 m  2 m  3 m  )  I\  £3  £  £2  7733  1 m  2 m  89  )  £2  £3  2 rn  3 m  1 m  and are either totally symmetric or anti-symmetric, depending on the value of the sum of the total angular momentum quantum numbers J, under any odd permutation of the columns  (  4  £2  £3  ml  m2  m3  (l)J( £  £  £2  1 m  3 m  2 m  )  (l)J(  )  £2  4  £3  2 m  1 m  3 m  £2  £3  2 m  m3  )  £3  £2  4  m3  2 m  1 m  The symbols also obey the orthogonality relations given by  (24+1)( £3=O m3=—3  4  £2  £  1 m  2 m  3 m  +1)( 3 (2t =—t 2 m 1 =— m  )  4  £2  £  1 m  2 m  3 m  (4 1 m  )  (4 3 m  £2  £  2 m  3 m  In general, the formulas which give the numerical value of the 3-j symbol in terms of the parameters are very complicated. A few cases exist, however, where this is simplified due to existing symmetries. We make use of only two of these in this work  ( (4  £2  0  £3  £  £  m  —m  0  o)  —  (_1)_m  —  /q—2t ) !q—24)!q—24)! )J12 1 1 ( (J+1)!  (J/2)!  ifjiseven  V  o} 0  ifJisodd  Note that this last result implies that all 3-j symbols which are characterized by vanishing magnetic quantum numbers are completely symmetric under any permutation of the columns. Therefore, the B symbols which we have defined in chapter 2  =  /(2% + 1)(2t 2 + fl(24 +  (4 1)1 0  £2  0  £3(  4  I I 0)\mi  £2  £3  1122  m3  are, by construction, completely symmetric under any permutation of the columns.  90  The Wigner 3-j symbols arise in many applications of quantum mechanics but, for our purposes, we will be concerned with their relationships with the spherical harmonics as the latter forms the basis with which we decompose our 6 dimensional particle into its 4 dimensional modes. The two relationships which we will make use of during this work are listed here without derivation.  The spherical harmonics form a closed set, i.e; there is a relation which allows us to write the product of any two spherical harmonics as the linear combination of other spherical harmonics. In terms of the 3-j symbols this is [51]  + 1)(2t+ 1)(2+  rn1(rn2() =  )  £=O m=-e  (  £  £2  £  1 m  m2  m  £  J \  0  0  0  )  y*rn(Q)  or, in terms of the B symbols,  yrni (,  rn2  (&, )  Ye(0,  B’7:  )  £=O m=—e  The other result that we make use of quite a bit is the Gaunt integral. This can be derived by multiplying the previous result with  (0, ) and integrating the volume element dfl  ±l)(2a+1) 2 y(24+1)(2e  f  (  =  £  1 m  sin 0d0d over the 2-sphere S 2  £2  £3 (1  2 m  )\0 3 m  In terms of the B symbols, this is  f  yrni (Q )yrn2 (9, )Y7 3 (0, )df  B17T  Note that the orthonormalization relation for the spherical harmonics has been used here  f  y(9  ‘  (9, )dQ  91  =  £2  £3  0  0  Appendix C The standard model of particle physics is the most accurate description to date of the strong, weak, and electromagnetic interactions that we observe in nature. Within the context of the theory, these three interactions are understood as arising due to the exchange of various spin 1 bosons amongst the spin 1/2 fermions which constitute all the matter that we directly observe. The Lagrangian for the model describes the most general renormalizable, locally gauge invariant field theory under the gauged symmetry group SUc(3) x SUL(2) x Uy(1). There are many good introductions to the standard model such as [32] [48]  [52] [53] [54]. In this chapter, however, we will only describe the particular sections of the standard model Lagrangian which directly apply to the present work.  The theory of the strong interactions, called quantum chromodynamics (QCD), is associated with the SUc(3) symmetry group and is described by the locally gauge invariant Lagrangian density  LSLJc(3)  where by G, a  = =  1, 2  —  O,G  —  +  =  GG 5 f 8 g 0 are the field strength tensors for the 8 spin 1 gluon fields given  8 and g 8 is the strong coupling constant. The structure constants,  fa&c,  define the Lie  algebra associated with the generators of the non-abelian 0 Sf1 (3) group  [,a )b]  r  2jfabcAC  where the generators, A”, are the Cell-Mann matrices. The Yang-Mills term in the Lagrangian leads to the various three and four point self interactions of the gluons. The second term describes the propagation and gauge interactions of the quarks, qja There are various summations occurring in this term, indexes the three generations of quarks while a, /3  =  j  =  1,2,3  1,2,3 are the color indices. The covariant derivative is  given by  =  ‘y”(Dj  =  7  -  i4G)  Note that there is no mass term for the quarks. The chiral symmetry forbids any bare mass terms for any fermions so we have to depend on spontaneous symmetry breaking to generate the quark, and other fermion, masses.  92  The electro-weak interactions provide a unified description of the electromagnetic and weak interac tions. These are described by the locally gauge invariant SUL(2) x Uy(i) Lagrangian density  +  £SUL(2)xUy(1) = £Gauge  £Fermion  +  EHigys  +  £Yukawa  The subscript L emphasizes the fact that only the left handed fermions interact with the gauge bosons associated with the non-abelian SUL (2) gauge group while the subscript Y is the label used to represent the hypercharge, the conserved noether charge associated with the abelian Uy (1) gauge group. The hypercharge, Y, is related to the electric charge,  Q=  Q,  and the third generator of the  , 3 r  via the relation  1,2,3, which are associated with SUL(2), and  which are  L 511  (2) gauge group,  Y+r . The gauge sector of the Lagrangian is 3  £Gauge =  where the field strength tensors W,,, i  =  —  associated with Uy (1), are given by  W  =  —  =  —  dflBV  —  YEijkW4Wt  . 1 1kB  and g is the SUL (2) coupling constant. The structure constants for SUL (2) are given by the totally anti-symmetric Levi-Civita symbol,  Eijk.  This then defines the Lie algebra of the SUL (2) generators as the  standard angular momentum algebra  [T,’I-J] = ieIjkTk  where the generators are related to the Pauli matrices as  Tt  -.  Since Uy (1) is an abelian gauge  group, the structure constants which define the Lie algebra associated with Uy(1) must be trivial. As with the gluons, the W bosons have three and four point self-interactions while the B 1. bosons, again due to the abelian nature of the Uy (1) gauge group, do not have any self-interactions.  The fermion sector of the Lagrangian is  Fermion = t  where  j,  E  QJL1I’QjL  + LJLmLJL +  again, indexes the generation.  93  UjRi],bUjR  + djRilMfR +  ejRi1PeR  The L, R subscripts represent the left and right chiral projections of the fermion fields given by  =  qL,R(x)  (17)q(x)  The left handed quarks and leptons are organized into doublets which transform under the SUL (2) gauge group  UjL  VJL  \  LJL=I  i  L=I 3 Q  diL)  ejL  while the right handed quarks and leptons are SUL (2) singlets. The hypercharges of the left handed fermions are given by  Q,L  1/6 and  —1/2 while, due to the relation between the hypercharge  LL  and electric charge, the hypercharges of the right handed fermions are equal to their electric charges. The covariant derivatives are given by  L 3 DQ  (a (a  DufR  = (s,.  —  i-B)  —  —  irW  —  iBi)  L 3 Q  irzW + iB) LJL  DdR  jR  = (s,.  + i-Bfl)  ( + ig’B) CjR where g’ is the Uy (1) coupling constant and we can now simply read off the gauge interactions between the fermionic content and the weak gauge bosons.  The Higgs sector of the Lagrangian is given by  £Higgs  where the  =  (D1)t (Dq)  —  V()  field transforms under SUL(2) as a doublet  =k Each component of the doublet,  and  °,  ‘°  transforms as a complex scalar under the Lorentz group.  94  Since the hypercharge is chosen to be Y  =  1/2, the covariant derivative is given by  (a  çb 1 D,  —  —  i9-B)  The absolute square of the covariant derivative then describes both the propagation of the the three and four point interactions of the potential be SUL (2) x  Cry (1)  field and  field with the weak gauge bosons. Requiring that the Higgs  invariant and renormalizable restricts V@) to the form  V@)  + A  =  Since p is a renormalizable parameter of the standard model Lagrangian, its value, in particular its sign, cannot be calculated from first principles, i.e; it must be measured through experiment. This is very unsettling since the assumption that p 2 < 0 is the complete explanation as to why electro-weak symmetry breaking occurs in the first place and the Riggs is the only standard model particle which has not been found, i.e; we can’t measure p yet.  Besides the gauge interactions, the  field also interacts with the fermions through the Yukawa  couplings  £Yukawa = —  >  + I’jLivkeJRj + h.c.  [rauta +  where i, j are generational indices, F are 3 x 3 matrices whose components are the constant Yukawa couplings, and h.c. stands for hermitian conjugate. It is necessary to perform a similarity transformation on the  4  field, given by  =  2 iT  so that the various flavours of quarks and leptons receive the correct masses  from the spontaneous symmetry breaking process.  The assumption that p 2 <0 in the Higgs potential leads directly to spontaneous symmetry breaking which induces a non-trivial vacuum expectation value (vev) for the one of the real scalar degrees of freedom in the  0 field  field. We then identify this vev with  and define this real scalar to be the Riggs field, ii. In  order to expand about the stable vacuum, the Higgs field is then shifted by the vev. In the unitary gauge, where the Goldstone bosom are all reinterpreted as the longitudinal components of the now massive gauge bosons, the Higgs potential then becomes a function of the actual Riggs field  V(h)  = jh2  +  where the vev is given by v =  95  +  Expanding about the stable vacuum in the other sectors of the standard model Lagrangian gives rise to masses for the fermions and the weak gauge bosons as well as mass mixing between the  and B, 1 gauge  bosons. Diagonalizing the mass mixing matrix then leads to the introduction of the more familiar mass eigenstates, the photon and the neutral Z boson. The standard model is a very intricate theory and a full discussion is beyond the scope of this work. For our purposes, we are interested specifically in the three point interactions between the Riggs and the fermions and weak gauge bosom (the Riggs does not interact with the gluons so we need not worry about them here). We will then conclude this section by listing the various Riggs interactions.  The interactions with the fermions are given by  Higgs—Fermion  —  — —  where we are summing over all fermions,  f.  %rnf_h Lj  Since nsf  <<  v for all fermions (the inequality isn’t so  severe for the top quark) the Riggs field couples very weakly to all the particles that are easy to produce in accelerators. This is exactly the reason as to why the Riggs has been able to stay hidden for so long, it preferentially couples to particles which are hard to produce. The Riggs interactions with the weak gauge bosons are given by  Higgs-Gauge  where W  = .Jz  (1474  =  —  iW4) are the charged W bosons and Z,. is the neutral Z boson.  96  


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