Detection Prospects of Doubly Charged Higgs Bosons from the Higgs Triplet Model at the LHC

Transcription

1 UPTEC F11 44 Examensarbete 3 hp Juni 11 Detection Prospects of Doubly Charged Higgs Bosons from the Higgs Triplet Model at the LHC Viveca Lindahl

2 Abstract Detection Prospects of Doubly Charged Higgs Bosons from the Higgs Triplet Model at the LHC Viveca Lindahl Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan Postadress: Box Uppsala Telefon: Telefax: Hemsida: In this thesis I explore the possibilities of detecting doubly charged Higgs bosons from the Higgs Triplet Model (HTM) at the Large Hadron Collider (LHC) at CERN. Higgs bosons are included into the Standard Model (SM) of particle physics in order to explain the origin of mass of the elementary particles. Even though the SM is considered to be a reliable starting point for any particle theory, no Higgs particles have to this day been found. There are therefore high expectations for the record-breaking energies of the LHC to lead to a Higgs discovery. The HTM produces seven different Higgs bosons, and among these we find the doubly charged ones. Because of their exotic charge, I focus on studying these Higgs particles in particular. To this purpose, I implement the full HTM theory as an alternative model option in standard particle physics Monte-Carlo software and then simulate LHC proton-proton collisions at a center of mass energy of 14 TeV. The investigated signal is defined as two like-signed leptons, four hard jets and missing energy in the final state. The main production mechanisms are pair-production and associated production with a singly charged Higgs. Since I choose to study a region of parameter space where the triplet vacuum expectation value is relatively large, the doubly charged Higgs decays into W's and the singly charged Higgs into WZ or tb. The results of the simulations show that the LHC could probe Higgs masses up to at least 3 GeV with an integrated luminosity of about 3 fb-¹. Handledare: Rikard Enberg Ämnesgranskare: Gunnar Ingelman Examinator: Tomas Nyberg ISSN: , UPTEC F11 44

3 Contents 1 Outline 4 Background 5.1 The Standard Model The Higgs Mechanism Neutrino Masses Beyond the Standard Model The Higgs Triplet Model Model Contents Constraints on the Parameters Minimization of the Potential Mass Matrices and Eigenstates Interactions Production and Decay of the Doubly Charged Higgs at the LHC Search for Doubly Charged Higgs Bosons at the LHC How to Probe Physics at the LHC Search Strategy Analysis Results Summary and Conclusions 4 Appendices A FeynRules Model File 43 B Feynman Rules for Higgs Vertices 5 B.1 Gauge-Higgs Three-vertices B. Fermion-Higgs Vertices B.3 Higgs-Higgs Three-vertices

4 1 Outline In this thesis I perform a phenomenological study of the Higgs Triplet Model (HTM), an extension of the Standard Model (SM) of particle physics. The HTM generates singly and doubly charged Higgs particles, which are of special interest since they do not appear in the SM. Hence, the aim of this work is to examine the possibilities of detecting such charged Higgs particles at the Large Hadron Collider (LHC) at CERN. To this end, the first task is to derive the analytical relations mass matrices, mixing angles and theoretical constraints needed for subsequent phenomenological studies. The second step is to calculate the Feynman rules, which describe the particle interactions of the theory. At this stage, it is necessary to specify which processes to simulate and what values the free parameters of the theory (masses, couplings, etc.) should be set to. By running the simulations under different configurations and developing analysis methods optimized for reducing the SM background and preserving the signal, the end goal is to devise a search strategy which has good prospects of detecting new physics in real-life experiments. The structure of the thesis is as follows. In section, I briefly introduce the SM and some of its limitations. 1 This motivates why studying the HTM, which is presented in section 3, is of interest. In the same section, the analytical expressions for the Higgs masses and mass eigenstates, as well as the minimization conditions for the Higgs potential, are derived. In section 4, I define a search strategy and method of analysis, and also present results from Monte-Carlo collider studies. Finally, conclusions are found in section 5. 1 There are numerous good reviews which do this in more detail, see e.g. [1,, 3]. 4

5 Background.1 The Standard Model The SM is a theory describing the interactions of all the fundamental particles that have been discovered this past century. This framework makes it clear that there is some underlying order ruling the world of elementary particles. On the highest level there are the fermions (half-integer spin particles) and bosons (spin zero particles). The fermions can then further be subdivided into quarks and leptons, and the leptons into charged leptons (electrons, e; muons, µ; and tauons, τ) and neutrinos (ν e, ν µ, ν τ ). Among the bosons there are the gauge bosons, the particles mediating force: gluons (strong force), W ± and Z (weak force), and the photon, γ (electromagnetic force). In the SM, there is also another type of boson, the Higgs boson, which is introduced to explain how the fundamental particles gain mass. Since the advent of particle accelerators, the SM has been thoroughly tested and proven to be highly successful. Even so, it is a deficient theory, being unable to fully explain all experimental observations and lacking compatibility with general relativity. It is therefore a common belief that, even though it fails to provide a Theory of Everything, the SM can be seen as a low-energy approximation of a more fundamental theory. From a more practical point of view, the SM is an essential building block of any more exotic particle theory. In more technical terms, the SM is a relativistic quantum field theory, describing the particles in our world by fields in space-time. The relativistic property states that the Lagrangian is invariant under the Poincaré symmetry group, the symmetry of special relativity. Moreover, the SM is a gauge theory, meaning that its Lagrangian is also invariant under another continuous symmetry group of local transformations of the fields. This locality allows the parameters of the transformation to depend on each point x in space-time, whereas for a global symmetry (like the Poincaré symmetry) the transformation is the same for all x. Imposing gauge symmetry has proven to be an attractive way to generate particle interactions within the theory. Furthermore, it implies the existence of a certain number of gauge bosons. Each symmetry group comes with a number of generators, which as the name suggests, are operators which generate the symmetry transformations. The number of gauge bosons that arise can be shown to be equal to the number of generators of the gauge symmetry. In the SM, the relevant gauge group is SU(3) C SU() L U(1) Y, which sums up to generators, and hence provides us with the gluons (8), the massive W ±, Z bosons (3) and the photon (1). Basically, SU(3) C is the gauge symmetry of quantum chromodynamics (QCD), the theory of the strong interaction, and SU() L U(1) Y is the symmetry of electroweak interactions, which is the most relevant one for Higgs boson physics. The chirality, or handedness, of a particle is a somewhat abstract property which states whether the particle transforms under a left- or right-handed representation of the Poincaré group. The SM can even more explicitly be described as a chiral gauge theory, that is, it treats left- and right-handed fermions differently. 3 Specifically, it has been shown experimentally that the W ± bosons interact only with left-handed fermions and right-handed anti-fermions. In group theoretic terms, this implies that particles of different handedness transform under separate representations of the gauge group. In particular, the left-handed fields pair up into a doublet and transform as a two-dimensional vector under SU() L, while the right-handed fields are singlets, meaning that they are invariant under the transformation (or equivalently, that all generators are zero in this representation). Considering their transformation properties under SU() L U(1) Y, the fermions of the SM can thus be organized as follows, ( ) ( ) EL i ν i =, QL i u i =, lr, i ur, i dr, i i = 1,, 3 (.1) l i L d i L For instance, the symmetry group of all rotations in three dimensions, denoted O(3), has three generators, corresponding to a rotation about each axis. The group of rotations in the plane, O(), has one generator, corresponding to the rotation about an axis perpendicular to the plane. 3 This fact explains the origin of the L, as in Left, in SU() L. 5

6 where ν i = {ν e, ν µ, ν τ } and l i = {e, µ, τ} are the leptons, and u i = {u, c, t} and d i = {d, s, b} are the quarks. The index i labels the three so called generations of leptons and quarks (the three different colors of quarks are left implicit). Note that the SM as it is stated does not contain any right-handed neutrinos, νr i. Such a particle would be sterile, i.e., have no interactions (except for gravity). This can be seen by using the definition of the weak hypercharge, Y, the generator of U(1) Y, Q = T 3 + Y/, (.) where Q is the electric charge and T 3 is the third component of weak isospin (one of the generators of SU() L ). With the hypothesized νr i being electrically neutral and an SU() L singlet (i.e, T i =, i = 1,, 3), this implies that also Y =. Thus, the νr i would be invariant under all gauge transformations. In more physical terms, this means it would not interact and for this reason it is usually excluded from the SM.. The Higgs Mechanism In the discussion so far, an important aspect of the SM has been left out. It turns out that it is impossible to simply hard-wire mass terms of the gauge bosons into the Lagrangian and still respect gauge symmetry. 4 In fact, the same holds true for the fermions, which have non-invariant mass terms of the form m f ψψ = m f (ψ L ψ R +ψ R ψ L ). A solution to both of these problems is provided by the Higgs Mechanism, in which our system undergoes spontaneous symmetry breaking (SSB), apparently losing part of its full symmetry by acquiring a non-symmetric ground state. This process is best illustrated by an example, which we might as well take to be the SM case. To this purpose, we add to our set of particles in equation (.1) a complex scalar SU() L doublet, Φ(x) = ( ) φ + (x) φ, (.3) (x) and a corresponding SU(3) C SU() L U(1) Y gauge invariant Lagrangian L Φ = D µ Φ V Φ, V Φ (Φ) = µ ΦΦ Φ + λ 1 (Φ Φ). (.4) Here, D µ is the covariant derivative, which is necessitated by the postulated gauge symmetry, and is for this particular symmetry given by D µ = µ igt i W i µ ig 1 Y B µ ig s T a G a µ, (.5) where T i, i = 1,, 3 are the SU() L generators, Y is the U(1) Y generator and T a, a = 1,,..., 8, are the SU(3) C generators. The W i and B are the gauge fields which give rise to the electroweak gauge bosons of the SM, while the G a are the gluon fields. Finally, g, g and g s are coupling constants which determine the strength of the interactions. The explicit form of the generators depends on which representation the fields transform according to. For Φ, we have TΦ i = σi /, where σ i are the Pauli matrices, Y Φ = 1, and TΦ a = (since Φ is defined to be an SU(3) C singlet). The classical ground state of the scalar field can be found by minimizing the potential V Φ with respect to the fields, φ + and φ. For the case µ Φ <, it can easily be verified that there is a continuous degeneracy of minima for field values satisfying Φ Φ = µ Φ /λ 1. This fact is illustrated in figure 1(a) for the simplified case when Φ is just a single complex scalar field φ (i.e., not a doublet). If µ Φ, the potential is positive definite and has only the unique minima at the origin, a point which is invariant under the gauge symmetry. This case will therefore not be of much interest to us here, since it will never lead to SSB. Also, the condition that the potential should be bounded from below requires λ 1. 4 Actually, renormalizability, which will however not be discussed further here, is the more important property in this context. 6

7 ΦI Φ I Φ Φ R Φ R V min (a) (b) Figure 1: (a): The degenerate set of minima of the well-known Mexican Hat Potential, V (φ) = µ Φ φ φ + λ 1 (φ φ), plotted in the complex plane. This potential has U(1) symmetry, that is, it is invariant under φ φ e iθ. (b): A simplified top-view of the potential in (a) showing the field transformations after SSB leading to the fields which oscillate about the vacuum state. The dotted, dashed and solid axes belong to the coordinate systems before any transformations, after a rotation, and after a rotation+translation, respectively. In the analysis so far, purely classical field theory has been employed in determining the minimum-energy configurations of the scalar fields. We connect to quantum theory by reinterpreting this ground state as the quantum vacuum of the universe, and by regarding the minimized scalar field value as a vacuum expectation value (VEV). Now, assuming that the scalar doublet acquires a VEV, Φ, at the degenerate minimum, the system is forced to choose a preferred direction the symmetry is spontaneously broken. In figure 1(a), this corresponds to the field choosing a particular phase to settle down with, even though each phase is equally likely. In whatever direction SSB occurs, we are thereafter free to make a global SU() L rotation of Φ to bring Φ completely into the neutral and real direction, i.e., the direction corresponding to the real component of φ. This simply means that we are allowed to rotate our coordinate system. The analogous procedure is illustrated in figure 1(b), where a rotation brings φ onto the real axis. Also, the particle masses and interactions are determined by studying the fields which fluctuate around the stable ground state, having vanishing VEV s. Thus, after the rotation we make a change of variables to these fields by shifting the origin to the chosen minimum. Applying these transformations we end up with a doublet of the form (keeping the variable name Φ) ( ) g Φ(x) = + (x) (v Φ + h(x) + ig (x))/, (.6) which has the VEV Φ = ( ) v Φ /, v Φ / = µ Φ /λ 1 R, (.7) that is, g + = g = h =, as we wanted. The fields g + /g and h correspond to the fields φ I and φ R, respectively, of figure 1(b). Upon expanding the Lagrangian of equation (.4), with Φ as in equation (.6), it will be discovered that there are mass terms for h of the form Mh h (where M h v Φ ), while for g ± and g there are none they are massless. These massless fields which appear as the result of SSB of a continuous symmetry are known as Goldstone bosons, and their presence is closely related to the geometry of the potential. A non-zero mass term is equivalent to a non-vanishing second derivative in that particular field direction. Turning again to figure 1(a) for visual aid, we see that field fluctuations in the radial direction (up the potential hill) generates 7

8 massive fields (h), while the tangential field direction (along the flat valley) corresponds to massless Goldstone bosons (g ±, g ). Up until now, we have not made full use of the gauge symmetry at hand. Before making the change of variables leading to equation (.6), it could have been remembered that there is nothing stopping us from doing a SU() L rotation of the fields which is different at each space-time point, x. The form of the potential would remain unchanged. Thus, we can make a rotation which follows the fluctuations of the Goldstone bosons such that not only Φ, but also Φ(x) is in the neutral and real direction. That is, Φ(x) = 1 ( ), (.8) v Φ + h(x) where g + (x) and g (x), and still, h =. This particular choice of gauge, eliminating the Goldstone bosons, is called unitary gauge. The remaining real scalar field, h, is physical and cannot be rotated away since it is the radial component of the fields (c.f. φ R in figure 1(b)).When quantized, this is the famous Higgs boson of the SM. It is now finally time to demonstrate the true virtue of the Higgs mechanism. By plugging in equation (.8) into the covariant derivative term of equation (.4) and collecting terms quadratic in the gauge fields we find ) where W µ ± = 1 (Wµ 1 iwµ D µ Φ = M W W +µ W µ + 1 M ZZ µ Z µ + non-quadratic terms, (.9) (gw 3 µ g B µ ) / g + g are the electroweak gauge and Z µ = bosons, which have now acquired masses M W and M Z v Φ! Upon counting the number of degrees of freedom involved, we find that there are just as many massive gauge bosons created as there were massless Goldstone bosons. Thus, it is commonly said that the Goldstone bosons are eaten by the gauge bosons, thereby gaining a mass. By Goldstone s theorem, there is one Goldstone boson for each generator of the symmetry that is broken during SSB. In the SM, we have SU(3) C SU() L U(1) Y SSB SU(3) C U(1) EM, (.1) where U(1) EM is the gauge symmetry of quantum electrodynamics (QED), the theory describing the electromagnetic force. We now do some counting, and see that we have gone from to generators, breaking 3 generators in the process. Thus, the Higgs mechanism does not only explain how the massive gauge bosons obtain a mass, but also why the photon and the gluons remain massless: they correspond to the unbroken gauge symmetries SU(3) C and U(1) EM, respectively. Similarly, as Φ develops a non-zero VEV, the charged fermions can gain a mass. This can be accomplished via the gauge invariant Yukawa interaction terms L Yuk,Φ = y ij l Ē LΦ i l j R yij Q d LΦ i d j R yij Q u Liσ i Φ u j R + h.c., (.11) where the various y s are Yukawa coupling constants (and h.c. will always stand for Hermitian conjugate ). Just as for the gauge bosons, the generated fermion masses are proportional to the doublet VEV, m f v Φ. Also note, that since there are no right-handed neutrinos νr i in the SM, the neutrinos cannot acquire masses in this way and consequently, they remain massless. For the record, we note that the above interactions are commonly written in terms of quark and charged lepton mass eigenstates. This is realized by diagonalizing the Yukawa matrices y u, y d and y l. Straightforward algebra then transforms the above expression into L Yuk,Φ = y j d u i L (V CKM) ij d j R φ+ y i d d i L d i R φ y i u u i L u i R (φ ) y j u d i L (V CKM )ij u j R (φ+ ) y i l ν i L l i R φ + y i l l i L l i R φ + h.c. (.1) 8

9 where V CKM is the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which determines to what extent quarks can interact across different generations, and yu, i yd i and y l i are the diagonal elements of the corresponding diagonalized Yukawa matrices. They are given by the expression y i f = mi f v Φ, f {u, d, l}, (.13) where m i f is the mass of f i, explicitly illustrating the fact that Higgs particles can be said to couple to mass. For these quantities in particular, the rule will be that their index should not be counted when applying the Einstein summation rule. In the situation above, this means that their index should be summed since it appears three (and not two) times..3 Neutrino Masses Beyond the Standard Model Despite the great successes of the SM, it provides an incomplete description of fundamental physics. Notably, it does not attempt to explain all known forces gravity is completely left out. Furthermore, there are a number of theoretical fine tuning problems present. Among these we find the mass hierarchy problem, the fact that the quantum corrections to the Higgs mass are much larger than the Higgs mass itself. In order to cancel these corrections, the parameters involved must be very finely tuned. The most relevant problem to us in this work however, is the fact that within the SM, neutrinos are massless. This is in strong disagreement with the fact that we now know, through neutrino oscillation experiments, that neutrinos indeed have non-zero masses [4]. An associated matter is the question why this mass is so small compared to the other fermion masses within the same generation. For instance, in the first generation of fermions, the electron, up and down quarks all have masses in the MeV region, while the electron neutrino has a mass on an ev scale. There are several ways to extend the SM such that it incorporates neutrino masses. Perhaps the most popular way is to apply some version of the seesaw mechanism, which also attempts to explain why the neutrino masses are so tiny. The main idea is to introduce new heavy particles into the theory in such a way that their masses keep a certain balance with the neutrino masses: the more massive these heavy states get, the lighter the neutrinos become. One possibility (known as type-i seesaw ) is to include right-handed neutrinos, whereby the neutrinos are allowed to have a Yukawa interaction with Φ just as the charged leptons and quarks in equation (.11). Hence, the neutrinos will gain a Dirac type 5 mass term, i.e., (M D ν ) ij νr i ν j L + h.c.. (.14) However, this is not the only mass generating term that would be possible in this case. There is no obvious reason why there should not also be a Majorana mass term such as 1 (MR ν) ij ν i R (ν j R )c + h.c. (.15) present. This term violates lepton number conservation, which is nevertheless only an accidental, not fundamental (i.e., postulated) U(1) symmetry of the SM. Thus, barring the special circumstances leading to additional symmetry in the form of lepton number conservation, one would in general expect neutrinos to have Majorana masses. Adding up the contributions from equations (.14) and (.15) we get a total neutrino mass term of the form (in matrix notation) 1 ( ) ( M D ) ( ) (ν L ) c ν νl ν R (M D ν ) T M R ν (ν R ) c + h.c.. (.16) 5 For a detailed description of Dirac vs. Majorana masses and a review of neutrinos in general, see e.g. [5, 6]. 9

10 The neutrino masses and mass eigenstates are found by diagonalizing the above neutrino mass matrix, which is in general complex and symmetric. In order to get the general picture of the seesaw mechanism, we will now look closer at the case of only one generation for which M R ν and M D ν are just complex numbers, and for simplicity these will be assumed to be real. The two eigenvalues of the mass matrix are then M ν1, = 1 MR ν ± 1 (M R ν) + (M D ν ). (.17) The condition that the right-handed neutrinos are heavy translates into M R ν M D ν, (.18) which when applied to equation (.17) gives us the approximate mass eigenvalues M ν1 (MD ν ) M R, M ν M R ν + (MD ν ) ν M R ν M R ν. (.19) Note that the mass eigenvalues in the general case are real but can have either sign. The physical masses however, are non-negative. This fact can always be made explicit in equation (.16) by properly choosing how to define the mass eigenstates. Now, the interesting fact to note from the above equation, is that that we have generated one light neutrino with mass M ν1 and one heavy neutrino with mass M ν, and that these masses are connected such that M ν suppresses the magnitude of M ν1. There is yet another way to add a Majorana mass term into the theory, and that is to equip the Lagrangian with a Yukawa term of the form L Yuk, = h ij (EL i)c iσ E j L + h.c., (.) where h is an arbitrary complex and symmetric matrix of Yukawa couplings, and is a complex scalar field which by gauge invariance must necessarily be an SU() L triplet, = (δ ++ δ + δ ) T. Furthermore, the weak hypercharge is constrained to Y =, since Y EL = 1 and L Yuk, should be gauge invariant, in particular under U(1) Y. Assuming that acquires a VEV = ( v / ) T, this Yukawa interaction will contain the neutrino mass term 1 ( h ij v )(νl i)c ν j L + h.c. = 1 (ML ν) ij (νl i)c ν j L + h.c.. (.1) The full neutrino mass term of equation (.16) then generalizes to 1 ( ) ( M L (ν L ) c ν M D ) ( ) ν νl ν R (M D ν ) T M R ν (ν R ) c + h.c., (.) with mass eigenvalues given by (again looking at the simplified case of only one generation and real matrix elements) M ν1, = 1 (ML ν + M R ν) ± 1 Now assume the hierarchy (c.f. equation (.18)) (M L ν M R ν) + (M D ν ). (.3) M R ν M D ν M L ν. (.4) Since M D ν v Φ and M L ν v, this is reasonable if the triplet VEV is much smaller than the doublet VEV (or really, if h ij v v Φ ). In section 3.3 it will be demonstrated that one way to enforce this condition is to demand that the triplet comes with a large mass coefficient. Under 1

11 these assumptions, the approximate mass eigenvalues are found to be (by Taylor expanding to second order in M D ν and first order in M L ν) M ν1 M L ν (MD ν ), M ν M R ν + (MD ν ) M R ν M R ν M R ν, (.5) which again illustrates the seesaw mechanism. This way to generate small neutrino masses by the inclusion of a heavy triplet scalar field is sometimes denoted type-ii seesaw. In summary, there are several reasonable scenarios which result in massive neutrinos, and far from all have been demonstrated above. It is therefore often an attractive approach to study minimal extensions to the SM. One way to do this is to only extend the fermion sector with righthanded neutrinos, corresponding to equation (.16). This work however, treats the case when the fermionic sector is kept minimal and a single complex triplet scalar,, is added to the scalar sector, The resulting theory is what is commonly denoted the Higgs Triplet Model, and it is described in detail in the following section. 11

12 3 The Higgs Triplet Model 3.1 Model Contents The Higgs Triplet Model (HTM) can be thought of as a minimal extension of the SM. Basically, the model can be constructed by extending the Higgs sector of the SM with a complex scalar SU() L triplet,. The full scalar sector of the HTM thus consists of, with weak hypercharge Y =, and the SM complex scalar doublet Φ, with Y Φ = 1: ( ) φ + Φ = φ, = δ ++ δ + δ. (3.1) In order for the HTM Lagrangian to be the most general one subject to the postulated SU() L U(1) Y gauge symmetry, it is also necessary to add a few additional terms to the SM Lagrangian. More specifically, the full HTM Lagrangian can be expressed as L HTM = L SM {}}{ L Gauge + L Fermions + L Φ + L Yuk,Φ + D µ V Φ, + L Yuk,, }{{} Extra HTM terms (3.) where L Gauge and L Fermions are the kinetic terms of the gauge bosons and fermions, respectively, and L Φ and L Yuk,Φ are the same as those defined in equations (.4) and (.11). Among the new HTM contributions are the additional scalar potential terms contained in V Φ,. Adding these to the SM V Φ, we get the total potential V = V Φ + V Φ,, which most generally can be parametrized as follows [7], V (Φ, ) = µ Φ (Φ Φ) + µ tr( ) + λ 1 (Φ Φ) + λ [ tr( ) ] + λ 3 det( ) + λ 4 (Φ Φ) tr( ) + λ 5 (Φ σ i Φ) tr( σ i ) ( ) 1 + k (Φ T iσ Φ) + h.c.. (3.3) Thus, there are five dimensionless parameters, λ 1,..., λ 5, and three dimensionful ones, µ Φ, µ and k. The squares on the µ s are there simply to remind us that they are of dimension mass-squared, and does not reflect any assumptions about their sign. In the above equation is in a representation: ( δ = + / ) δ ++ δ δ + /. (3.4) In L HTM we also find the additional Yukawa interaction term, L Yuk,, which is defined by equation (.). The triplet kinetic term, D µ, gives rise to both gauge boson scalar interactions and, assuming the triplet acquires a non-zero VEV, gauge boson mass terms. The covariant derivative, D µ, is given by the general formula in equation (.5), with Y = and T a =. The SU() L generators, T i, are given by the three-dimensional generalizations of the Pauli matrices, 1 T i = 1 1 i 1 1 1, i i, 1 i 1, (3.5) where the appropriate normalization factors have been included. 1

13 3. Constraints on the Parameters The Yukawa term L Yuk, in equation (3.) is of particular significance for the HTM since it can generate neutrino masses without including right-handed neutrinos, the existence of which there is no convincing evidence. Straightforward expansion of this term yields L Yuk, = h ij (E i L )c iσ E j L + h.c. = h ij [ (ν i ) c P L ν j δ 1 ( (l i ) c P L ν j + (ν i ) c P L l j) δ + (l i ) c P L l j δ ++ ] + h.c., where P L = (1 γ 5 )/, is the left projection operator. A couple of observations can be made regarding the above equation. First of all, lepton number symmetry is explicitly broken by its presence. To see this, we can make an attempt to conserve lepton number. L Yuk, then forces us to set L =, while the k (Φ T iσ Φ) term of the potential requires L = (since L Yuk,Φ implies L Φ = ). Thus, we fail to conserve L. Secondly, we see that if δ acquires a VEV δ = v /, we obtain the neutrino mass matrix (c.f. equation (.1)) (3.6) M ij ν = h ij v. (3.7) This matrix can be diagonalized using the unitary Maki-Nakagawa-Sakata (MNS) matrix (see e.g. [8, 9]) diag(m 1, m, m 3 ) = V MNS M νv MNS, (3.8) that is, the neutrino mass eigenstates ν i are related to the flavor states ν i by the relation ν = VMNS ν. By eliminating M ν from equation (3.8) using equation (3.7), followed by squaring and taking the trace of the resulting equation, one finds, h ij = 1 v m i. (3.9) i,j There are several experimental constraints involved here. Cosmological observations provide an upper bound of m i.3 ev (99.9% CL) [1], which via the above equation bounds the magnitude of the combination v h ij. Roughly, therefore, the following relation should be satisfied: i h ij v.1 ev = 1 1 GeV. (3.1) Furthermore, there are upper bounds for h ij coming from lepton flavor violating decays. Hence, to produce values for the neutrino masses and the elements of V MNS that are consistent with the currently favored ones, v must have a lower bound, say, v 1 ev [7, 11]. There is also an upper bound on v coming from measurements of the ρ parameter, defined by ρ = M W M Z cos θ W, (3.11) where θ W is the weak mixing angle. In the SM, ρ 1 at tree-level 6, consistent with the data. In the HTM, ρ is less than one at tree-level, ρ = 1 ε 1 + 4ε, (3.1) where ε v /v Φ. Experimentally, ρ is known to have a value very close to one, ρ exp = (95% CL) [1]. This quoted value has been fit to experiment such that if the SM were true one would ideally measure ρ exp = 1. The observation that ρ 1 thus forces ε to be very small, ε.1. Alternatively, using v Φ 46 GeV, this implies v 1 GeV. Attempts have been made to explain the hierarchy v v Φ by e.g. a two-loop radiative mechanism [13] or by introducing extra dimensions [14]. It can also be accounted for by applying a type-ii seesaw mechanism, requiring the triplet mass coefficient to be large, as will be demonstrated in section In quantum field theory, tree-level refers to calculations in perturbation theory where only connected Feynman diagrams without loops are included. A brief introduction to Feynman diagrams can be found in section

14 3.3 Minimization of the Potential We will now study the potential defined in equation (3.3) in more detail, and derive the minimization conditions needed for SSB to take place. First of all, it can be noted that, requiring V to be Hermitian forces the coefficients λ 1,..., λ 5, µ Φ and µ to be real. Now assume that the fields acquire VEV s, Φ and, at a minimum of V. What restrictions on the coefficients does this imply? And also, is it possible to find minima for which SSB does not break U(1) EM, which is necessary for the conservation of electric charge? Following a similar procedure as in section., we therefore make a global SU() L rotation of the fields such that has its VEV in the direction of its neutral component, = = δ v /, (3.13) that is, δ ++ = δ + = and δ = v /. A necessary condition for V to be at a local minimum for these field values is that all the first order partial derivatives vanish, V ψ =, for ψ = φ +, φ, δ ++, δ +, δ. (3.14) Φ= Φ, = In particular, for ψ = δ ++, equations (3.13) and (3.14) imply k φ + =. (3.15) The dimensionful coefficient k is necessary to keep non-zero to avoid the addition of an extra Goldstone boson (see discussion in [15]). Hence, φ + =, and the entire charged sector has a vanishing VEV. This leaves the electromagnetic U(1) EM symmetry unbroken after spontaneous symmetry breaking, as desired. Accordingly, in addition to equation (3.13) we now also have Φ = ( ) φ = ( ) v Φ /. (3.16) By the reasoning above we can now, without loss of generality, look for local minima of the potential subject to the constraint that the charged fields vanish. Thus, we extract only the neutral part of V, V n = µ Φ φ + µ δ + λ 1 φ 4 + λ δ 4 + (λ 4 + λ 5 ) φ δ ( ) k φ φ δ (3.17) + h.c.. We can here make some observations regarding the coefficient k. Even if we at first let k be arbitrary and complex, we can always make k positive and real by absorbing its phase into the trilinear term Φ T iσ Φ of the potential V (by redefining the fields). The relative phase of Φ and appears only in this term and thus, the rest of the potential is left unaffected. Now, assuming k >, we note that (3.17) can only have a minimum when the combination φ φ δ is real and positive 7 which gives, φ φ δ = φ e iα δ e iβ = φ δ = α = β, (3.18) at the minimum. For any minimum, we therefore still have som freedom left in defining our fields, as long as their phases satisfy the above constraint. If we e.g. make a global U(1) Y transformation by an angle α, φ Y Φ =1 φ, δ Y = δ, (3.19) 7 This can be realized by assuming that the potential has a minimum for φ φ δ = re iθ, and then noting that φ φ δ + h.c = r cos(θ) r, with equality only for θ = + πn. 14

15 the resulting VEV s of the fields are real and non-negative. It will therefore always be assumed here that this is the case, without loss of generality. The algebra has now been simplified somewhat and we are left with the task of minimizing the function V n (x, y) = µ Φ x + µ y + λ 1 x 4 + λ y 4 + (λ 4 + λ 5 )x y k x y. (3.) with respect to x = φ and y = δ. The stationarity conditions then read V n x =, V n y =, at (x, y) = (v Φ/, v / ), (3.1) which amounts to (for v Φ, v ), µ Φ + λ 1 v Φ + 1 (λ 4 + λ 5 )v λ kv =, (3.) µ + λ v + 1 (λ 4 + λ 5 )v Φ 1 λ kv Φ =, (3.3) where λ k k/v is a dimensionless parameter, introduced for convenience. Using equations (3.) and (3.3), the number of free parameters of the potential can thus be reduced from ten: µ Φ, µ, λ 1,..., λ 5, k, v Φ and v ; to eight: λ 1,..., λ 5, λ k, v Φ and v ; by eliminating µ Φ, µ and using λ k instead of k. In reality however, v Φ and v will be related, since they both contribute to the masses of W and Z, which are known experimentally. Besides the minimization conditions derived above, there are a number of constraints on the vacuum structure, that is, the geometry of the potential. Specifically, these include: i) The squared masses of the scalar particles should be non-negative, or equivalently, the vacuum should be at a local minimum of the potential. 8 ii) The vacuum should be stable (at a global minimum) or at least meta-stable/long-lived. iii) The potential should be bounded from below. Demanding that the above conditions are satisfied will restrict the extent of parameter space, providing for example theoretical bounds on the Higgs masses. Especially conditions ii) and iii) are inherently non-trivial, since the potential lives in a 1 dimensional space (made up out of 5 complex fields). For a recent study of theoretical constraints in the HTM, see [16]. Finally, it is interesting to note that if we require the triplet mass to be heavy, meaning µ v Φ, equation (3.3) reduces to (since also v v Φ to make ρ 1) v k v Φ µ, (3.4) which means that v is guaranteed to be small relative to v Φ without demanding that the dimensionful parameter k is particularly small. This also demonstrates the workings of the type-ii seesaw mechanism, mentioned in section.3. However, for a phenomenological study of the HTM, it might be more interesting to consider lower energy scales where the Higgs particles are light enough to be detected at the LHC. Thus, one could instead look at the case µ v Φ, which, referring again to equation (3.3), allows for λ k = k/v 1. This condition can generate Higgs masses of the order of v Φ, which will become evident in the following section where the mass relations of the Higgs particles are derived. 8 Since the squared masses are equal to the coefficients of the terms quadratic in the fields, they constitute the (diagonalized) Hessian of the potential. Thus, a negative eigenvalue of the Hessian, implies that the potential cannot be at a minimum (but instead at a maximum or saddle point). 15

16 3.4 Mass Matrices and Eigenstates Having found the minimization conditions of the potential, we now shift the fields about their VEV s to obtain the more physically interesting fields, which fluctuate about the origin. That is, we make the change of variables, φ = (φ R + v Φ + i φ I)/, δ = (δ R + v + i δ I )/, (3.5) where δr and δ I are real scalar fields. The charged fields are left unchanged since they already have zero VEV s. Making the above substitutions and also eliminating µ Φ and µ using equations (3.) and (3.3), the mass matrices can be obtained by collecting all terms in V that are quadratic in the fields. That is, V = (Ψ ) T M ++Ψ ++ + (Ψ ) T M +Ψ (Ψ R) T M RΨ R + 1 (Ψ I) T M IΨ I (3.6) + non-quadratic terms, where Ψ ±± = δ ±±, Ψ ± = (φ ± δ ± ) T, Ψ R = ( φ R δ R )T and Ψ I = ( φ I δ I )T, and the convention being used is that ψ (ψ + ) for any charged field ψ +. We then extract the scalar fields with well-defined masses by using a set of bases consisting of mass eigenstates which diagonalize the mass matrices. In this way, one finds seven Higgs particles, which are commonly denoted H ±±, H ±, H, h and A, and three massless Goldstone bosons, G ± and G. This diagonalization procedure reduces the above equation to V = M H ++H H ++ + M H +H H M h h h + 1 M H H H + 1 M A A A + non-quadratic terms. (3.7) The mass matrices together with the corresponding masses and mass eigenstates will now be explicitly listed. The doubly charged fields, δ ±±, are themselves mass eigenstates. In order to comply with the coming notation, these mass eigenstates will be denoted H ±±. Their mass matrix, or simply just mass in this case, is given by M ++ = M H ++ = 1 v Φ (λ k λ 5 + ε λ 3 ). (3.8) The singly charged fields, φ ± and δ ±, have a mass matrix given by ( M + = vφ(λ ε ε/ ) k λ 5 ) ε/, (3.9) 1/ and the mass eigenstates and masses (eigenvalues) arising from this matrix are found to be G ± 1 ε = 1 + ε φ± ε δ±, (3.3) and H ± = ε ε φ± ε δ±, (3.31) M G + =, (3.3) M H + = 1 v Φ(λ k λ 5 )(1 + ε ). (3.33) 16

17 Here, G ± are the Goldstone boson which give mass to the W ± gauge bosons. These will be set to zero when unitary gauge is applied. The mass matrix belonging to φ I and δ I is ( ) M I = vφλ ε ε k. (3.34) ε 1/ This matrix gives rise to yet another massless Goldstone boson, G, responsible for the Z boson mass, as well as a pseudoscalar Higgs field, A, These have masses G = A = 1 ε 1 + 4ε φ I ε δ I, (3.35) ε ε φ I ε δ I. (3.36) M G =, (3.37) M A = 1 v Φλ k (1 + 4ε ). (3.38) A few observations can be made concerning the expressions which have been derived so far. i) Since ρ 1 requires ε 1, we see that H ± δ ± and A δ I. Also, H±± = δ ±±, so there are five triplet-like Higgs fields. ii) These triplet-like Higgs fields, H ±±, H ± and A, all have masses which are scaled relative to v Φ by the parameter λ k. If λ k is, say, of the order of unity, these masses can be comparable to v Φ 46 GeV. iii) For a fixed λ k, and to first order in ε, the mass splitting between H ++, H + and A, is determined by λ 5. The hierarchy ordering is set by the sign of λ 5. This opens up for a possible decay chain among the Higgs fields, where heavier scalars decay into lighter ones. Note that, even though the squared mass splitting is only dependent on λ 5 to first order in ε, the mass differences will get smaller as λ k increases. This leads to a mass degeneracy among the triplet-like fields for large enough λ k, which is illustrated in figure, where all the Higgs masses are plotted as a function of the parameter λ k. The final mass matrix arises from the neutral fields scalar fields, φ R and δ R, and turns out to be ( ) M R = vφ λ 1 (λ 4 + λ 5 λ k )ε 1 (λ 4 + λ 5 λ k )ε λ k + λ ε. (3.39) The mass eigenstates are with corresponding masses Mh = 1 ( m 11 + m M H = 1 h = cos θ φ R + sin θ δ R, (3.4) H = sin θ φ R + cos θ δ R, (3.41) (m 11 m ) + 4(m 1 ) ), (3.4) ( ) m 11 + m + (m 11 m ) + 4(m 1 ), (3.43) 17

18 Masses GeV h H A H H Λ 5.4 Masses GeV h H A H H Λ 5.4 Figure : The Higgs masses as a function of λ k. The left and right figures are set up identically except for the value of λ 5, the sign of which determines the mass hierarchy of the the triplet-like fields A, H + and H ++. To the left, λ 5 is chosen (quite arbitrarily) to.4, while to the right, λ 5 =.4, flipping the mass ordering. The remaining parameters are set to: v = 1 GeV, λ 1 = 1.5, and λ i =.4, i =, 3, 4. where m ij denotes the components of M R. Note that M h mixing angle θ [ π, π ] satisifies the relation M H. In the above equations, the tan θ = m 1 m 11. (3.44) m The sign of θ depends on the sign of m 1. More exactly, m 1 > θ <, and m 1 < θ >. For the special case of m 1 =, the mass matrix is diagonal and hence there is no mixing between the doublet and triplet fields. In this case, if m 11 > m, then M H = m 11 and H = φ R (up to a phase), i.e., a pure doublet field. By inspecting equation (3.41), it is clear that θ = ±π/ for this case. On the other hand, if m 11 < m, then M H = m and H is a pure triplet, implying θ =. For the case m 11 = m and m 1, where equation (3.44) is not defined, there is maximal doublet-triplet mixing, i.e. θ = ±π/4. This flipping of the doublet-triplet composition of h and H can be seen in figure. In the figure, there are three lines following each other these are the triplet-like fields and one line crossing the others this is the doublet dominated field. As can be seen, the line corresponding to A lies extremely close to the line of either h or H, making them indistinguishable in the figure. This tendency to degeneracy is a general feature of the mass-spectrum which arises due to the hierarchy ε 1. At the cross-over point (which is not really a point when zooming in, but a region where the lines of h and H smoothly pass each other), M h M H, and there is maximal doublet-triplet mixing as stated earlier. As λ k grows, H becomes more and more triplet field dominated and its mass is nearly degenerate with M A. Below the cross-over point, the roles are reversed, and instead h gets increasingly triplet-like, with a mass very close to M A. 3.5 Interactions Probability Calculations in Particle Physics In a collision of elementary particles, the probability for a specified initial state to result in a certain final state is measured by the cross section, σ, of the interaction, a quantity measured in units of area (often in terms of barn, b, where 1b = 1 8 m ). The cross section is an intrinsic property 18

19 of the particles and their interaction. However, to say something about the absolute interaction frequency, one must take into account the effect of the luminosity, L, of the accelerator, that is, how many particles are passing each other per unit time and area at the interaction point. The number of interactions during a time t is then given by [17] t N = σ L dt. (3.45) The cross section can be calculated once the Lagrangian, which contains all the information about the interactions in the particle theory, has been specified. Mathematically, the cross section is expressed as a function of (the square of) a transition probability amplitude, a matrix element, which is usually a quite complicated multidimensional integral. The method of handling the calculations is to apply perturbation theory, where the transition amplitude is expanded into an, in general, infinite sum of amplitudes. Each term in this sum can be graphically represented by a Feynman diagram (see e.g. figure 3). More physically, one can consider each Feynman diagram to represent one of all the possible ways for the initial state to evolve into the final state. The common way to handle the mathematical expressions needed for calculating cross sections is thus to add up the Feynman diagrams involved. The mapping from a diagram to an analytical expression is defined by the Feynman rules associated with the Lagrangian. According to the above procedure, one can then go from an interaction, to a sum of diagrams, to a sum of mathematical expressions (integrals) to finally, a cross section Calculation of the Feynman Rules In this work, I calculate the Feynman rules for the HTM using the Mathematica package Feyn- Rules [18], specifically, FeynRules 1.6beta and Mathematica 7./8.. FeynRules comes with a number of translation interfaces, which allow the user to export the Feynman rules to software which can calculate cross sections and perform Monte-Carlo simulations. This makes it a useful tool for doing phenomenology on theories beyond the SM, such as the HTM. From the user end, what has to be done to get started with FeynRules, is to write (or download) a so called model file, which contains the definitions of the particles, parameters (masses, coupling constants, etc.), gauge symmetries and so on, of the model. The Lagrangian also has to be provided, and can be put into the same model file. Since the HTM is basically the addition of an extra triplet and some new interactions to the SM, the process of writing the model file is simplified. The SM model file has been validated and is available for free download at the FeynRules homepage. What is then essentially left to do is to add the new particles, parameters and Lagrangian terms of the HTM. What input should be given to the model file? The free parameters specific to the HTM are all the λ s, v, v Φ and the elements of the triplet Yukawa matrix h. As mentioned in section 3.3, the values of v Φ and v are related to each other and constrained by precision measurements of electroweak SM quantities. Specifically, this fact can be expressed as ( 1 M v Φ = W sin θ W ε v ), (3.46) 1 + ε e v Φ where e is the absolute value of the electric charge. Thus, v Φ and v cannot vary independently of each other if e, M W and θ W are chosen to be input parameters. By the form of the equations involved, I find it convenient to use the quantity ε as the free parameter in the calculations, in favor of v Φ or v. Of course, ε is only free in the sense that we may allow it to vary in an interval given by the uncertainties in determining ρ. Furthermore, it might be more natural to give input in terms of masses and mixing angles, which are more closely related to observations than the λ s are. This also makes it trivial to guarantee that the masses are non-negative. Accordingly, the final input parameters to the model are chosen to be: M H +, M H ++, M A, M H, M h, ε, θ and h. To realize this, the mass relations, equations (3.8), (3.33), (3.38), (3.4), (3.43), and the relation 19

20 for θ, equation (3.44), are inverted, yielding λ 1,..., λ 5, λ k as functions of the Higgs masses, θ and ε. The result is as follows, MA λ k = 1 + 4ε vφ, (3.47) MH λ 5 = λ k 1 + ε vφ, (3.48) λ 3 = ε λ 5 1 ε λ k + MH ++ ε, (3.49) v Φ λ = 1 [ ( M 4ε cos θ H M ) ( h M vφ + H + Mh ) ] vφ λ k, (3.5) ( M H + Mh ), (3.51) λ 1 = ε λ 1 4 λ k + 1 λ 4 = λ k λ 5 tan θ ε v Φ (ε λ λ k λ 1 ). (3.5) The λ s are written in the order they would have to be calculated without writing them out explicitly as functions of the Higgs masses and θ. Obviously, these explicit functions can straightforwardly be obtained by performing the substitutions, starting from the top. The HTM parts of the model file are attached in appendix A. These are the lines of code that need to be added to (or modified in) an SM model file in order to get the full HTM model file. The file is written in Mathematica language with some additional functions provided by the FeynRules package. It is seen to be fairly pedagogically organized and the contents of the different sections should be more or less self-explanatory. The parameters which have the property ParameterType -> External are the free parameters of the model, determined by user input. The Higgs masses however, are given as external parameters in the particle section. The parameters labeled Internal are calculated using the external ones and should be placed in order of dependence. Also, only unitary gauge is being considered so including the Goldstone bosons G and G ± is not strictly necessary. The Feynman rules for the vertices which are particularly interesting from a phenomenological point of view the gauge-higgs three-vertices, fermion-higgs vertices, and the Higgs-Higgs threevertices are listed in appendix B. These rules seem to agree with those listed in e.g. [11, 16]. However, when comparing them to the ones of [15], there are discrepancies. For instance, [15] lists the vertex AW H +, which I am not able to reproduce. It should also be noted that the mass matrices derived in the previous section do not agree with those of [15]. Hence, there appears to be errors present in their calculations, as has previously been pointed out by [19]. 3.6 Production and Decay of the Doubly Charged Higgs at the LHC The existence of doubly charged Higgs bosons, H ±±, has the potential of giving rise to especially clear detection signatures. The production and decay mechanisms of H ±± which are relevant for hadron colliders such as the LHC are therefore summarized below H ±± Production Some of the H ±± production channels which are potentially of importance at the LHC are: i) Pair-production: q q γ, Z H ++ H, ii) Associative production with H ± : q 1 q W ± H ±± H, iii) Associative production with W ± : q 1 q W ± H ±± W,