SPECIF'IC DECAY WIDTHS IN THE LEFT-RIGHT SUPERSYMMETRIC MODEL

Stavros Antonopoulos

A Project in

The Department of

P hysics

Presented in Partid Fulfillment of the Requirements for the Degree of Master of Science at

Concordia University Montréal, Québec, Canada

March 1997

@ Stavros Antonopoulos, 1997

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Abstract

We review bridy the Standard Model (SM), Left-Right Model (GR) and Minimal

Supersymmetric Standard Model (MSSM). We study their specific symmetry breaking

mechanisms and partide contents. Findy, we caldate specific interaction vertices of

doubly charged Higgsinos in the Left-Right Supersymmetric mode1 (GR SUSY) and

their decay widths. A possible edension of this work is proposed.

The author would Iüe to express his gratitude to his supenisor Dr. M. Rank for

suggesting the topic, her guidane and patience. Specid thanks to my f d y and my

uncle for always supporting me during d i f f id t periods. Finally, to aU those pwple fKmi

the Physics Department and elsewhere who advised and guided m e in many occasions.

Without all the abave this work wodd not have been wmpleted.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Lagragian Formulation 2

. . . . . . . . . . . . . . . . . . . 1.2 GWS mode1 and symmetry breaking 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Higpp mechanism 7

. . . . . . . . . . . . . . . . . . . . . . . . 1.4 Beyond the Standard Mode1 12

2 LefbRight Symmetry 14

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Description of the mode1 15

3 Supersymmetry 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 18

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Supersymmetric Algebra 21

. . . . . . . . . . . . . . . . . . . . . . . 3.3 Particle content and notation 23

. . . . . . . . . . . . . . . . . . . . 3.4 Structure of the MSSM Lagrangian 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Chargino Mixing 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Introduction 27

. . . . . . . . . . . . . . . . . . . . . 3.5.2 The Chargino Lagrangian 27

3.5.3 Feynmanrules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Neutdino Mixing 33

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Analyticrrsults 35

4 Left-Right Supersymmetry 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 SymmetryBreaking 42

. . . . . . . . . . . . . . . . . . . . . 4.2 Mass eigaistates of Vector Bosons' 43

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Gaugino-Higgsino Mmng 47

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Chugino Mixing 47

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Neutralino mixing 50

5 Chargino Decay Widths 52

. . . . . . . . . . . . . . . . . . . . . . 5.1 Chugindepton-slepton vertifes 52

. . . . . . . . . . . . . . . . . 5.2 Chsrginenmtralino-Higgs bason vertices 55

. . . . . . . . . . . . . . . . 5.3 Chargineneut raiin01V'tot boson vertices 60

5.4 DecayWidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion 67

A Spinor notation and conventions 68

B Decay width sample caiculation 73

List of Tables

. . . . . . . . . . . . . . . . . . 3.1 Most common supersymmetric partners 24

3.2 Ordering of positive mass eigenvalues of charginos and neutralinos in

three possible p regions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Matter and Higgs field content of the GR mode1 . . . . . . . . . . . . . 41

5.1 Total decay widths for chargino decays into lepton. s-lepton and H i w . 65

5.2 Totd decay widthr for chargino and neutralino decays into Vector bosons . 66

vii

List of Figures

3.1 Feynman rules for some characteristic chargin0 intezutions in the MSSM. 32

5.1 Chatginelepton-slepton vertices. . . . . . . . . . . . . . . . . . . . . . 54

5.2 Charginc+Neutralino-Higgs boson vertices in the LR-ÇUSY model. . . 58

5.3 C h a r g i n ~ N e u t n i s boson vertices in the LR-SUSY model (cont'd). 59

5.4 Charginc+NeutralineVator boson vertices in the GR-SUSY model. . . 62

5.5 C h a r g i n ~ N e t n V t o boson vertices in the L-R-SUSY model (cont'd). 63

B.1 Decay diagram with momenta and vertex factor. . . . . . . . . . . . . . 73

viii

Chapter 1

Introduction

Ever since the time of Democritus, and perhaps earlier, people have tried to understand

the basic building blocs of matter. Things have changed dramatically with the discovexy

of Quantum Mechanies and Special Relativity at the beginning of the century.

Later on, Dirac combined both theories to derive his wave equation for the electron

which can be considered the basis for a new era dominated by Quantum Electrody-

namics (QED) and its extension, Quantum Field Theory (QFT).

We have also gone a long way trying to understand the forces acting on the par-

ticles. The k t step, arising fiom the experiments of Oërsted and Faraday, were

MaxweU's famous equations which put under the same roof, so to speak, electricity

and magnetism. The main characteristic of these equations is that they satisfy the

secalled Lorentz space-time transformations. Historically this waa the main reapon

that led Einstein to formulate specid relativity, which ever since has been the cor-

nerstone of al1 theories that attempt to describe any aspect of the physical world. In

the 1960's Glashow, Weinberg and Salam published a series of papers [l, 2, 31 which

successfdly unified electromagnetism with the weak force (electmweak) and meiveci

the Nobel prize for their work. The strong force is included with electrowd in an

1

SCI(3)c x SU(2)L x U(l)v mode1 but there still remain a lot of things to learn about

strong interactio~~since most of our results are based on approximations. The above

gauge theory is the scwded Standard Mode1 (SM). We will clarify some of the termi-

nology later on.

The SM is undoubtedly a very successfd theory. It has b e n able to accurately

explain almost all experimental results, especidy in the fermionic sector. For example

the calculations of the theoretical value of the anomalous magnetic moment of the

electron agrees with the experimentally obtained result to one part in log [4]! The

predictive power of the SM has dm been provecl, the gauge bosons being the best

example, their existence (and their masses) wete postulated as an integral part of the

t heory in the eady 1970's almost 10 years before their experimental detection at CERN

in 1983.

Despite its phenomenological successes the SM has theoretid problems and this is

the main reason why theorists believe that it is merely a low energy limit of a larger

unified theory. The SM contains at least 19 arbitrary paameters so it is quite far h m

every theoretician's dream: a single unified thwry with at most one undetermined

coupling constant.

1.1 Lagrangian Formulation

At present the most widely used field theory formulation is the so called Lsgtangian

which, as is classical physics, is based on the principle of least action [5, 6). We make

the transition fiom a point particle to a field through the replacement : z(t) + #(xp)

and proceed to find the equation of motion by minimhing the action:

L is called the Lagrangian density but we usudy refer to it as the Lagrangian. The

minimization of the action is expressed mathematically through the farniliar Euler-

Lagrange equations:

As an illustrative example let us apply the above ideas to the Lagrangian:

We then have:

and replacing in eq. (1.1) we get:

Therefore, starting with eq. (1.2) we recovered the Klein-Gordon equation. In a

gives Dirac's equation (where we treat Jt and 3 independently) and

gives Maxwell's quations with a source j?

Without formally deriving this assertion we can state the two basic d e s to follow

in order to make the connection between the t m s in the Lagrangian and the Feynman

d e s of perturbation theory [7]:

1. The terms in the Lagrangian that are quadratic in the fields,

e-p.: c$~, &b, $(i7,,W - m)$, . . . , determine the relevant

propagatorsl.

2. The other terms Qve the interaction vertices. In particular after taking away

the fields from a specific term of iL: what is le& is the interaction vertex. For

example, if t = - + ie&y'$~, the vertex factor for the interacting fields

is: iey*.

By carefdly inspecting any Lagangian we notice that it is invariant under a ph-

transformation:

d(2) + ek+(z)' a r 'R

since, &b [i eiQ&$ and 4 + cg"$.

If we extend this invariance requirement to any Lagrangian we can show that it

implies the existenceof a wnserved c u m t and in particular a wnserved total charge Q.

Incidentally this is an example of Noether's theorem which states that invariance under

translations, time displacements, rotations etc. leads to conservation of momentum,

energy and angular momentum respectively. The above invariance is called global gauge

invan'crnc$.

'For example, taking the inverse and multiplying by -i of the krm (ir,W - m)JI = (#- m)$ gives i/W- m) t iw+ m ) / v - m2) wbich is one of the k n o m fomu of an s = ) field pmpagabr.

'1t is actudy a ph- invariance, but the word gauge bas ban artablished due to hisforicd musons.

It wodd be interesting if we could generalize the above taasformation (1.3) so that

the parameter a is a function of every space-time point, i.e. a E a(z). The problem

arises with the transformation of the derivative: $$ + eh(=)&$ + ie ia(* )~&a. We

can clearly see that the last tenn breaks the invariance of C . W e could remedy this

situation with the construction of r covaràant derivative D, which has an added vector

field term -ieA, : D, s 4 - ieA,. We further demand that A, transforms as follows:

where the last term of eq. (1.4) will cancel the unwanted term of the derivative. There-

fore the new Lagrangian containing the covariant derivative is invariant under the U(1)

phase transformation ( 1.3), and takes the form:

Further on, regarding A, as the photon field we can add a (gauge invariant) term

corresponding to its kinetic eneqgc - f FJ'"F'. Therefore the final form of the QED

Lagrangian, which describes the interactions of electrons wit h photons, is3:

This is indeed a remarkable result since the simple requirement of a U(1) gauge in-

variance led us to an interding-field Lagrangian. The most general of the

U(1) transformation is: eia(~)Q, where Q is the charge operator with eigendues:

Qc = -1,Q, = +3, etc., and this gives rise to the

netic curent: jy = 47r r~$ i which applies to all

=Note that, aa wpected, the photon fidd is masdese s ina would break the gauge invariance

genetal form of the elechomag-

charged motter particles. The

the addition of a mass term -. m2A:

electromagnetic eharacter of the above group is emphasid by using the subscript em:

W l ) c m -

1.2 GWS modeland

In 1956, in an attempt to put some

syrnmetry breaking

order in a vast collection of experimental obser-

vations and results, Lee and Yang [SI p r o p d that weak interactions violate pa+it#.

Soon after, their work was confirmed independently by Wu [9] and Garwin [IO] in a

series of experiments.

Parity violation implies the non-existence of 'right-handed' neutrinos. With this in

mind we try t o find internal symmetries of the Lagrangian that apply between particles

that have the same space-time properties. The first step is to arrange the leptons in

left and right-handed families as follows, the left-handed fields:

and the right-handed ones:

e ~ , PR, T ~ *

Then, in a similar to the previous section analysis we have the following groups of

SU(2)& : $' + ei qr)T~,, & + $R where T E fo and JILIR can be any member of the above familiesS. Next, we in-

troduce the folIowing vector fields: W W = (W1, w, W) and BC, that couple to the

'Parity (leftright symmetry) wau aiways thought to be one of the collbcfved symmetriee of nature. =The reader shodd be wamai that in the literstute th- are smail variations in the cafncie~ts of

the exponents. Hem we dopt the conventions of ref. [7)-

corresponding cwrents in a similar way that A' couples to the electromagnetic current:

The generators of the two groups satisfy the GeU-Mann-Nishijima relation:

which translated into the current form implies: j,O" = Jz + ijz. Later we will show

in a more transparent way how the electromagnetic current is embedded in the two

currents. The new covariant derivatives needed due to the introduction of the four

massless fields W H , Bfi can be shown to be:

J ~ R : D" = (4'-d544)7

: D , = ( i & - f ~ w , - g ' $ B , ) .

Therefore the new Lagrangian is:

where :

w, = a,w, - auw, - gw, x w,.

1.3 The Higgs mechanism

The Higgs mechanism or spontaneous symmetry breaking mechanism 'gives' masses to

the massless fields introduced above without breaking gauge invariance [4, 6, 71. This

idea is not new to physics. For example, a ferromagnetic crystal extended to infinity

7

is described by a rotationaiy invariant Lagrangian. In the &round state though, dl

spins allign in a particular (=dom) direction thenfore breaking the original rotational

symmetry. In the electroweak SCI(2)L x U(l)y model the spontaneous breaking of

symmetry is induced by a wmplex isospin doublet, the so d e d Higgs field:

w i t h Q = 1 , 0 , T 3 = f f a n d Y = 1 .

The next step is to add the appropriate kinetic and potmtid terms in the La-

where :

W e choose the v0cuu.m expectation d u e (vev) of the Higgs field to be :

This is the most nahval and economicéd choice since we want the vacuum (4') to be

invariant under U(1 j, transformations and thus the photon to remain massless. For

this particular choice, since Tt = -) and Y = 1, replacing in eq. (1.7) we get the

following:

therefore the tansformed field remaios the same as the original.

Before going furthet, it is a gwd idea to gather dl the fields mentioned so far and

write down the full Lagrangian of the Standard Mode1 :

L a t = d c ~ ' ( i 4 - + W p - b f ~ ~ ) d r

+Giry(ia - bpr)tlR

-twwwCy - i w 4 B B,

+ I(ia - +w, - dpPw( ' - ~ ( 4 )

-(Gi$çd+n + Gdr4ctl>n + h-ce),

where 1 l a = ( )+( ). The first two terms represent quark and lepton kinetic energies

(depending on the contents of the qb fields) and their interactions with the gauge bosons.

The third term is a58ociated with the gauge boson kinetic energies and self interactions

while the fourth represents the masses and couplings of the gauge bosons to the Hi-

field. Finally, the last term is similar to the previous one since it is related to the fermion

masses and their couplings to Higgs. Gi and Ga are coupling constants similar to g

and g', whose values are not predicted by the model. Later we will see the importance!

of these terms in determinhg the various masses. Let us now substitute the vev of the

Higgs field 6 into the relevant term of C,:

then eq. (1.12) becorna:

Notice that by diagonalizing the mixing matrix we get two new neutral fields. There-

fore, after symmetry bnaking, we end up with four fields, three messive and one

massless: Wz, W;, 2'' and A'. Assuming normalization of the original fields we have

to normalize the new ones:

Comparing the mass terms in the above part of the Lagrangian with the ones

expected for charged and neutral bosons we have :

1 Mwt = iug,

Ma = +dsl+E',

MA = 0.

And we define the Weinberg ongle through the relation :

It is interesting to note that the resdt MA = O is not a prediction of the model

since the original Lagrandan had a massless photon. The GWS model though fixes

the parameter p r Mg/ M ~ C O S ~ ~ ~ , which specifies the relative strength of the neutrd

and charged eumats in weak interactions to be qua1 to unity. This result has been

confirmecl by experiment (within a small error).

The above mechanism gives masses to the gauge b o e s ; let us now see how fermions

acquire mass in the SM. It can be shown that terms like -m& or iM2B'B, are not

gauge invariant, we can however use t h e same Higgs doublet as before and the last

term of eq. (1.11) in order to generate quark and lepton masses. In particdar, in order

to get a mass term for the electron field we have:

. Ce = -Gee-L&R + h.c.

To break the symmetry we use the Higgs doublet with the following choice of its

vacuum expectation value:

where h(x) is the neutral Higgs field and we have gauged away the three other fields6.

Replacing in Ce we get :

or details sce Ch. 14 of rcf. [7].

11

Incidentally, this is one example of the free parameters that have to be put 'by haad'

in the SM. We know, for example, tbat the maps krm is of the form: -TT$+, which

means that Gev/2 = m.. T h d o t e we have to experimentally meanire m. in order to

determine the value of Ge. Notice that the coupling of the second temi is - m./v cz

10-~ so i t is quite s m d to produce a detectable signal and thedore evidence for the

Higgs boson.

We proceed in a similar way to get the quark masses; the only difference is that we

have to have am SU(2) transformed Higgs doublet in order to give mass to the upper

member of the quark doublet:

Note that the Lagrangian can be written in diagonal form in order to include Linear

combinations of flavor eigenstates.

As a final remark, we should emphasize that the Higgs sector is the least investigated

part of the SM mainly because, a9 we saw above, the Higgs couplings are proportional

to the fermionic masses and the ody 'easilyn produced famions ate the light ones:

e-, u, etc.

1.4 Beyond the Standard Model

As we mentioned in the introduction, the SM cannot be the end of the story s i n e

it builds on many assumptions and leaves many fundamental questions unanswered.

W e do not attempt hem to give a detailed list of alI the drawbacks that the SM has

but merely overview some of them to justify the search for new models equipped with

solutions to some (or poasibly aU) of its problems.

One of the most obscure aspects of the SM is the Higgs sector. As we saw in the

previous section, the choice and the vacuum expectation value of the Higgs boson is

to tdy ad hoc, its mass diverges quadrat idy if one includes hjgher order corrections,

and h d y there is not even a trace of a possible experimental detection that would

support its existence. Secondly, the SM is based on the three gauge groups that we

mentioned before which are basically postulated on a purely phenomenological basis.

In order for a thwry to be complete it would have to provide an understanding of the

origin of these groups along with the values and wrrelations of its coupling constants.

Associated with this is the (arbitrary) choice of assigning left-handed fermions to dou-

blets and right-handed ones to singlets which is done to fit experimental observations

such as beta decay, the value of electric charge, etc. The above is related to the e

called genenrtion pmblem which basically asks the fundamental questions of why there

are only three (?) families and why their masses and mixings follow a hiexarchical pat-

tern. induding to the above the poasibili ty of a non-zero neutrino mass, CP violation,

quantization of electric charge, etc. then a new mode1 or at least an extendeci one

becomes a neccessity.

Finally, if we are to believe that a unified theory exists then we have to look for ways

to incorporate gravity into the SM or look for a t o t d y different thaoiy. In wntrast to

all other interactions, gravity is totally outside t h e framework of gauge theories since

General Relativity does not follow fiom the principle of local gauge invariance and at

the same time General Relativity incorporates, wntrary to all other interactions, the

principle of equivalence?

7For din&rent modcb th& are sugga~td as a rernedy for the various p m b l e ~ ~ , together with th& own drawbacks, the infvested d e r is tefermi ta Table 1 on rd. [Il].

Chapter 2

Left-Right Symmetry

According to the standard model ail interactions except the weak, respect ail spacetime

symmetries. i t w o d d appear thexdore naturai to try and extend the SM to make it

left-right symmetric. But symmetry, although intuitive, is not the only concern. There

axe three main reasons [Il, 12,131 for trying to incorporate right handed bidoublets in

order to extend the SM in the SU(2)L x x U(1) group; the so-called Left-Right

Mode1 (GR).

The h t reason is trying to understand parity violation, one approach would be to

assume that nature is intrinsically left-right symmetric and the observed asymmetry

takes place after some breakdown at low energies, that is the vacuum is nominvariant

under parity. This is why left-right symmetries involve some sort of parity breakdown

mechanism that takes e&ct at some energy scale Mp.

Another reason, closely associateci with the first, is the neutrino mass which has

initiated many controversial experiments [14] and debates in the last decade. As tm

physical problems involving 'the missing m a s of the universe, galaxy formation, etc.,

would be easily resolved if the neutrino indeed has a mass in the eV rmge [15]. If

m, # O then the l&-right symmetric model is the m a t natural framework to incorp*

rate it. It ha9 been shown [16] that such a particle in a left-right symmetric model is

a Majorana' particle with mass:

Lastly, d h e r nason is the lack of any physical interpretation of the U(l)v symme

try [ I f ] , mainly due to the multiplicity of values of Y in the SM: Y (uL) = - 1, Y (ez) =

-2, Y(uL) = 5, etc. In a Idt-nght mode1 U ( l ) becomes the B-L generator which is the

only anorndy-free quantum number left ungauged. With this inclusion the weak gauge

group becomes: SU(2)L x x U(l)B-L and a similar to the Gell-Mann-Nishijima

formula holds: B-L

Q = & + I ~ + - Then VB-&) = -1 for all leptons and +i for quarks of al l generations and helicities.

2.1 Description of the model

In defining the components of our model, we follow the same path as in the SM with

the addition, of course, of right handed isospinors. For one generation we have :

with quantum numbers

'A Mojomna particle ia identid ta its own antiparticle [7j.

15

The third number in parenthesis is (B - L) and, as we saw above, corresponds to the

U(1) generator.

The fermion-gauge h interaction part of the Lagrangian is given by :

where in the last term of eq. (2.3) we imply both left and right contributions. Similarly,

the fermion kinetic energy terms are:

Notice that if we want the Lagragian to be invariant under a left-right interchange,

it immediately follows that: g~ = g~ G g.

One of the possible minimal Higgs sets required to break the symmetry 'down to'

U(1), is the foilowing: two triplets

wit h quantum numbers (1,0,2) and (0,1,2) respect ively, and one bi-doubiet

with (1/2,1/2,0). Using the above fields we can break the symmetry in the following

MW 2 W)--

With our particular choie of Higgs multiplets the parity P and SU(2) syrnmete can

be broken at the same energy scale: Mp = MW,.

The vev's of the Higgs fields axe chosen to be:

and

where eia is a CP violating phase.

With this background information we c m go on and derive the charged and neutrd

gauge boson masses in a similar way as we did for the SM. Also, certain arguments [Il]

can show that is < vn and v t *: K which is essential in understanding why certain

physical parameters like, for example, the mass of the neutrino are s m d compared to

others. - Next, we wiU wnsider the Supersymmetric model (SUSY) where we will apply

many of the ideas introduced here to specific calculations. It should be understood

that the left-right mcdel introduced here is a mmplete model of its own, but in our

case it will be used as a basis for a more extensive model based on Supmymmetry,

the secalled L e W i g h t Supersymmetric model (GR S USY).

Chapter 3

Supersymmetry

3.1 Introduction

We will attempt heteby to introduce the basic concepts of supersymmetry and try to

explain and andyze some of its predictions. This short introduction can by no means

be considered either an extensive or in depth approach to the subject. It is based

maidy on teferences [18, 19,201 and the reader is enmuragecl to consult them.

According to our presuit understanding of subatomic physics all particles have

fermiois as their basic constituents (quarks) or they themselves are considered elemen-

tary (leptons). On the other hand all known forces are mediated by bosonic carriers

(7, Wh, ZO, g).' Therefore, if we ever hope to unie particles and interactions into a sin-

gle self-consistent thwry we will probably have to somehow unie bosons and fermions.

The basic idea of supesymmetry is that it transforms bosons into fermions and vice

versa; we will later discuss the baie features of supersymmetry generators and some

of the impkations of their dgebra

Supersymmetry (SUSY) predicts the existence of a supctportner for all known par-

'Gravitons have not yet ban obarvai, poseibly due to th& weak strength, but it ia believed that they exkt and have rrph 2.

ticles as we wiil show below. The total angular momentum 6) of the superpartnet

differs by half a unit fimm the original field. Thedore the superpartners of bosons

are fermions and vice versa AI1 the other quantum numbew are the same as those of

the original partner (color, lepton number, etc.). SUSY too, inwrporates a symmetry

breaking mechanism, similar to the SM and L R Model, which breaks the mms degen-

eracy between supersymmetric partnets. After symmetry breaking the fields are mass

eigenstates and ne can further p r d with possible generation mixings (super CKM

matrix) and even supersymmetric particle mixing between scalar leptons, quarks, etc.

One of the most serious problems of the SM which is ideriteci in the GR mode1

is the ecalled gauge hierordiy pmblem (GHP). It rnanifests itself when we try to

calculate the masses of the Higgs particles. By doing SO, using perturbation series

beyond tree level, we get quadratic divergences which would push the masses to the

order of Plank s d e (Mp - l O l S GeV) unless the perturbation terms cancel to 26

decimal places(!) [13, 311. Supersymmetric models on the other hand do not have

such a problem since, as we saw above, for every fermionic(bosonic) loop there is

a corresponding bosonic(fefmionic) with the same mass and coupling strength but

opposite sign. Therefore, as shown in eq. (3.1) below, they canc J each other out2:

AU particles in a supersymmetric theory obey a continuous global symmetry called

'In ~~~ apenymmetry, due to nonquaüty of the rn-, the didivgeneas do appcar but are not quadratic.

R-symmetry. Certain conditions can break the continuity but nev&heless a d i s m t e

R-symmetry almat alnrays appliea which in tum gives rim b a conssved quantum

number callecl R-prit% defhed to be:

All 'ordinary ' particles have R = +1 and ail supersymmetric partners have R = - 1. R is a multiplicative quantum number which, combinecl with the above observation,

gives rise to two fimdamental consaquenas on any supersymmetric model:

Firstly, all supersymmetric particles must be produced in pairs since at any present

experiment al situation incoming states consist only of ordinary particles, therefore the

final state has to contain an even number of supersymmetric particles.

Secondly, there must b& a lightcst and stable SUSY particle since any deeaying

super-particle would no t be able to decay only to non-supersymmet ric particles due to

R-parity conservation3.

It is remarkable that supersymmetry has occupied so many theorists over the years

without a hint of experimental evidence. The two main reasons for this is the potential

resolution of the gauge hierarchy problem and a possible inclusion of gravity through

local supersymmetry. But there is an additional reason for hoping that one day we will

detect some sign of supersymmetry: according to out models almost al1 supersymmehic

partners, after symmetry breaking, acquin masses which are, in generd, much heavier

than ordinary particles.

=The photino (y), wbicb is the photon's mperpartner, is one of t$e potmible candidates.

3.2 Supersymmetric Algebra

The generator(s) of supemymmetric transformations interchange fermions into bosons

and vice versa, acwrding to the defining relations [4, 191:

Q Ifermion) = Iboson)

Q lboson) = Ifermion)

Applying the unitary operator LI, which represents a 2rr rotation, on the fist relation

above we have:

UQ Ifennion) = U Iboson) = UQU-'U Ifennion) .

But we know that for fermionic statës:

U [fermion) = - [fermion) ,

t herefore UQU-lU lfermion) = U [boson) = Iboson)

= Q [fermion) = -QU I f d o n )

+ UQUo1 = -Q.

And similarl y for the second equation.

W e see that a 27r rotation of Q is accompanied by a change in sign; it a n be

shown that for any Lorentz transformation the behavior of the Q's is that of a spinor

operator. On the other hand, it can also be shown that Q operators are invariant under

translations which implies that:

[Q,E]=[Q,PI=O (3-3)

and the antioommutator of two symmetry generators is also a symmetry geaerator. Let

us for exampk consider the eigenvdues of {Q, Qt) applied to any state 1s):

(4 CO, 9') 1s)

= ($1 QQ+ 1s) + (4 8 '0 1s) (3-4)

= 19' Is)12 + 10 1412 2 0

and since 1s) repfesents any state, the above H d t i a n operator will have positive

definite eigenvalues, unless Q O.

The anticommutator of Q and Qt is a linear combination of the energy and me

mentum operators:

which suggests the manection of supersymmetry with space-time trandormations and

therefore supergravity. We can dso show that by summing over all generators of our

mode1 the momentum terms cancel and we are Ieft with one term proportional to the

energy :

For a sensible theory, we want our energy eigenvalues to be bounded from above

and therefore (3.4) implies that: 7 > O. As usual, we denote by 10) the state with the

lowest eigenvalue (vacuum). Awording to eq. (3.4), if Q (O) = Qt 10) = O then E (O) = O

and vice versa. This means that any state Is), except the vacuum, will have at least

one 'superpartnet' state: Q 1s) or Qt 1s) with a spin number that diff' by $ from

the original state and they are all members of the same supermultiplet. Combining

this result with eq. (3.3) ne see that supersymmetric partnes must have the same

mass. We know that this resuit is not supported by experiments, which leads us to the

conclusion that if supersymmetry is to be a reaLizoble theory we must provide i t with

22

a symmetry breaking m d m i s m , which although lifts the mass degeneracy it does not

affect the structure of the multiplets . IiiRaent models have different numbers of generators (N) but all of the Q's have

four components since, as we showed above, they are spinor operators. T h d o r e ,

N = 1 supersymmetry is the simplest example of such a theory and it is d e d the

Minimal Supersymrnetnc Standard Mode1 (MSSM). Other candidates are N = 4 and

N = 8 (supergravity) which have advantages and disadmtages: finiteness, particle

spectra etc. For example, for theories wit h N > 1 we would get an expression analogous

to (3.5):

Zn this work we are exclusively working with N = 1 supersymmetry.

3.3 Particle content and notation

Supersymmetric partners of ordinary particles are denoted by an overhead (-). Often,

weak eigenstates will mix in a similar to the SM fashion, giving mass eigenstates which

in Table 1 we put on a separate column. Fermionic superpartners of bosons are named

with a suflix -in0 and bomnic superpartners of fermions are named with a prefix s d a r

(or simply s-). Table 1 lists the m a t wrnmon superpartners and their names4.

A word of caution is appropriate here: the following table is only rrpnsentafiue of

the particle content in the MSSM. The reader should be warned of the existence of

different notations and multiple mixing mechanisms (an example of which we s h d see

in a following seetion) which give rise to different physical puticles (mas eigenstates).

'We wiil not indude maan eîgenstates with sp&c coupürigs but the reader should be aware of their exiskna.

Or- particles Supersymmetric partners

9 9 gluino 8 gluino

W* W* wino

@ higgsino 2& charginos

K higgsino

7 9 phot ino ZO ZO zino 2 neutralinos @ - @ himino Ha I f a higgsino

Table 3.1: Most cornmon supersymmetric partners.

Incidentally, assuming the photino (7) is the iightest SUSY-particle, experimentd

searches look for photino production which would show up as substantial fractions of

the total missing energy and momentum since, unlike neutrinos, its production is not

associated with any lepton. (e.6 for the reaction: l* + If 7 in the P r a t frame and

neglecting the mass of the lepton, haif the en- of the event escapes).

3.4 Structure of the MSSM Lagrangian

Below we give the Lagragian tems n&ded for an SU@) x U(1) mode1 of bmken

supersymmetry [18,21]. Supetsymmetric Yang-Mills theories contain two basic types

of fields: gauge mdtipleb (Aa, 5) in the adjoint representation of a gauge group G

and motter mdtiplcts (Ai, d>i) in some reptesentation R of G. The Ai's are complu

scalar fields, the Vc's spin4 4 vector fields and h d y A and 9 are twecomponent

fermions (see Appendix A). Employing the sumrnation convention throughout we have

the following four main terms:

1. Kinetic energy terms

2. Self interactions of gauge multiplets simila

three and four-gaugeboson vertices [7] :

x to the SM giving the f d a r

where we borrowed the notation from the SM non-Abelian field strength tensor:

F' 4V; - &Vz - fa&,v'. And the gaugino-gauge field interactions:

a r -6 Cg, = ig jakA O A (3.7)

where faa are the structure constants of G.

We should note that there is no interaction between the U(1) singlets A' and v:, therefore in this particular case we set fak = 0.

3. Gauge and matter multiplets interaction term which is important for this

work:

where Ta is the Hermitian group generator

that for U(1) multiplets we have to pedorm

v: 4 $ 9

in the representation R of G. Note

the following replacements:

gT. =t + where K is the U(1) quantum number of the matta field (hypercharge).

4. Finally, the fermion mpru te- and Yukawa interactions are given by:

The ordinary smhr potential

Where the superpotential W

define:

Another important piece needed

is some cubic fundion of the scalar fields and we

is the sop-Jupersymmetry bnokàng terms which

do not introduce quadratic divergences to the unrenormalized theory. The possible

explicit terms are [22]:

With A2 and A3 being (gauge invariant) quadratic and cubic combinations of the scalar

fieldss. The terms and are Majorana mars terms for the gauginos, corresponding

to the groups G and U(1) respectively. W e shall see the importance of the last two

terms in (3.13) later in our discussion of neutraiinos.

5Nok that Q[ ] and Sf an the d a d haginary parts respectively.

3.5 Chargino Mwng

3.5.1 Introduction

Before showing explicitly the Lagrangian tenns responsible for the &ng of chargeci

gauginos and hiwinos, it is appropriate to Say a feu words about the model:

AS we mentioned buore!, we deal with possibly the simplest model of broken SU(2) x

U(1) supersymmetry with the minimal Higgs set required, namely two Higgs doublets

Hl and H2 with vev's:

where V I # va.

In order to obtain the Feynman d e s we list the appropriate inteaction terms and

then shift the scalar fields: Hi -+ Hi +- (Hi), i = 1,2 (unitary gauge). Findy, we

switch from two to four-wmponent notation and we use the definitions of the mass

eigenstates (physical particles).

3.5.2 The Chargino Lagrangian

The Lagrangian term for chargino mixing is the follow ing:

+ M X + X - - && $& +.h.c.. (3.15)

The first texm in eq. (3.15) cornes h m (3.8) keeping only the a = 1,2 contributions

and taking 4 2 as the SU(2) group generator for the representation R, where u are

the Pauli matricea. We aiso d the relation: A* = &(A1 i X 2 ) .

The second term cornes h m the sofbSUSY breaking part of the Lagrangian (3.13)

where again we kept only the a = 1,2 contributions and dehed M 2&. The lrst

term cornes from an additional term in the superpotential of the form: p ~ j ~ i ~ i . Let us now try and iewrik the above part in a more compact form, with the aid of

the following definitions:

$$ z (-iA+, $ha), $: z ( - iA0,

m& I f#(u: + v;).

Further, we can easily verify that:

Where:

W e then define the mass eigenstates as:

where Lr and V are unitaw and obey the following relation:

MD being diagonal with nonnegative entries: a, M-, where by convention we choose

M+ > M-. Ftom the above defmition we obtain a m a s term form:

Lc = -I 2 [$-x$+= + t l > + ~ ~ + ~ T ] + h . ~ .

Now, due to eq. (3.20) and the unitarity condition we have:

and so eq. (3.21) becornes:

L, = -k 2 [~-([I.)-IM,~+~ + ++(va)-1 M ~ ~ - ~ ] + h.=.

- - -1 t [+-PM.~+~ + + + v ~ M , ~ - T ] + h . ~

= -5 [X-M,~ +T + X + ~ D X - T ] + h.e.

= - [ x ~ ( ~ D ) q x : ] + h.c.

where we used the f h t relation of (A.12).

If we now define the four-component Dirac spinors:

and use eq. (A.13) we can write (3.23) as foliows:

From (3.20) we uui easily derive that:

which, by computing the eigenvectors of XtX and XXt, can give us the elements of

V and U' respectively.

We list here some of the formdaa given in refis. [la, 231 for the resulting d u e s

(under the assumption that M and p are r d ) , the interested reader can find a mon

29

involved treatment in the references above.

1 = - { M ~ + + 2rnL & [(M~ - + 4m', cos2 28,

2

+ (A@ + p2 + 2 M p sin 2&)] ') . (3.27)

FUher, parametrizing U and V with respect to a parameter 4 we get U = 0- and

V = O+ (or V = 030+ if ddX < O =), where:

sin 4* -sin 4* cos&

The above formulas attack the general case. In order to get explicit and simple expres

sions for the charginos it would be instructive to consider the special case: p = O and

v, = u2. With this choice 4 = #+, wllectively denoted as 4 and

The defining relations (3.19)- and (3.24) will give:

3.5.3 Feynman iules

W e will now try to show the connection between the relevant Lagrangian terms, the

four-component spinors and the derivation of Feynman d e s . We wiU take as an

example the foilowing A -tl> - A term (see (3.8)) which is a supersymmetric version of

the familiar W* eu term in the SM:

'Under tbii condition both nniltïng m i r a fi* are paitive.

Keeping only the a = 1,2 contributions we have

C A = hg [dtiP + C + P ] 1ÿ1 L; + h.c.

= &g[(X1 - U2)$:Li + (A1 + iX2)qbl L;] + hoc.

= ig [~+e,fi* + A - v , ~ - X+ë,g - A-P'~,] ,

where Table 14 of ref. [18] h a been used and the notation conventions of Appendix A.

Following our usual stmtegy, we wili try to wnvert eq. (3.30) into four-component

notation with the aid of the following definitions:

In particular, using (A.14), eq. (3.30) becomes:

Expressing the above in terms of the chargino mars eigenstates we get:

We can then uvlily read the vertex factors for these specific interactions in the MSSM

which appear in 3.33, they are shown explicitly in Figure 3.1. Note that the arrows

indicate lepton and flavour quant um-number flow for lep tons and quarks respectively.

Findy, for the last two diagrams the charge conjugation matrix C a p p a s explicitly

in the vertex elements iastead of the chargino spinors: 2;.

Figure 3.1: Feynman rules for some characteristic chargino interactions in the MSSM.

Similarly to the chargin0 case, the contributions for the neutralinwnixing Lagrangian

piece corne from the A - d, - A terms of (3.8) with a = 3, the additionai superpotentid

term and the relevant sofbSUSY b d n g terms, where we have defineci M' z 2&. We

should note here that some supersymmetric models, incorporated into GUT scenarios,

predict that M and M' are proportional [la, 23):

similar to the 0, prediction. These models also predict that M = (2/d)a, where M~

stands for the gluino masa and g. the SU@), coupling constant. Let us show explicitly

the derivation of the A - $ - A te rms (a = 3):

LN = igI/Z(oij/fi) (X+j H;) + &~g'yibG(~'+j~.) + h . ~ .

Wit h this defini tion LN becomes:

1 LN = -- ($O) Y@ + h.e. 2

33

w here

M' O -mz sin 0, sin ew m, cos O, sin dw O M m. sin 8, COS gw -mz COS &, COS ew

-mz sin B, sin fiw rn, sin 8, cos 8 , O -P m, cos 0, sin 8 , -nas cos 8, cos ew -/r O

(3.38)

The neutralino mass eigenstatcs are d&ed by the equation:

where N is a unitary matrix satisfying the relation :

Notice that Y is a symmetric matrix which is due to the Majorana nature of the

particles', therefore we only need one diagonaiizing matrix. Squaring the neutdno

rnixing matrix No we get (compare with eq. (3.20)):

W e can then rewrite (3.37) as:

Summation over repeated indices is implied. This is clearly a mass term for the 2:

Majorana spinors

In an attempt to get a feeling for thare results we consider the following simplitying

assumptions which yield a massive photino (L,) whose mass satidies: #M = M. W e

? ~ t can be shown that for four-component Majotana spinors the foilowing holds: :;(II -p)xg = &l .p)*!

then redefine our basis as:

In this basis the new miacing matrix simplifies to:

and the mass eigenstates are definecl by:

Rom eq. (3.43) we see that we only need to diagonalize the non-trivial 2 x 2 submatrix

of the matrix Y, and by doing so we get the diagonalizing matrix:

Nt= ( cos4 sin q5 -isin4 icosq5

The fact that we have four neutralinos in the MSSM means that we caa go on and

make further simplifying assumptions or try and see the dfect of different limits on

spinor wmponents and the corresponding Feynman rules (181.

3.6.1 Analytic results

We will not show any numerical results for the diagonalizing matrices U,V and Y since

it is beyond the scope of this work. The interested d e r is refmed to ref. [24] for

an extensive numerical analysis. We will however, for the sake of completeness, show

some approximate analytic remilts below.

With the assumption that IM f pl, lM'f pl > m,, which is a rather reasonable one

remembering that the origins of the masses M and M' are h m the SORSUSY break-

ing terms and p h m the additional superpotential, then the folIowing approlQmate

relations hold [24, 25

and

For the above we used the assumption that IMpI > rn: sin 2/3, where

or equivdently that detX > O, since sin 2P = 2 sin cos P = 2 cos B, sin 8,. Using the

above matrices to diagonalize X, we h d y get the following elements of M.:

Note that if detX < O then we have to replace sgn(p) with -1.

Finally, the neutralino mixing matrix is:

1 m s sin 20w(hf8+r sin 2 0 ~ ) 2(Mf-M)(r2-Mlz)

I m', ain 3Uw(M+r sin 20u) 2(M-M')(g-W)

Similarly to the chargino case we use the above result to obtain the (approximate)

neutralino masses:

n 5 ( M f + p sin 24 , ) sin2 Ow rnq " Mt+ Me-fia 1

Before we close our short introduction to the chargino and neutralino m a s eigenstates,

we should mention that, except from (3.34) which is trne only if we impose SUSY-

GUT, there are no other 'tight' constraints on the different parameters associated with

the mixing matrix. Obviously different models put different constraints and make

assump t ions about the ordering of the states. We should emphasize that the column

37

Table 3.2: Ordering of positive mass eigenvalues of chatginos and neutralina in t k possible p regions.

arrangement of U,V and Y is directly relatecl to the mass ordering of the states (ordered

with increasing mass rg . r n ~ > me), simply because they are constructed this way.

So, for example, if any chargino or neutraino: 2: (i = 2,3,4) is produced i t will cascade

tiii the lightest+supusymmetric-particle8 (LSP) 2: is pmduced.

Finally, from (3.49) we clearly see that the relative size of m a and mg is a rather

sensitive function of p. It is therefore instructive to show the mass ordering in the thra

distinct regioos as shown in Table 3.2, always under the assumption that M f < M .

Ref. [26] gives an extensive numerical analysis of the mass hieranhy and different

potential scenarios that may apply. Here we just skimmed the s u r f " by borrowing

their table aod mentionhg the t h n e different regioos. As a la& remark on this topic

we shouid Say that in the limiting cases: M' E M, lpl 2: M and lpl ci M f , we get

degenerate eigenvalues and therefore we should use degenerate perturbation theory.

-- ~p

8As WC have oan bdare is one of the posoible candidates for the LSP.

Chapter 4

Left-Right Supersymmetry

We saw in the previous chapter that SUSY solves some of the problems of the MSSM

but unfortunately not an. One possible scenario is SUSY GUTs which are quite suc-

ceshil in predicting unification of the coupling constants and give relationships be-

tween parameters of the SM. ARer having studied the basic characteristics of both

the L R and MSSM it would therefore be natural to try and combine the two in a

fully left-right supe~ymmetric model (LR SUSY). The proposed gauge group is the

SU(2)L x x U(l)B-t [13,24,27]. LR SUSY d d be a SUSY GUT but it could

dso be incorporated in SUSY GUTs arnong with other attractive features. As with

a l l newly proposed models the problem that arises is simply how to distinguish LR

SUSY h m the MSSM. The main direction should be to look for signals of particles

that are absent from the MSSM. Later we will see that the model predicts the existence

of doubly charged scalars and higgsinos which, as we saw in the previous chapter, they

do not exist in the Standard Supersynunetric model.

As before, every field bas a mperpartner denoted by a (-) over it and the same

name conventions .s in the N = 1 supersyxmnetric model apply. The mode! inchdes

the following types of fields:

1. Left and Nght motter fie(& are assigned to doublets similady to the L-R model

and their bosonic supapartners to singlets as shown on Table 4.1.

2. The SU(2)Ls gauge fields WEn are the vector boson triplets -with ALa being *

their superpartnets. Similady, the V N is the U(l)B-L singlet vector boson and

Av its superputna'.

3. Finally, the Hi- sector of the theory consists of two bidoublets: responsible

for both the up and down quark masses, two triplets AL,R, and two additional

triplets 6b,R which do not play any role in spontaneous symmetry breaking but

are introduced to cancel triangle anomalies which would occar without their

presence[28].

In Table LR-fields we list the partide content of the model and their quantum numbers.

A wmplete and detailed description of the Lsgrangian can be found in refs. [13,24,27].

We will not include it in this work since it is quite involved and we will only use the

required parts for specïfic mechanisms. Let us however list the individud parts of the

Lagrangian with a brief description of their function [27]:

where the fmt and second terms deai with the gauge ahd matter fields (kinetic energy

terms, self interactions, etc.), the third is the Yukawa piece involving self interactions

of the mstter and Higgs multiplets. V is the scalar potential and CiOlt is the soR

supersyrnmetry-brea.king part which, among other things, gives Majorana mass to the

gauginos. W e have. revicwed al1 these terms when describing the MSSM in the previous

lThe gauge bo#ins are the only exception to the notation rule.

Fields Matter

Higgs

A,, = (*$+ A 1 -+ -++ 1 A -z* L a

-- ( g- &.ta - -+'- Y), Table 4.1: Mattei and Higgs field content of the GR model.

chapter.

left and

The main düfimmce is that the left-right supersymmetric model contains both

right parts dong with the extra fields (Higgs sector, etc.). For illustrative

purposes let us write down the gauge terms of the Lagangian:

L- = -LWL 4 ~ r v Wr + $xLüfl(ap - igTacl,)AL

-f WLWr + )XRafl(a, - igT,G.,)Ar

-1v 4 w' V'W + )X,ü,'6,XV

and the sojt S U S Y - b d n g ones:

(Compare the above equation with (3.13)).

Below we will examine the mechanisms of symmetry breaking, which are quite

similar to the L R model.

W e clearly see from the gmup structure of the model that it contains three gauge

symmetries and therefore three coupling constants: gLYgR and gv. By requiring the

first stage of symmetry breaking to involve Parity brealring at a mass sa l e Mpy we

end up with g' = gn and the gauge fields WE massless. The second step is to b d

the 'right-symmetricy part, Le. SU(2)R x U(l)B-L to U(l)v, which is accomplished

through the vev of A.. It should be noted tthat by appropriately choosing the Higgs

multiplets both syrametries can be broken at the same scaie: M p = MW,. Findy, since

certain arguments indicate that the breating scale of supersymmetry is close to the

weak s d e [13] in this model we break the 'lefbsymmetric' part, i.e., x U ( l ) y

to U(1)- togetlrer with SUSY through the non-zero vev. of O:

(SUSY) s[1(2)~ x su(2)R x (I(l)B-L x p

<'** w, .

In an analogous to the standard mode1 fshion, we ailow non-zero vacuum expectation

values only for the neutral Higgs fields. In partidar the vev's of the Higgs multiplets

are: O O

(A.) = ( * ) 9 (A') = ( 6 L . R ) = 0,

4.2 Mass eigenstates of Vector Bosons

ki the e s t stage of symmetry breaking we generate masses for W z , Wz and V . The

relevaat term in the Lagrangian [28] is:

If we now substitute the vev of AR and simplify, we get:

From the fùst tenn of eq. (4.4) above, we clearly see that the mass of the chargeai

nght-handed gauge boclon, which is expeekd to be of the form M&~Y: WG, is:

Also, notice that we can mite the second term of (4.4) in the matrix form:

which is off-diagonal and therefore suggests mixing of the Wm and V, fields. By

diagonalizing the matrix we see that it ha9 the eigenvaiues: (gg+4g:) and 0. Proceeding

further by identifhg the linear combinations of the original fields and V, with the

physical fields Z,'R and B, we expect mass terms of the form: f WzZ' and tMiBz .

Following similar steps as we did in the standard mode1 (and normalizing the fields)

we finaily get:

with mass:

and similarly for the massless field B, we get:

For the second stage of symmetry breaking the relevaat term in the Lagrangian [13,28]

is:

We would nonnally follow exactly the

of the two Higgs fields but there is an

same procedure as above and replace the vev's

important observation to be made before. The

44

first stage is energetidy much higher than the second and therefore the right-handed

vector bosons U=i, and &, decouple tmm the second stage leaving only the massless

B, contribution. With this in mind, eq. (4.9) simplifies to:

1 1 1 (;9,dv + i g R d v ) 4-11 + q ~ t 1 (ig,03v + igRww;) a,r 4

where, again, we have defined:

From (4.7) and (4.8) we can write vR in terms of the two neutrd fields:

Using the above we get the approximate resdt:

where :

Notice that this is not a strict equality since we ignored ail terms wntaining the 'heavy7

2, field. Comparing the f k t tenn of eq. (4.1 1) with that of eq. (4.4) we immediately

recognize that it is a mass term for the charged vectot bosons, similady to the right-

handed case above, Therefore:

Findy, we see that the mixing matrix of (4.11) is off diagonal which again suggests

that TL and B, combine to gïve physid mass eigenstates. Either by diagonalizing

the matrix or by simply rewriting the second tenn we get:

Identifying the above with the expected m a s k r m s for two neutral physical fields:

) M$ZzL + M i X , we get the equivaient expression to relations (4.7) and (4.8):

which has mass :

Finally, the second term of (4.13) gïves the familiar photon field:

4.3 Gaugino-Higgsino Mixing

Let us now introduce the part which consists of the gaugino and higgsino mixing terms

which corne h m the familiar A - $ - A term of the Lagrangian. The terms involved

are the soft supetsymmetry-bnsking term and the s d a r potential [13, 24, 27):

- i c / 2 ~ r [(a AL)t(g,o AL + 2gJv)o - A=] + h.c. LGH -

4.3.1 C hargino Mixing

Next, we substitute the vacuum expectation values of the Higgs fields fiom eq. (4.2)

into eq. (4.16). Keeping as usual the tems involving charged fields, we get:

r c = { ix , ( f i g R u R A l + g,&&)

+ j~:~'tc,& + ~ A : ~ R s ~ ~ + i A t g ~ ~ &

+ 4mJ:X; + 4mnX$A,

+ P&& + p&U) + h-C*

Following the same principles as in the MSSM we rewrite the rest of the chargui

terrns in the foilowi,ng way:

In this mode1 we d e f i e the following fields:

++ ' ( - i ~ f , -ai, Jt, @, At, di),

$- (-a;, -A;, k, &, A;, A;). With these definitions the mixing matrix is:

Notice that in the GR SUSY we have six charginos, instead of two. The mass eigen-

states are defined as follows:

With U and V being unitary matrices such that

where MD is a diagonal matrix with non-negative entries. With the above definitions

we can rewrite eq. (4.18) as:

o r using the fmt equation of (A.13) and define the following four component Dirac

eq. (4.23) becumes:

Cc = -Migiziy

where summation over i is implied.

Since Mo is the mass matrix it is required to mntain only nonnegative entties. We

foiIow the same path as in the minimal SUSY case and wnsider the following relations:

Therefore, we have to consider the eigenvalue problem for- XtX and XXt in order to

compute the dements of the diagonalizing matrices U' and V.

The eigenvalue problem is mmplicated and must be solved numer idy . We wiil

not quote any results since they are not particularly illuminating, Saif [24] has done an

extensive work on both analytic and numerical solutions to the masses and elements

of the diagonalizing matrices and the interested reader is encourageci to consult this

work.

4.3.2 Neutrabo mixing

In order to obtain the neutrrüno part of the Lagrangian we once mon replace the vev's

of the Higgs fields into (4.16). but this time we keep only the neutral terms:

C H = - - 2~O,~,)v& + h . ~ .

+ i&(AsR- . \ % L ) ~ @ u +hoc.

- 1 ~0 - az( &a - ~%&&,g + h-c- (4.27)

- - 0 x 0 + X O A O + mL(x:x", + i:I!) + mn(XI, . , ,) + m v ( L x v + LL)

-0 -0 + BI (+la& + au&) + hac- As usual we define:

(c-i~z, -iA:, -At, &, t&, t&, 4%. g). (4.28)

Then eq. (4.27) can be written as:

where

Y =

The neutralino mass eigenstates are d&ed by the usual form:

xoT = NP*,

with N being a unitary matrix which satisfies the relation:

W Y N - l = N ~ .

ND is a diagonal matrix with positive elements. Notice that since we are dealing with

neutral fields we only need one diagonalkation matrix N, which obeys the relation:

Chapter 5

Chargino Decay Widths

After having introduced the basic mechanimis of the Mt-Right mode1 and its particle

content we can now proceed and calculate specific decay widths of singly or doubly

charged charginos. As usud, we first have to isolate the relevant terms of the GR La-

grangian and calculate the vertex factors of the interactions. It is then straightrorward

to calculate the different decay widths.

5.1 Chargino-lepton-slèpton vertices

Let us focus on the Ydawa piece of the Lagrangian and partidarly on the tem:

Replacing the explicit field stnicture on the above we get :

Now, usiog the fad that éLfl are scalars, the identity = f'jc and some trivial

algebra we end up with the following form:

The next step will be to convert eq. (5.3) into Ccomponent notation. In order to do

this we define the following fields1:

With this at hand and using eqs. (A.14) and (A.16), we have :

=++ c - (xr ) PLeéL = = iAt+eLé,,

Using the above definitions eq. (53) becornes :

From the second and third terms of (5.6) we can easily read off the interaction vertices

which aire shown in Figure 5.1. Notice that the charge conjugation matrix C is written

explicitly at the vertex factor. In Appendk B we give al1 the details of the decay-width

calculat ion.

See eq. (C27) in 118).

Figure 5.1: Chargindepton-slep ton vertices.

- 54

5.2 C hargino-neut ralino-Higgs boson vert ices

For the vertices involving the H i g p bosons we focus on the following terms of the

Expanding the above wing the explicit field structure we get:

If we now expand further out expression and keep only the chargcd X terms, we end up

with the following:

DeJining the 4 component spinors:

and following the diagondization conventions of Huitu et al. [29], we have:

&R P,= N'x: Bdore continuing, there are two points worth noting in the above formulas:

1. For compactness7 we imply summation over repeated indices.

2. Notice that th- is no expression similar to (5.12) for i\O, since it is not included

in our neutralino basis d>O.

Therefore, replacing the above definitions in the expression (5.9) we get:

r, = {iagL (~ ; ! ;~AoA- + &A+x+ - A++A+x- - vGXt~++~+) t

+i&& ( K ~ ~ ~ A O X - + N&~:A+x+ - A++A+x- - G x ~ +A++x+) R ) (5.13)

Notice that we defincd: (Zt)= gr. Lastly, using (5.15) and (5.14) the final form of

the Lagrangian is:

= O

i& W U ? '+P ,~?A; + i y l D P&At CC = { ( + .l.j

+ u ; ~ ~ T P&+A: - iV$J'12T P&A:+)

+i&& ( i r ( s ~ ~ ~ $ ; P& A: + + N & I ~ % p&j+a+, (5.16)

+ ~ $ 2 : P ~ ~ : + A : - i&U.$g; P ~ ~ ; A ; + ) }

+h.c.

From (5.16) we can easily read off the interaction vertices which are shown in Fi- 5.2.

2: z; J , ZQR -l w v: ~2 (1 - y),

Figure 5.2: ChargineNeutraüneHiggs boson vertices in the L-R-SUSY model.

58

Figure 5.3: Chargin+NeutralineHiggs boson vertices in the GR-SUSY mode1 (cont'd).

5.3 Chargino-neutralino-Vector boson vertices

Finally, let us look at terms of the Lagrangian that give us the Chargino-neutralin*

Vector boson interaction:

Again, we keep only the charged temm of (5.17). Afkr some algebraic steps we get:

Making the usual replacements witb the charginos and neutralinos we have:

- g T 7 ~ ~ L B ~ = ztü,,~;. Therefore we can rewrite Cc in four component notation as follows:

-++w-p - i ~ ; ~ y p L ~ w l l + ' ] - i K ~ ~ ~ y ' ~ r X n

Finally, fmm expression (5.21) we can easily read off the interaction vertices which are

shown in Figures 5.4 and 5.5.

Figure 5.4: Chargino-Neutralino-Vixtor boson vertices in the LR-SUSY model.

62

Figure 5.5: Chargin-Neutralino-Vector boson vertices in the LR-SUSY mode1 (cont'd).

63

5.4 Decay Widths

W e are now in the position to calculate the lowart order decay widths for the dif-

ferent supenrymmetric interactions. The details of a sample caldation are given in

Appendix B, in Tabla 5.1 and 5.2 we wiU simply quote the resuits which are expressed

with respect to the initial partide's test frame.

Decav Wldth

Table 5.1: Total decay widths for chagino decays into lepton, s-lepton and Higgs.

Decay Widt h

Table 5.2: Total decay widths for chargino and neutraüno decays into Vector bosons.

66

5.5 Conclusion

As we mentioned in the introduction, the Standard Model, successhtl as if is, is most

probably not the ha1 aiiswer of theoretical physics for explainhg and unifying the dif-

ferent manifestations of matter and fundamental forces of Nature. The gauge hierarchy

problem (GHP), CP and Parity violation, seemingly unrelated structure constants, etc.

are some of the problems the SM is f a d with. DXerent models have been proposed

in an attempt to remedy some or possibly all of the above drawbacks. We reviewed

some of these models and saw their differences and their similarities. We then intro-

duced the E R SUSY model which, although retains aii of the SM'S desirable features,

yet it provides solutions to many of the problems mentioned above (e.g.: GHP, Pa.rity

viotat ion.).

After explainhg the basic structun of the model we have calculated interaction

vertices among various doubly and singly charged -charginos and neutralinos. Fûrther,

we used these interaction vertices to c a l d a t e speWfic decay widths.

In order to extend and complete this work the decays wodd have to be evaiuated

numeridy. This requires that certain values be given to some of the fke constants

that wodd corne fiom different scenarios and restrictions found in the literature. By

doing so we wodd be able to comment on what to expect for realistic decay rates.

Since the doubly charged Higgses and Higgsinos are not restricted by the theory, then

is a chance that they are quite light and therefore we can hope that they may provide

a sign for G R SUSY and enhance production rates. We believe and hope that further

study wili concentrate on téis direction.

Appendix A

Spinor notation and conventions

Let M c S L(2, C), then the seif-representation is deiined as:

D ( M ) M, VM r SL(2, C).

The elements of the representation space tiansform as1:

Similady, the representation D ( M ) (MmoL)* is epiuafent to the complex conjugate

self representation M' and P is its rep-ntation space. The 1" elernents are dotted

(right-handed) Weyl spinors, whereas the elements of F are undotted (left-handed)

Weyl spinors. W e now define the direct s u m of the two sp- F and p" as:

E E F ~ F ,

where E is the four-dimensional complex representation spaee of Dirac spinors. There-

fore, if cc F and i j c F* then:

'The spinor notation and riubscriptsupemcript conventions in this work are b d maidy on reG.[l8, 301.

is a Dirac spinor. Further, we define a representation of SL(2, C) on E by the map:

Therefore,

Notice that the dotted and undotted indices refer to different (non-quivalent) repre-

sentations and should not be confirsed. Writing the index structure of the Dirac spinor

explicitly we get:

i1>= ( 2 ), wherea=1 ,2andt= i , i . (A-3)

In a similar fashion, the spinors and ifC" trandotm under M* and M-' respectively.

We now define the following conventions:

gw E dîag(1, -1, -1, -l),

P" = ( E , PI, (A-4)

up = ( l p ) , iTw E (1, -m).

In the sxalled Weyl bais or chiml representation the 7 matrices an as foilows:

and furt herrnore: -1 O

where

Writing the complete index stmcture of expression (A.l) we see that any 4 x 4 matrix

î acting on Dirac four-spinors m u t have the following index s t ~ c ~

With this background materid at hmd let us rewrite the Dirac equation as follows:

( 7 " ~ r - m)J, = O

Therefore the Dirac equation in two-camponent notation bemmes:

(üp#)*(, = mijœ, ( ~ ~ p l l ) ~ ~ > l b = me. (A.10)

Ln order to raise and lower the spinor indices we define the antisymmetric tensor E"

which plays the role of a metris in the spinor space F:

So, for example, e = ea@cfi and similarly for the dotted indices: ii, = E,$. Findy

we define the familiar left and right projection operators:

and so if PL,,& we have + = (;:)O

2Actually, €ME-' = M-a r c GL(2, C). For furtha details see rd.[30].

We wiU now state some characteristic and usefd identities of the component spinors:

s€ = qa€e = €?h

qcTq z qOuf'&, = -ce'+ Let us for the saùe of clarity pmve the second formula above:

Using the above relations we can prove the following set of equations which are used

in 'translating' the four-component into twcxomponent notation:

$ 1 ~ = 916 +

- -)i*, u w h = rl, 0.~4 - %o"f1.

W e clearly see that the indices 1 and 2 label different four-component spinors and

t heir two-component subspinon. Using specific combinat ions of the above equat ions

we get the following relations which are useful in building four-component spinors from

Lagrangians expressed in twemmponent notation:

&pLa = mG,

The charge conjugated spinor gC is given by $c = C tLT where (in the chiral represen-

tation) :

A Majomnu spinor has by definition the property tbc = J>, and so, according to (A.16),

= (. Therefore a Majorana spinor has the general fom:

(A. 17)

(A. 18)

Appendix B

Decay widt h sample calculation

Let us now, for illustrative purpapes, have a closer look at a specific decay. We wiil use

as an exampb the proeap: Xi + zjAO,i > j. The tnclevel diagram for this pro-

is shown in Figure B below together with the vertex factor. The letters in parenthaies

denote the particle momenta

Figure B.l: Decay diagram with momenta and vertex factor.

The amplitude for this process is clearly:

Squaring the amplitude we get:

= kp~wTr(7'7~ + 7u7'}

= & , . p " 8 r

= 8 k a p .

Here we used the standard trace theorems, see for example r d [7]. Now, ~ d n g

to out momenta definitions we know that: p = k + k'. Therefore,

a p - k = ; ( $ + k a - K a ) =) (mt+mj l - M:).

Working in the initial particle's test frame (center of m a s fiame) and using the appro-

priate formula (311, we get the foliowing decay width:

where X is the kinematic factor [26, 311.

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