Mssm Model

The Minimal Supersymmetric Standard ModelVersion 1.9.175

June 24, 2011

Jorge C. Romao1

1Departamento de Fsica, Instituto Superior Tecnico

A. Rovisco Pais, 1049-001 Lisboa, Portugal

Abstract

We write down the mass matrices and couplings needed for Minimal Supersym-

metric Standard Model (MSSM) calculations. These include the mass matrices for

all the particles in the model, the charged and neutral current couplings of the spin12 particles with the gauge bosons, and the couplings of the charged and neutral

scalars with the spin 12 particles. To fix our notation, we have also included the

couplings of the gauge bosons with themselves, although these are the same as in

the SM.

Contents

1 Introduction and Motivation 1

2 SUSY Algebra, Representations and Particle Content 2

2.1 SUSY Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Simple Results from the Algebra . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 SUSY Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 How to Build a SUSY Model 5

3.1 Kinetic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Self Interactions of the Gauge Multiplet . . . . . . . . . . . . . . . . . . . . 6

3.3 Interactions of the Gauge and Matter Multiplets . . . . . . . . . . . . . . . 6

3.4 Self Interactions of the Matter Multiplet . . . . . . . . . . . . . . . . . . . 7

3.5 Supersymmetry Breaking Lagrangian . . . . . . . . . . . . . . . . . . . . . 7

3.6 RParity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 The Minimal Supersymmetric Standard Model 8

4.1 The Gauge Group and Particle Content . . . . . . . . . . . . . . . . . . . . 8

4.2 Field Strengths and Covariant Derivatives . . . . . . . . . . . . . . . . . . 10

4.3 The Superpotential and SUSY Breaking Lagrangian . . . . . . . . . . . . . 11

4.4 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.5 The Constrained Minimal Supersymmetric Standard Model . . . . . . . . . 12

5 Mass Matrices in the MSSM 13

5.1 The Chargino Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 The Neutralino Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.3 Neutral Higgs Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.4 Charged Higgs Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.5 Lepton Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.6 Quark Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.7 Slepton Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.8 Sneutrino Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.9 Squark Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Couplings in the MSSM 21

6.1 Gauge Self-Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.1.1 V V V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.1.2 V V V V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.2 Charged Current Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.3 Neutral Current Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

i

6.4 3-point gauge boson couplings to scalars . . . . . . . . . . . . . . . . . . . 26

6.4.1 Gauge boson couplings to sfermions (V f f) . . . . . . . . . . . . . . 27

6.4.2 Gauge boson couplings to Higgs (V HH) . . . . . . . . . . . . . . . 27

6.5 4-point gauge boson couplings to scalars . . . . . . . . . . . . . . . . . . . 27

6.5.1 Gauge boson couplings to sfermions (V V f f) . . . . . . . . . . . . . 27

6.5.2 Gauge boson couplings to Higgs (V V HH) . . . . . . . . . . . . . . 27

6.6 Scalar couplings to fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.6.1 Charged scalars couplings to fermions . . . . . . . . . . . . . . . . . 27

6.6.2 Neutral scalars couplings to fermions . . . . . . . . . . . . . . . . . 29

6.6.3 Final formulas for the scalar couplings to fermions . . . . . . . . . . 30

6.6.4 Final formulas for the Higgs couplings to SUSY fermions . . . . . . 32

6.6.5 Final formulas for the Higgs couplings to SM fermions . . . . . . . 33

6.7 Trilinear scalar couplings with Higgs bosons . . . . . . . . . . . . . . . . . 34

6.7.1 Higgs Sfermion Sfermion . . . . . . . . . . . . . . . . . . . . . . 34

6.8 Quartic scalar couplings with Higgs bosons . . . . . . . . . . . . . . . . . . 37

6.9 Quartic sfermion interactions . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.10 Higgs couplings with the gauge bosons . . . . . . . . . . . . . . . . . . . . 37

6.10.1 Higgs Gauge Boson Gauge Boson . . . . . . . . . . . . . . . . . 37

6.10.2 Higgs Higgs Gauge Boson Gauge Boson . . . . . . . . . . . . 37

6.11 Higgs boson self interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.11.1 Higgs Higgs Higgs . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.11.2 Higgs Higgs Higgs Higgs . . . . . . . . . . . . . . . . . . . . . 38

6.12 Ghost interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.12.1 Ghost Ghost Gauge Boson . . . . . . . . . . . . . . . . . . . . . 40

6.12.2 Ghost Ghost Higgs . . . . . . . . . . . . . . . . . . . . . . . . . 40

A Changelog 41

i

1 Introduction and Motivation

In recent years it has been established [1] with great precision (in some cases better than

0.1%) that the interactions of the gauge bosons with the fermions are described by the

Standard Model (SM) [2]. However other sectors of the SM have been tested to a much

lesser degree. In fact only now we are beginning to probe the selfinteractions of the

gauge bosons through their pair production at the Tevatron [3] and LEP [4] and the

Higgs sector, responsible for the symmetry breaking has not yet been tested.

Despite all its successes, the SM still has many unanswered questions. Among the various

candidates to Physics Beyond the Standard Model, supersymmetric theories play a special

role. Although there is not yet direct experimental evidence for supersymmetry (SUSY),

there are many theoretical arguments indicating that SUSY might be of relevance for

physics below the 1 TeV scale.

The most commonly invoked theoretical arguments for SUSY are:

i. Interrelates matter fields (leptons and quarks) with force fields (gauge and/or Higgs

bosons).

ii. As local SUSY implies gravity (supergravity)it could provide a way to unify gravity

with the other interactions.

iii. As SUSY and supergravity have fewer divergences than conventional filed theories,

the hope is that it could provide a consistent (finite) quantum gravity theory.

iv. SUSY can help to understand the mass problem, in particular solve the naturalness

problem ( and in some models even the hierarchy problem) if SUSY particles have

masses O(1TeV).As it is the last argument that makes SUSY particularly attractive for the experiments

being done or proposed for the next decade, let us explain the idea in more detail. As

the SM is not asymptotically free, at some energy scale , the interactions must become

strong indicating the existence of new physics. Candidates for this scale are, for instance,

MX O(1016 GeV) in GUTs or more fundamentally the Planck scaleMP O(1019GeV).This alone does not indicate that the new physics should be related to SUSY, but the so

called mass problem does. The only consistent way to give masses to the gauge bosons and

fermions is through the Higgs mechanism involving at least one spin zero Higgs boson.

Although the Higgs boson mass is not fixed by the theory, a value much bigger than

< H0 > G1/2F 250 GeV would imply that the Higgs sector would be strongly coupledmaking it difficult to understand why we are seeing an apparently successful perturbation

theory at low energies. Now the one loop radiative corrections to the Higgs boson mass

would give

m2H = O( 4

)2 (1)

1

which would be too large if is identified with GUT or P lanck. SUSY cures this problem

in the following way. If SUSY were exact, radiative corrections to the scalar masses

squared would be absent because the contribution of fermion loops exactly cancels against

the boson loops. Therefore if SUSY is broken, as it must, we should have

m2H = O( 4

)|m2B m2F | (2)

We conclude that SUSY provides a solution for the the naturalness problem if the masses

of the superpartners are below O(1 TeV). This is the main reason behind all the phe-nomenological interest in SUSY.

In the following we will give a brief review of the main aspects of the SUSY extension of

the SM, the socalled Minimal Supersymmetric Standard Model (MSSM). Almost all the

material is covered in many excellent reviews that exist in the literature [5, 6].

2 SUSY Algebra, Representations and Particle Con-

tent

2.1 SUSY Algebra

The SUSY generators obey the following algebra

{Q, Q} = 0 (3)

{Q, Q

}= 0 (4)

{Q, Q

}= 2 () P (5)

where

(1, i) ; (1 i) (6)and , , , = 1, 2 (Weyl 2component spinor notation). The commutation relations

with the generators of the Poincare group are

[P , Q] = 0

[M , Q] = i () Q

From these relations one can easily derive that the two invariants of the Poincare group,

P 2 = PP

W 2 = WW

(7)

2

where W is the PauliLubanski vector operator

W = i2M

P (8)

are no longer invariants of the Super Poincare group. In fact

[Q, P2] = 0

[Q,W2] 6= 0 (9)

showing that the irreducible multiplets will have particles of the same mass but different

spin.

2.2 Simple Results from the Algebra

From the supersymmetric algebra one can derive two important results:

A. Number of Bosons = Number of Fermions

We have

Q|B >= |F > ; (1)NF |B >= |B >Q|F >= |B > ; (1)NF |F >= |F > (10)

where (1)NF is the fermion number of a given state. Then we obtain

Q(1)NF = (1)NFQ (11)

Using this relation we can show that

Tr[(1)NF {Q, Q}] = Tr [(1)NFQQ + (1)NFQQ]

= Tr[Q(1)NFQ +Q(1)NFQ]

= 0

But using Eq. 5 we also have

Tr[(1)NF {Q, Q}] = Tr [(1)NF 2P] (12)

This in turn implies

Tr(1)NF = #Bosons #Fermions = 0

showing that in a given representation the number of degrees of freedom of the

bosons equals those of the fermions.

3

B. 0|H|0 0From the algebra we get{

Q1, Q1}+{Q2, Q2

}= 2Tr ()P

= 4H (13)

Then

H =1

4

(Q1Q1 +Q2Q2 +Q1Q1 +Q2Q2

)(14)

and

0|H|0 = (||Q1 |0 ||2 + ||Q1 |0 ||2 + ||Q1 |0 ||2 + ||Q2 |0 ||2) 0 (15)

showing that the energy of the vacuum state is always positive definite.

2.3 SUSY Representations

We consider separately the massive and the massless case.

A. Massive case

In the rest frame {Q, Q

}= 2m (16)

This algebra is similar to the algebra of the spin 1/2 creation and annihilation

operators. Choose | such that

Q1 | = Q2 | = 0 (17)

Then we have 4 states

| ; Q1 | ; Q2 | ; Q1Q2 | (18)

If J3 | = j3 | we show in Table 1 the values of J3 for the 4 states. We notice

State J3 Eigenvalue

| j3Q1 | j3 + 12Q2 | j3 12

Q1Q2 | j3

Table 1: Massive states

that there two bosons and two fermions and that the states are separated by one

half unit of spin.

4

B. Massless case

If m = 0 then we can choose P = (E, 0, 0, E). In this frame{Q, Q

}= M (19)

where the matrix M takes the form

M =

(0 0

0 4E

)(20)

Then {Q2, Q2

}= 4E (21)

all others vanish. We have then just two states

| ; Q2 | (22)

If J3 | = | we have the states shown in Table 2,

State J3 Eigenvalue

| Q2 | 12

Table 2: Massless states

3 How to Build a SUSY Model

To construct supersymmetric Lagrangians one normally uses superfield methods (see for

instance [5, 6]). Here we do not go into the details of that construction. We will take

a more pragmatic view and give the results in the form of a recipe. To simplify matters

even further we just consider one gauge group G. Then the gauge bosons W a are in the

adjoint representation of G and are described by the massless gauge supermultiplet

V a (a,W a ) (23)

where a are the superpartners of the gauge bosons, the socalled gauginos. We also

consider only one matter chiral superfield

i (Ai, i) ; (i = 1, . . . , N) (24)

belonging to some N dimensional representation of G. We will give the rules for the

different parts of the Lagrangian for these superfields. The generalization to the case

where we have more complicated gauge groups and more matter supermultiplets, like in

the MSSM, is straightforward.

5

3.1 Kinetic Terms

Like in any gauge theory we have

Lkin = 14F aF a +i

2aD

a + (DA)DA+ iDPL (25)

where the field strength F a is given by

F a = Wa W a gfabcW bW c (26)

and fabc are the structure constants of the gauge group G. The covariant derivative is

D = + igWaT

a (27)

In Eq. 25 one should note that is left handed and that is a Majorana spinor.

3.2 Self Interactions of the Gauge Multiplet

For a non Abelian gauge group G we have the usual selfinteractions (cubic and quartic)

of the gauge bosons with themselves. These are well known and we do write them here

again. But we have a new interaction of the gauge bosons with the gauginos. In two

component spinor notation it reads [5, 6]

LW = igfabc abW c + h.c. (28)

where fabc are the structure constants of the gauge group G and the matrices were

introduced in Eq. 6.

3.3 Interactions of the Gauge and Matter Multiplets

In the usual non Abelian gauge theories we have the interactions of the gauge bosons

with the fermions and scalars of the theory. In the supersymmetric case we also have

interactions of the gauginos with the fermions and scalars of the chiral matter multiplet.

The general form, in two component spinor notation, is [5, 6],

LW = gT aijW a(i

j + iAi Aj

)+ g2

(T aT b

)ijW aW

bAiAj

+ig2T aij

(ajA

i

aiAj

)(29)

where the new interactions of the gauginos with the fermions and scalars are given in the

last term.

6

3.4 Self Interactions of the Matter Multiplet

These correspond in non supersymmetric gauge theories both to the Yukawa interactions

and to the scalar potential. In supersymmetric gauge theories we have less freedom to

construct these terms. The first step is to construct the superpotential W . This must be

a gauge invariant polynomial function of the scalar components of the chiral multiplet i,

that is Ai. It does not depend on Ai . In order to have renormalizable theories the degree

of the polynomial must be at most three. This is in contrast with non supersymmetric

gauge theories where we can construct the scalar potential with a polynomial up to the

fourth degree.

Once we have the superpotential W , then the theory is defined and the Yukawa interac-

tions are

LY ukawa = 12[

2W

AiAjij +

(2W

AiAj

)ij

](30)

and the scalar potential is

Vscalar =12DaDa + FiF

i (31)

where

Fi =W

Ai

Da = g AiTaijAj (32)

We see easily from these equations that, if the polynomial degree of W were higher than

three, then the scalar potential would be a polynomial of degree higher than four and

hence non renormalizable.

3.5 Supersymmetry Breaking Lagrangian

As we have not discovered superpartners of the known particles with the same mass, we

conclude that SUSY has to be broken. How this done is the least understood sector of

the theory. In fact, as we shall see, the majority of the unknown parameters come from

this sector. As we do not want to spoil the good features of SUSY, the form of these

SUSY breaking terms has to obey some restrictions. It has been shown that the added

terms can only be mass terms, or have the same form of the superpotential, with arbitrary

coefficients. These are called soft terms. Therefore, for the model that we are considering,

the general form would be1

LSB = m21Re(A2) +m22 Im(A2)m3(aa +

aa)+m4 (A

3 + h.c.) (33)

1 We do not consider a term linear in A because we are assuming that , and hence A, are not gauge

singlets.

7

where A2 and A3 are gauge invariant combinations of the scalar fields. From its form, we

see that it only affects the scalar potential and the masses of the gauginos. The parameters

mi have the dimension of a mass and are in general arbitrary.

3.6 RParity

In many models there is a multiplicatively conserved quantum number the called Rparity.

It is defined as

R = (1)2J+3B+L (34)With this definition it has the value +1 for the known particles and 1 for their su-perpartners. The MSSM it is a model where Rparity is conserved. The conservation

of Rparity has three important consequences: i) SUSY particles are pair produced, ii)

SUSY particles decay into SUSY particles and iii) The lightest SUSY particle is stable

(LSP).

4 The Minimal Supersymmetric Standard Model

4.1 The Gauge Group and Particle Content

We want to describe the supersymmetric version of the SM. Therefore the gauge group is

considered to be that of the SM, that is

G = SUc(3) SUL(2) UY (1) (35)

We will now describe the minimal particle content.

Gauge FieldsWe want to have gauge fields for the gauge group G = SUc(3) SUL(2) UY (1).Therefore we will need three vector superfields (or vector supermultiplets) Vi with

the following components:

V1 (,W 1 ) UY (1)V2 (a,W a2 ) SUL(2) , a = 1, 2, 3V3 (gb,W b3 ) SUc(3) , b = 1, . . . , 8

(36)

where W i are the gauge fields and , and g are the UY (1) and SUL(2) gauginos

and the gluino, respectively.

LeptonsThe leptons are described by chiral supermultiplets. As each chiral multiplet only

describes one helicity state, we will need two chiral multiplets for each charged

8

lepton2. The multiplets are given in Table 3, where the UY (1) hypercharge is defined

Supermultiplet SUc(3) SUL(2) UY (1)Quantum Numbers

Li (L, L)i (1, 2,12)Ri (R, cL)i (1, 1, 1)

Table 3: Lepton Supermultiplets

through Q = T3+Y . Notice that each helicity state corresponds to a complex scalar

and that Li is a doublet of SUL(2), that is

Li =

(Li

Li

); Li =

(Li

Li

)(37)

QuarksThe quark supermultiplets are given in Table 4. The supermultiplet Qi is also a

doublet of SUL(2), that is


Qi (Q, Q)i (3, 2, 16)Di (dR, dcL)i (3, 1, 13)Ui (uR, ucL)i (3, 1,23)

Table 4: Quark Supermultiplets

Qi =

(uLi

dLi

); Qi =

(uLi

dLi

)(38)

Higgs BosonsFinally the Higgs sector. In the MSSM we need at least two Higgs doublets. This is

in contrast with the SM where only one Higgs doublet is enough to give masses to

all the particles. The reason can be explained in two ways. Either the need to cancel

the anomalies, or the fact that, due to the analyticity of the superpotential, we have

to have two Higgs doublets of opposite hypercharges to give masses to the up and

down type of quarks. The two supermultiplets, with their quantum numbers, are

given in Table 5.

2We will assume that the neutrinos do not have mass.

9


H1 (H1, H1) (1, 2,12)H2 (H2, H2) (1, 2,+12)

Table 5: Higgs Supermultiplets

4.2 Field Strengths and Covariant Derivatives

For the derivation of the Feynman rules for the interactions it is important to fix the

notation for the the field strenghts and covariant derivatives. Here we just write the

SUL(2) UY (1) part. We have

LGauge = 14 F aF a BB (39)

where

F a = Wa W a gabcW bW c (40)

B = B B (41)

where 123 = 1. The covariant derivative reads

D = + i ga

2W a + i g

Y2B (42)

where our normalization for the hypercharge is

Q = T3 + Y (43)

After symmetry breaking (see below) we have

W 3 = sin WA + cos WZ

B3 = cos WA sin WZ (44)

with

e = g sin W = g cos W ; ;

g

g= tan W (45)

We can then write the covariant derivative in the more useful form

D = + i

[g2

(0 W+0 0

)+

g2

(0 0

W 0

)

+g

cos W

(T3 sin2 WQ

)Z + eQA

](46)

10

4.3 The Superpotential and SUSY Breaking Lagrangian

The MSSM Lagrangian is specified by the Rparity conserving superpotential W

W = ab

[hijU Q

ai UjH

b2 + h

ijDQ

biDjH

a1 + h

ijEL

biRjH

a1 Ha1 Hb2

](47)

where i, j = 1, 2, 3 are generation indices, a, b = 1, 2 are SU(2) indices, and is a com-

pletely antisymmetric 2 2 matrix, with 12 = 1. The coupling matrices hU , hD and hEwill give rise to the usual Yukawa interactions needed to give masses to the leptons and

quarks. If it were not for the need to break SUSY, the number of parameters involved

would be less than in the SM. This can be seen in Table 6.

The most general SUSY soft breaking is

LSB = M ij2Q Qai Qaj +M ij2U UiUj +M ij2D DiDj +M ij2L Lai Laj +M ij2R RiRj +m2H1Ha1 Ha1+m2H2H

a2 H

a2

[12Msss +

12M+ 1

2M + h.c.

]+ab

[AijUh

ijU Q

ai UjH

b2 + A

ijDh

ijDQ

biDjH

a1 + A

ijEh

ijEL

biRjH

a1 BHa1Hb2

](48)

Theory Gauge Fermion Higgs

Sector Sector Sector

SM e, g, s hU , hD, hE 2,

MSSM e, g, s hU , hD, hE

Broken MSSM e, g, s hU , hD, hE ,M1,M2,M3, AU , AD, AE , B

m2H2 , m2H1, m2Q, m

2U , m

2D, m

2L, m

2R

Table 6: Comparative counting of parameters

4.4 Symmetry Breaking

The electroweak symmetry is broken when the two Higgs doublets H1 and H2 acquire

VEVs

H1 =

( 12[01 + v1 + i

01]

H1

), H2 =

(H+2

12[02 + v2 + i

02]

)(49)

with m2W =14g2v2 and v2 v21 + v22 = (246 GeV)2. The full scalar potential at tree level

is

Vtotal =i

Wzi2 + VD + Vsoft (50)

The scalar potential contains linear terms

Vlinear = t01

01 + t

02

02 (51)

11

where

t1 = (m2H1

+ 2)v1 Bv2 + 18(g2 + g2)v1(v21 v22) ,t2 = (m

2H2 +

2)v2 Bv1 18(g2 + g2)v2(v21 v22) (52)

One can determine in the tree-level approximation the minimum of the scalar potential

by imposing the condition of vanishing tadpoles in Eq. 52. Oneloop corrections change

these equations to

ti = t0i ti + Ti(Q) (53)

where ti, with i = 1, 2, are the renormalized tadpoles, t0i are the tree level tadpoles

given in Eq. 52, ti are the tadpole counter-terms, and Ti(Q) are the sum of all oneloop

contributions to the corresponding onepoint functions with zero external momentum.

The contribution from quarks and squarks to these tadpoles in our model can be found

in ref. [8]. In an on shell scheme we identify the tree level tadpoles with the renormalized

ones. Therefore, to find the correct minima we use Eq. 52 unchanged, where now all the

parameters are understood to be renormalized quantities. Eqs. (52) can be solved for B

and up to a sign. We have

2 = 12m2Z +

m2H2 sin2 m2H1 cos2 cos 2

B = 12sin 2

(m2H1 +m

2H2

+ 22)

(54)

It can be shown that necessary conditions for the existence of a stable minimum are

(B)2 > (m2H1 + 2)(m2H2 +

2)

m2H1 +m2H2

+ 22 > |B| (55)

Notice that Eq. 54 only makes sense if 2 > 0 and, as we shall see in Section 5.3, also

B > 0 because is related with the mass squared, m2A, of the physical CPodd state.

4.5 The Constrained Minimal Supersymmetric Standard Model

We have seen in the previous section that the parameters of the MSSM can be consid-

ered arbitrary at the weak scale. This is completely consistent. However the number of

independent parameters in Table 6 can be reduced if we impose some further constraints.

That is usually done by embedding the MSSM in a grand unified scenario. Different

schemes are possible but in all of them some kind of unification is imposed at the GUT

scale. Then we run the Renormalization Group (RG) equations down to the weak scale

to get the values of the parameters at that scale. This is sometimes called the constrained

MSSM model.

12

Among the possible scenarios, the most popular is the MSSM coupled to N = 1 Super-

gravity (SUGRA). Here at MGUT one usually takes the conditions:

At = Ab = A A ,B = A 1 ,m2H1 = m

2H2

= M2L =M2R = m

20 ,M

2Q =M

2U = M

2D = m

20 ,

M3 = M2 =M1 =M1/2 (56)

The counting of free parameters3 is done in Table 7

Parameters Conditions Free Parameters

ht, hb, h , v1, v2 mW , mt, mb, m tan

A, m0, M1/2, ti = 0, i = 1, 2 2 Extra free parameters

Total = 9 Total = 6 Total = 3

Table 7: Counting of free parameters in the MSSM coupled to N=1 SUGRA

where we introduced the usual notation

tan =v2v1

(57)

It is remarkable that with so few parameters we can get the correct values for the param-

eters, in particular m2H2 < 0. For this to happen the top Yukawa coupling has to be large

which we know to be true.

5 Mass Matrices in the MSSM

5.1 The Chargino Mass Matrices

The charged gauginos mix with the charged higgsinos giving the socalled charginos. In

a basis where +T = (i+, H+2 ) and T = (i, H1 ), the chargino mass terms in theLagrangian are

Lm = 12(+T , T )

(000 MMMTCMMMC 000

)(+

)+ h.c. (58)

where the chargino mass matrix is given by

MMMC =

M2 12gv212gv1

(59)and M2 is the SU(2) gaugino soft mass. It is sometimes more useful to write this mass

matrix in terms of the physical masses, that is

MMMC =

M2 2mW sin 2mW cos

(60)3For one family and without counting the gauge couplings.

13

The chargino mass matrix is diagonalized by two rotation matrices UUU and VVV defined by

Fi = UUU ij j ; F

+i = VVV ij

+j (61)

Then

UUU MCMCMCVVV1 =MdiagCM

diagCMdiagC (62)

where MdiagCMdiagCMdiagC is the diagonal chargino mass matrix. To determine UUU and VVV we note that(

MdiagCMdiagCMdiagC

)2= VVVM CM

CMCMCMCMCVVV

1 = UUUMCMCMCMCMCMC(U

UU)1 (63)

implying that VVV diagonalizesM CMCMCMCMCMC and U

UU diagonalizesMCMCMCMCMCMC . In the previous expres-

sions the Fi are two component spinors. We construct the four component Dirac spinors

out of the two component spinors with the conventions4,

i =

(Fi

F+i

)(64)

5.2 The Neutralino Mass Matrices

In the basis 0T = (i,i3, H11 , H22 ) the neutral fermions mass terms in the Lagrangianare given by

Lm = 12(0)TMMMN

0 + h.c. (65)

where the neutralino mass matrix is

MMMN =

M1 0 12gv1 12gv20 M2

12gv1 12gv2

12gv1 12gv1 0

12gv2 12gv2 0

(66)and M1 is the U(1) gaugino soft mass. The neutralino mass matrix can be written in

terms of the physical masses

MMMN =

M1 0 mZ sin W cos mZ sin W sin 0 M2 mZ cos W cos mZ cos W sin

mZ sin W cos mZ cos W cos 0 mZ sin W sin mZ cos W sin 0

(67)

This neutralino mass matrix is diagonalized by a 4 4 rotation matrix NNN such that

NNN MMMNNNN1 = diag(mF 0

1, mF 0

2, mF 0

3, mF 0

4) (68)

4Here we depart from the conventions of ref. [5, 6] because we want the to be the particle and not

the antiparticle.

14

and

F 0k = NNNkj 0j (69)

The four component Majorana neutral fermions are obtained from the two component

via the relation

0i =

(F 0i

F 0i

)(70)

5.3 Neutral Higgs Mass Matrices

The quadratic scalar potential includes

Vquadratic =12[01,

02] MMM

2P 0

[0102

]+ 1

2[01 ,

02] MMM

2S0

[0102

]+ (71)

where the CP-odd neutral scalar mass matrix is

MMM 2P 0 =

B tan + t1v1 BB B cot + t2

v2

(72)This matrix also has a vanishing determinant after the tadpoles are set to zero, and the

zero eigenvalue corresponds to the mass of the neutral Goldstone boson. The mass of the

CP-odd state, usually called A, is

m2A =2B

sin 2(73)

The relation between the lagrangian states and the mass eigenstates is, as can be easily

verified, [G0

A0

]=

[ cos sin sin cos

][0102

](74)

that can be written in matrix form as

P 0 = RPRPRPP 0 (75)

where, P 0T = (G0, A0), P 0T = (01, 02) and the rotation matrix is

RPRPRP =

[ cos sin sin cos

](76)

for future reference we note that

01 = RPRPRP11G

0 +RPRPRP21A

0

02 = RPRPRP12G

0 +RPRPRP22A

0 (77)

15

The neutral CP-even scalar sector mass matrix in Eq. 71 is given by

MMM 2S0 =

B tan +m2Z cos2 + t1v1 B m2Z sin cos Bm2Z sin cos B cot +m2Z sin2 + t2v2

(78)As before we define the rotation matrix as

S0 = RSRSRSS 0 (79)

where S0T = (h0, H0), S 0T = (01, 02). For future reference we note that

01 = RSRSRS11h

0 +RSRSRS21H

0

02 = RSRSRS12h

0 +RSRSRS22H

0 (80)

For completeness we note that this (orthogonal) matrix is normally parameterized by an

angle in the following way

RSRSRS =

[ sin coscos sin

]. (81)

5.4 Charged Higgs Mass Matrices

The mass matrix of the charged Higgs bosons follows from the quadratic terms in the

scalar potential

Vquadratic = [H1 , H

2 , ] M

2HM2H

M2H

[H+1H+2

](82)

where the charged Higgs mass matrix is

MMM2H =

B tan +m2W sin2 + t1v1 B+m2W sin cos B+m2W sin cos B cot +m

2W cos

2 + t2v2

(83)The relation between the lagrangian states and the mass eigenstates is, again as in Eq. (74)[

G

H

]=

[ cos sin sin cos

][H1H2

](84)

that can be written in matrix form as

S = RS

RS

RS

S (85)

where, ST = (G, H), S T = (H1 , H2 ) and the rotation matrix is

RS

RS

RS

=

[ cos sin sin cos

](86)

for future reference we note that

H1 = RSRSRS11G

+RS

RS

RS21H

H2 = RSRS

RS12G

+RS

RS

RS22H

(87)

16

5.5 Lepton Mass Matrices

In the superpotential in Eq. 47 the Yukawa couplings are matrices in generation space.

For the case of the leptons it is possible to start with the matrix hE already diagonalized.

However, for some applications, could be useful to have a general matrix. Here we consider

an arbitrary matrix. In 2component spinor

notation the relevant mass terms in the Lagrangian are

LM = v12(hE)ij

Li

cLj

v12(hE)ij

Li

cLj (88)

where are the interaction eigenstates. The 4component spinors are

=

LcL

(89)and therefore we have

LM = MEMEMEPR MEMEMEPL

= LMEMEMER RMEMEMEL (90)

where

(MEMEME)ij =v12(hE)ij (91)

To diagonalize the mass matrix MEMEME we need different rotations for the left handed and

right handed components. We introduce 5

R = RRRRRR

R ; L = R

LRLRL

L (92)

Then

RLRLRLMEMEMER

RRRRR =MdiagEM

diagEMdiagE or R

LRLRLMdiagEM

diagEMdiagE R

RRRRR =MEMEME (93)

where MdiagEMdiagEMdiagE is a diagonal matrix. The rotation matrices are obtained by noticing that

MEMEMEMEMEME = RLR

LRL (

MdiagEMdiagEMdiagE

)2RLRLRL and MEMEME

MEMEME = RRRRRR (

MdiagEMdiagEMdiagE

)2RRRRRR (94)

that tell us that RLRLRL diagonalizes MEMEMEMEMEME

and RRRRRR diagonalizes MEMEME

MEMEME . For future refer-

ence we write the relations between the mass and the interaction eigenstates. We have

Ri =(RRRRRR

)jiRj ; Ri = Rj

(RRRRRR

)ji

Li =(RLRLRL

)jiLj ; Li = Lj

(RLRLRL

)ji

(95)

5I changed here the convention in comparison with previous versions. Now the convention is uniform

for all fields and it is consistent with the convention used in SPheno [9].

17

5.6 Quark Mass Matrices

In 2component spinor notation the relevant terms in the Lagrangian are

LM = v12(hD)ij d

Li d

cLj

v22(hU)ij u

Li u

cLj + h.c. (96)

where the primed states are again the interaction eigenstates. In 4component spinor

notation with the definitions

d =

dLdcL

; u =uLucL

(97)we get

LM = dLMDMDMDdR uLMUMUMUuR + h.c. (98)where

(MDMDMD)ij =v12(hD)ij ; (MUMUMU )ij =

v22(hU)ij (99)

To obtain the eigenstates of the mass we rotate the quark fields through

dR = RdRRdRRdR d

R ; dL = R

dLRdLRdL d

L ; uR = R

uRRuRRuR u

R ; uL = R

uLRuLRuL u

L (100)

For future reference we write the relations between the mass and the interaction eigen-

states. We have

qRi = (RqRRqRRqR)

ji qRj ; q

Ri = qRj (R

qRRqRRqR)ji

q = (d, u)

qLi = (RqLRqLRqL)ji qLj ; q

Li = qLj (R

qLRqLRqL)ji (101)

Then

RdLRdLRdLMDMDMDR

dRRdRRdR =MdiagDM

diagDMdiagD and R

uLRuLRuLMUMUMUR

uRRuRRuR =MdiagUM

diagUMdiagU (102)

where MdiagD(U)MdiagD(U)

MdiagD(U) are a diagonal matrices. These are diagonalized by noticing that

RdLRdLRdLMDMDMDMDMDMD

RdLRdLRdL=(MdiagDM

diagDMdiagD

)2; RdRR

dRRdRMDMDMD

MDMDMDRdRRdRRdR=(MdiagDM

diagDMdiagD

)2RuLRuLRuLMUMUMUMUMUMU

RuLRuLRuL =

(MdiagUM

diagUMdiagU

)2; RuRR

uRRuRMUMUMU

MUMUMU RuRRuRRuR =

(MdiagUM

diagUMdiagU

)2(103)

Before we close this section let us write down the Cabibbo-Kobayashi-Maskawa (CKM)

mixing matrix with our conventions. The couplings of the W with the quarks are

LCC = g2W d

Li

uLi g2W+ u

Li

dLi (104)

Then in terms of the mass eigenstates the charged current Lagrangian reads

LCC = g2VVV CKMij

W dLj

uLi g2VVV CKMij W

+ uLi

dLj (105)

where the CKM matrix VVV CKM is defined through

VVV CKM = RuLRuLRuLR

dLRdLRdL

(106)

18

5.7 Slepton Mass Matrices

In the unrotated basis i = (eL, L, L, eR, R,

R) we get

LM = 12 MMM

2 (107)

where

MMM2 =

M2LLM2LLM2LL M2LRM2LRM2LRM2RLM

2RLM2RL M

2RRM2RRM2RR

(108)and

M2LLM2LLM2LL =

12v21 h

Eh

TE +M

2L 12 (2m2W m2Z) cos 2

M2RRM2RRM2RR =

12v21 h

TEh

E +M

2R (m2Z m2W ) cos 2

M2LRM2LRM2LR =

v12AE

v22hE

M2RLM2RLM2RL = M

2LRM2LRM2LR

(109)

We define the mass eigenstates

= RRR (110)

which implies

i = RRRji j (111)

The rotation matrices are obtained from

RRR (Mdiag

Mdiag

Mdiag

)2RRR =MMM

2 (112)

In most applications the matrices in Eq. 108 are real and therefore the rotation matrices

RRR are orthogonal matrices.

5.8 Sneutrino Mass Matrices

In the unrotated basis i = iL we have

LM = 12 MMM2 (113)

where

MMM2 = M2L +

12m2Z cos 2 (114)


= RRR (115)

which implies

i = RRRji j (116)

19


RRR (MdiagM

diag

Mdiag

)2RRR =MMM

2 (117)

In most applications the matrix in Eq. 114 is real and therefore the rotation matrices RRR

are orthogonal matrices.

5.9 Squark Mass Matrices

In the unrotated basis ui = (uLi, uRi) and d

i = (dLi, d

Ri) we get

LM = 12 uMuMuMu2 u 12 dMdMdMd2 d (118)

where

MqMqMq2 =

M2qLLM2qLLM2qLL M2qLRM2qLRM2qLRM2qRLM

2qRLM2qRL M

2qRRM2qRRM2qRR

(119)with q = (u, d). The blocks are different for up and down type squarks. We have

M2uLLM2uLLM2uLL =

12v22 h

Uh

TU +M

2Q +

16(4m2W m2Z) cos 2

M2uRRM2uRRM2uRR =

12v22 h

TUh

U +M

2U +

23(m2Z m2W ) cos 2

M2uLRM2uLRM2uLR =

v22AU

v12hU

M2uRLM2uRLM2uRL = M

2uLRM2uLRM2uLR

(120)

and

M2dLL

M2dLL

M2dLL

=1

2v21 h

Dh

TD +M

2Q 16 (2m2W +m2Z) cos 2

M2dRR

M2dRR

M2dRR

= 12v21 h

TDh

D +M

2D 13(m2Z m2W ) cos 2

M2dLR

M2dLR

M2dLR

=v12AD

v22hD

M2dRL

M2dRL

M2dRL

= M2dLR

M2dLR

M2dLR

(121)


q = RqRqRq q (122)

which implies

qi = RqRqRqji qj (123)


RqRqRq (MdiagqM

diagqMdiagq

)2RqRqRq =MqMqMq

2 (124)

In many applications the matrices in Eq. 119 are real and therefore the rotation matrices

RqRqRq are orthogonal matrices.

20

6 Couplings in the MSSM

In this section we give a list of all the MSSM couplings. The following table is a guide.

Name Type Equation

Gauge VVV Eq. (128)

Self-Interaction VVVV Eq. (130), Eq. (131)

3-Point Gauge V ff Eq. (140), Eq. (150)

Coupling V f f

V Eq. (140), Eq. (150)

V HH

3-Point Higgs Hff Eq. (183)

Coupling Hff Eq. (192)

H Eq. (177)

HV V

HHH Eq. (203)

Other 3-Point ff Eq. (168)

4-Point V V f f

Coupling HHV V

HHHH Eq. (209)

f fHH

ff f f

Ghost V

H

Table 8: Couplings in the MSSM. V , f f and H are generic names. In particular H

includes the Goldstone bosons.

6.1 Gauge Self-Interactions

The gauge sector of the MSSM is exactly the same as in the SM. We present it here both

for completeness and to fix our notation.

6.1.1 V V V

From Eq. 39 we get

LV V V = 12 g(W

a W a

)abcW bW c (125)

Now using

W 3 = sin WA + cos WZ

21

W =W 1 iW 2

2(126)

we obtain

L = i e[ (

W W

)W+A (W+ W+ )WA

(A A)W+W]

+i g cos W

[ (W

W

)W+ Z (W+ W+ )W Z

(Z Z)W+W]

(127)

which gives the following Feynman rule for the vertices,

i gV

[g(p2 p3) + g(p3 p1) + g(p1 p2)

](128)

V

W+

Wp1

p2

p3

where V = A(Z) and gA = e, gZ = g cos W .

6.1.2 V V V V

For the quartic selfinteractions we have

LV V V V = 12 g2[(W+ W

+) (W W

) (W+ W)2]+g2 cos2 W

[W+ W

Z

Z W+ W ZZ]

+e2[W+ W

A

A W+ WAA]

+e g cos W[W+ W

(Z

A + ZA) 2W+ WAZ]

(129)

which gives the following Feynman rules for the quartic vertices,

i g2[2 gg gg gg] (130)

W+

W+

W+

W+

i gV gV[gg + gg 2 gg] (131)

W+

W+

V

V

22

where V = A(Z) and gA = e, gZ = g cos W .

6.2 Charged Current Couplings

Using two component spinors and following the notation of ref. [5] the relevant part of

the Lagrangian can be written as

L = gW[(3 + 3

) 1

2

(H02

H+2 + H1

H01

) 1

2

(Li

Li + dLi

uLi)]

+ gW+

[(+ 3 3

) 1

2

(H+2

H02 + H01

H1)

12

( Li

Li + uLi

dLi)]

(132)

To obtain the couplings in four component notation we first write Eq. 132 in terms of the

mass eigenstates in two component notation, Fi and F0i . In order to do that we recall

that for the neutralinos

i = NNNi1F 0i i = NNN i1F 0i i3 = NNNi2F 0i i3 = NNN i2F 0iH01 = N

NNi3F 0i H01 = NNN i3F

0i

H02 = NNNi4F 0i H

02 = NNN i4F

0i

(133)

while for the charginos

i = UUUj1Fj i = UUU j1FjH1 = U

UUj2Fj H

1 = UUU j2F

j

i+ = V V V j1F+j i+ = VVV j1F+jH+2 = V

V V j2F+j H

+2 = VVV j2F

+j

(134)

We obtain then

L = gW[F 0i

F+j

(NNN i2V

V V j1 12NNN i4V

V V j2

)+ Fj

F 0i

(NNNi2UUU j1 1

2NNNi3UUU j2

) 1

2

(Li

Li + dLi uLi

)]+ gW+

[F+j

F 0i

(NNNi2VVV j1 1

2NNNi4VVV j2

)+ F 0i

Fj

(NNN i2UUUj1 1

2NNN i3U

UUj2

) 1

2

(Li

Li + dLi uLi

)](135)

23

Finally using

F 0i F+j = j PR0i

Fj F 0i =

j

PL0i

F 0i Fj =

0i

PLj

F+j F 0i = 0i PRj

Li Li =

Li

PLLi

Li Li = Li

PLLi (136)

we get

L = gW[j

(OLjiPL +O

RjiPR

)0i

12

(Li

PLLi + d

Li

PLuLi

)]+ gW+

[0i

((OLji)

PL + (ORji)

PR)j

12

( Li

PLLi + u

Li

PLdLi

)](137)

where

OLji =

(NNNi2UUU j1 1

2NNNi3UUU j2

)ORji =

(NNN i2V V V j1 + 1

2NNN i4V

V V j2

)(138)

Finally we rotate the leptons and quarks to the mass eigenstates using the relations6

uLi = RuLRuLRuLijuLj u

Li = uLj (R

uLRuLRuL)ij

dLi = RdLRdLRdLijdLj d

Li = dLj

(RdLRdLRdL

)ij

Li = RLRLRLijLj

Li = Lj

(RLRLRL

)ij

Li = RLRLRLijLj

Li = Lj (R

LRLRL)ij

(139)

to get

L = gW[j

(OLjiPL +O

RjiPR

)0i

12

(Li

PLLi + VVVCKM

ijdLj

PLuLi)]

+ gW+

[0i

((OLji)

PL + (ORji)

PR)j

12

(Li

PLLi + VVVCKM

ij uLiPLdLj

)](140)

where the CKM matrix VVV CKM was defined in Eq. 106.

6Notice that as the neutrinos are massless we can rotate them by the same matrix as the leptons.

Then the charged current for neutrinos and leptons will remain diagonal.

24

6.3 Neutral Current Couplings



L = 12

(gW 3 + g

B) (

H+2 H+2 H01 H1

) gW 3

(+ + )(141)

12

(gW 3 + gB) (H02 H02 H01 H01) (142)+1

2

(gW 3 + gB) Li Li + 12 (gW 3 + gB) Li Li gB cLi cLi(143)+1

2

(gW 3 13gB) uLi uLi + 12 (gW 3 13gB) dLi dLi (144)+2

3gB ucLi

ucLi 13 gB dcLi dcLi (145)To obtain the couplings in four component notation we first write Eq. 145 in terms of the

mass eigenstates in two component notation, Fi and F0i . We also use

gW 3 = eA +g

cos W

(1 sin2 W

)Z (146)

gB = eA gcos W

sin2 WZ (147)

We get in two component notation

L = eA[VVV ikV

V V jkF+i

F+j UUU ikUUUjkFi Fj Li Li + cLi cLi

+23uLi

uLi 13 dLi dLi 23 ucRi ucRi + 13 dcRi dcRi]

+g

cos WZ

[12(NNN i4N

NNj4 NNN i3NNNj3) F 0i F 0j

+(12UUU i2U

UUj2 + UUU i1UUUj1 sin2 WUUU ikUUUjk)Fi

Fj

+(1

2VVV i2V

V V j2 VVV i1V V V j1 + sin2 WVVV ikV V V jk)F+i

F+j

12 Li

Li + (12 sin2 W ) Li Li + sin2 W cLi cLi

+(12+ 2

3sin2 W )uLi

uLi + (12 1

3sin2 W ) dLi

dLi

+ 23sin2 W u

cLi

ucLi 13 sin2 W dcLi dcLi]

(148)

Now using the unitarity of the diagonalization matrices we get, still in two component

spinor notation,

L = eA[ (

Fi Fi F+i F+i

)+(Li

Li cLi cLi)

25

23

(uLi

uLi ucLi ucLi)+ 1

3

(dLi

dLi dcLi dcLi)]

+g

cos WZ

[12(NNN i4N

NNj4 NNN i3NNNj3) F 0i F 0j

+(12UUU i2U

UUj2 +UUU i1UUUj1 sin2 W ij)Fi

Fj

+(1

2VVV i2V

V V j2 VVV i1V V V j1 + sin2 W ij)F+i

F+j

12 Li

Li + (12 sin2 W ) Li Li + sin2 W cLi cLi

+(12+ 2

3sin2 W )uLi

uLi + (12 1

3sin2 W ) dLi

dLi

+ 23sin2 W u

cLi

ucLi 13 sin2 W dcLi dcLi]

(149)

Finally we get in four component notation7

L = eA[i

i + i i 23 ui ui + 13 di di

]+

g

cos WZ

[120i

(OLij

PL +ORijPR

)0j +

i

(OLij

PL +ORijPR

)j

+12i

(1

2PL +

12PR)i

+

f=,u,d

f i

[(If3 +Qf sin2 W

)PL +Q

f sin2 W PR

]fi

](150)

where

OLij = 1

2(NNN i4N

NNj4 NNN i3NNNj3)

ORij = (OLji)

OLij = 1

2UUU i2U

UUj2 +UUU i1UUUj1 sin2 W ij

ORij = 1

2VVV j2V

V V i2 + VVV j1V V Vi1 sin2 W ij (151)

and If3 and Qf are, respectively, the weak isospin and the charge of fermion f = , u, d.

6.4 3-point gauge boson couplings to scalars

In this section we give the 3-point couplings of gauge bosons with the scalars.

7We can see that the neutral current interaction of the leptons and quarks is diagonal in generation

space after we perform the rotation into the mass eigenstates.

26

6.4.1 Gauge boson couplings to sfermions (V f f)

We write the Lagrangian as

L =i gXYZff

f XfY Z

+ i eQf fXfXA

+[i gXY

Wfff XfYW

+ h.c]

(152)

where in the last term Qf Qf = 1. We have

gXYZff

= gcos W

[RRR

(f)Xi RRR

(f) Y i

(T f3 sin2 WQf

)+RRR

(f)X,i+3RRR

(f) Y,i+3

( sin2 WQf)]gXYWff

= g2RRR

(f)Xi RRR

(f) Y i (153)

where i = 1, 2, 3. This in turn corresponds to the following Feynman rules.

i (p1 + p2) gXY

Zff(154)

fY

f XZ

p1

p2

i eQf (p1 + p2) (155)

fX

f XA

p1

p2

i (p1 + p2) gXYWff

(156)

fY

f XW

p1

p2

6.4.2 Gauge boson couplings to Higgs (V HH)

6.5 4-point gauge boson couplings to scalars

6.5.1 Gauge boson couplings to sfermions (V V f f)

6.5.2 Gauge boson couplings to Higgs (V V HH)

6.6 Scalar couplings to fermions

6.6.1 Charged scalars couplings to fermions



L = Lgauge + LY ukawa (157)

27

where the gauge part is determined by the gauge group and quantum numbers of the

matter multiplets and is given by

Lgauge = H+1[g2(i3) H1 +

g2(i) H1 g (i) H01

]+ H+2

[ g

2(i3) H+2

g2(i) H+2 g (i+) H02

]+ Li

[g2(i3) Li +

g2(i) Li g (i) Li

]+ Ri

[g

2 (i) cLi

]+ uLi

[ g

2(i3) uLi

g

32(i) uLi g (i+) dLi

]+ uRi

[4g

32(i) ucLi

]+ dLi

[+

g2(i3) dLi

g

32(i) dLi g (i) uLi

]+ dRi

[ 2g

32(i) dcLi

]+ h.c. (158)

The Yukawa part is derived from the superpotential W by use of Eq. 30. In order to use

this equation it is helpful to write down explicitly the superpotential. We have

W = (hE)ij L1i RjH21 (hD)ij Q1i DjH21 (hU)ij Q2i UjH12+ (hE)ij L

2i RjH

11 + (hD)ij Q

2i DjH

11 + (hU)ij Q

1i UjH

22 (159)

Then in two component spinor notation the Yukawa part of the couplings of charged

scalar to fermions is

LY ukawa = (hE)ij LicLjH01 (hE)ij RjLiH01 + (hE)ij Rj LiH1 + (hE)ij H1 LicLj (hD)ij dLidcLjH01 (hD)ij dRjdLiH01 + (hD)ij dRjuLiH1 + (hD)ij H1 uLidcLj (hU)ij uLiucLjH02 (hU )ij uRjuLiH02 + (hU)ij uRjdLiH+2 + (hU)ij H+2 dLiucLj+ (hD)ij uLid

cLjH

1 + (hU)ij dLiu

cLjH

+2

+h.c. (160)

Now this can be written in four component spinor notation in terms of the mass eigen-

states. We will do this separately for Lgauge and LY ukawa. We obtain8

Lgauge = H1[(

g2UUUA2 (NNNB2 + tan WNNNB1) g UUUA1NNNB3

)APR

0B

]+ H2

[( g

2VVV A2 (NNN

B2 + tan WNNN

B1) g VVV A1NNNB4

)APL

0B

]8We still keep the unrotated fields for the leptons and quarks. The final rotation will be done in

Section 6.6.3.

28

+ RRRX,i X

[g2(NNNA2 + tan WNNN

A1)

0APL

i g UUU B1 cBPL i

]+ RRRX,i+3

X

[g2 tan W NNNA1 0APR

i

]+ RuRuRuX,i u

X

[ g

2

(NNNA2 +

13tan WNNN

A1

)0APLu

i g VVV B1 BPLdi

]+ RuRuRuX,i+3 u

X

[4

3

g2tan W NNNA1 0APRu

i

]+ RdRdRdX,i d

X

[g2

(NNNA2 13 tan WNNNA1

)0APLd

i g UUU B1 cBPLui

]+ RdRdRdX,i+3 d

X

[23

g2tan W NNNA1 0APRd

i

]+ h.c. (161)

and

LY ukawa = (hE)ijNNN A3(RRR

X,i X

jPL

0A +R

RRX,j+3 X

0APL

i

)+ (hE)ij

(UUU A2R

RRX,j+3 X

cAPL

i +H

1

jPL

i

) (hD)ijNNNA3

(RdRdRd

X,i dX d

jPL

0A +R

dRdRdX,j+3 dX

0APLd

i

)+ (hD)ij

(UUU A2R

dRdRdX,j+3 dX

cAPLu

i +H

1 d

jPLu

i

) (hU)ijNNNA4

(RuRuRu

X,i uX u

jPL

0A +R

uRuRuX,j+3 uX

0APLu

i

)+ (hU)ij

(VVV A2R

uRuRuX,j+3 uX

APLd

i +H

+2 u

jPLd

i

)+ (hD)ij UUU

A2R

uRuRuX,i uX d

jPL

A + (hU)ij VVV

A2R

dRdRdX,i dX u

jPL

cA

+ h.c. (162)

In Section 6.6.3 we will give the final formulas.

6.6.2 Neutral scalars couplings to fermions



L = Lgauge + LY ukawa (163)where the gauge part is

Lgauge =(H01) [ g

2(i3) H01 +

g2(i) H01 g (i+) H1

]29

+(H02) [ g

2(i3) H02

g2(i) H02 g (i) H+2

]+ i

[ g

2(i3) i + g

2(i) i g (i+) i

]+ h.c. (164)

and the Yukawa part is

LY ukawa = (hE)ij LicLjH1 (hE)ij H01LicLj (hD)ij H01dLidcLj (hU)ij H02uLiucLj+h.c. (165)

Now this can be written in four component spinor notation in terms of the mass eigen-

states. We will do this separately for Lgauge and LY ukawa. We obtain

Lgauge =(H01) [ g

2

(NNN A2 tan WNNN A1

)NNN B3 0APL

0B gVVV A1UUU B2 APLB

]+(H02) [ g

2

(NNN A2 tan WNNN A1

)NNN B4 0APL

0B gVVV A2UUU B1 APLB

]+RRRX,i

X

[ g

2

(NNNA2 tan WNNN A1

)0APL

i gVVV A1 APLi

]+h.c. (166)

and

LY ukawa = (hE)ij H01 jPLi (hD)ij H01 djPLdi (hU)ij H02 ujPLui+ (hE)ij UUU

A2R

RRX,i X jPLA

+h.c. (167)

In Section 6.6.3 we will give the final formulas.

6.6.3 Final formulas for the scalar couplings to fermions

In this Section we present the final formulas for the scalar couplings to fermions. To be

more precise we are going to only to write down those for the couplings fermionsfermion

chargino and fermionsfermionneutralino. These leave out the couplings of the neutral

and charged Higgs bosons with the fermions. These can be read from Eqs. (161), (162),

(166) and (167) not forgetting that we still have to perform the rotation into the Higgs

bosons mass eigenstates.

We interaction lagrangian can be written as

L = X i[CL()iAXPL + C

R()iAXPR

]A + X i

[CL()iAXPL + C

R()iAX PR

]cA

30

+ uX di

[CL(d)iAXPL + C

R(d)iAX PR

]A + dX ui

[CL(u)iAX PL + C

R(u)iAX PR

]cA

+ fX f i

[N

L(f)iAX PL +N

R(f)iAX PR

]0A

+ h.c. (168)

where the indices i, A,X apply to generations, charginos (or neutralinos) and sfermions,

respectively. All repeated indices are summed over and f = , , u, d. The coefficients C

and N can be read from Eqs. (161), (162), (166) and (167). We list them below. CL()iAX = (hE)kj UUU

A2R

RRX,kR

RRRRRij

CR()iAX = gVVV A1RRR

X,kR

LRLRLik

(169)

CL()iAX = 0

CR()iAX = gUUUA1RRR

X,kR

LRLRLik + (hE)

kj UUUA2R

RRX,j+3R

LRLRLik

(170)

CL(d)iAX = (hD)kj UUU

A2R

uRuRuX,kR

dRRdRRdRij

CR(d)iAX = gVVV A1RuRuRu

X,kR

dLRdLRdLik + (h

U )kj VVV A2R

uRuRuX,j+3R

dLRdLRdLik

(171)

CL(u)iAX = (hU )kj VVV

A2R

dRdRdX,kR

uRRuRRuRij

CR(u)iAX = gUUUA1RdRdRd

X,kR

uLRuLRuLik + (h

D)kj UUUA2R

dRdRdX,j+3R

uLRuLRuLik

(172)

and for the N s

N

L()iAX = g

2 tan W NNN

A1R

RRX,k+3R

RRRRRik (hE)kj NNNA3RRR

X,kR

RRRRRij

NR()iAX =

g2

(NNNA2 + tan WNNNA1

)RRR

X,kR

LRLRLik (hE)kj NNNA3RRR

X,j+3R

LRLRLik

(173)

N

L()iAX = 0

NR()iAX =

g2

(NNNA2 tan WNNNA1

)RRR

X,kR

LRLRLik

(174)

N

L(u)iAX =

4

3

g2tan W NNN

A1R

uRuRuX,k+3R

uRRuRRuRik (hU)kj NNN A4RuRuRu

X,kR

uRRuRRuRij

NR(u)iAX =

g2

(NNNA2 +

13tan WNNNA1

)RuRuRu

X,kR

uLRuLRuLik (hU)kj NNNA4RuRuRu

X,j+3R

uLRuLRuLik

(175)

N

L(d)iAX =

2

3

g2tan W NNN

A1R

dRdRdX,k+3R

dRRdRRdRik (hD)kj NNN A3RdRdRd

X,kR

dRRdRRdRij

NR(d)iAX =

g2

(NNNA2 13 tan WNNNA1

)RdRdRd

X,kR

dLRdLRdLik (hD)kj NNNA3RdRdRd

X,j+3R

dLRdLRdLik

(176)

31

6.6.4 Final formulas for the Higgs couplings to SUSY fermions

In this Section we present the final formulas for the Higgs couplings to supersymmetric

fermions (charginos and neutralinos). These can be read from Eqs. (161), (162), (166)

and (167).

We interaction lagrangian can be written as

L =A[D

L(S+)ABi PL +D

R(S+)ABi PR

]0B S

i + h.c.

+1

20A

[EL(S0)ABi PL + E

R(S0)ABi PR

]0B S

0i +

A

[FL(S0)ABi PL + F

R(S0)ABi PR

]B S

0i

+1

20A

[EL(P 0)ABi PL + E

R(P 0)ABi PR

]0B P

0i +

A

[FL(P 0)ABi PL + F

R(P 0)ABi PR

]B P

0i (177)

where the indices i, A,B apply to the Higgs, and charginos (or neutralinos), respectively.

All repeated indices are summed over. The coefficients D,E and F can be read from

Eqs. (161), (162), (166) and (167). We list them below 9.D

L(S+)ABi =

[ g

2VVV A2 (NNN

B2 + tan WNNN

B1) g VVV A1NNN B4

]RRR

(S+) i2

DR(S+)ABi =

[g2UUUA2 (NNNB2 + tan WNNNB1) g UUUA1NNNB3

]RRR

(S+) i1

(178)

EL(S0)ABi =

12

[ gNNNA2NNN B3 + gNNN A1NNN B3 gNNNB2NNN A3 + gNNNB1NNN A3

]RRR

(S0)i1

+12

[+ gNNNA2NNN

B4 gNNN A1NNNB4 + gNNNB2NNN A4 gNNN B1NNNA4

]RRR

(S0)i2

ER(S0)ABi =

(EL(S0)ABi

)(179)

EL(P 0)ABi = i2

[ gNNNA2NNNB3 + gNNNA1NNNB3 gNNNB2NNNA3 + gNNN B1NNNA3

]RRR

(P 0)i1

i2

[+ gNNNA2NNN

B4 gNNNA1NNNB4 + gNNNB2NNN A4 gNNN B1NNNA4

]RRR

(P 0)i2

ER(P 0)ABi =

(EL(P 0)ABi

)(180)

FL(S0)ABi = g2

(VVV A1UUU

B2RRR

(S0)i1 + VVV

A2UUU

B1RRR

(S0)i2

)FR(S0)ABi = g2

(VVV B1UUUA2RRR

(S0)i1 + VVV B2UUUA1RRR

(S0)i2

) (181)9The rotation matrices RRR(S

0) and RRR(P0) are real orthogonal matrices. However SPheno[9] considers

RRR(S+) to be in general complex, so we follow this convention.

32

FL(P 0)ABi = i

g2

(VVV A1UUU

B2RRR

(P 0)i1 + VVV

A2UUU

B1RRR

(P 0)i2

)FR(P 0)ABi = i g2

(VVV B1UUUA2RRR

(P 0)i1 + VVV B2UUUA1RRR

(P 0)i2

) (182)6.6.5 Final formulas for the Higgs couplings to SM fermions

In this Section we present the final formulas for the Higgs couplings to Standard Model

fermions. These can be read from Eqs. (162) and (167).

We write the interaction lagrangian as 10

L =Si j(GL()ijk PL +G

R()ijk PR

)k + S

i dj

(GL(qq)ijk PL +G

R(qq)ijk PR

)uk

+ i

(H

L()ijk PL +H

R()ijk PR

)i S

0i k

+ ui

(H

L(u)ijk PL +H

R(u)ijk PR

)uj S

0k + di

(H

L(d)ijk PL +H

R(d)ijk PR

)dj S

0k

+ i

(IL()ijk PL + I

R()ijk PR

)j P

0k

+ ui

(IL(u)ijk PL + I

R(u)ijk PR

)uj P

0k + di

(IL(d)ijk PL + I

R(d)ijk PR

)dj P

0k (183)

The coefficients G,H and I can be read from Eqs. (162) and (167). We list them below. GL(qq)ijk = (hD)kjRRR

(S+) i1 RLRLRL

(u) kk RRRRRR

(d)jj

GR(qq)ijk = (h

U)jkRRR

(S+) i2 RRRRRR

(u) kk RLRLRL

(d)jj

(184)

GL()ijk = (hE)kjRRR

(S+) i1 RLRLRL

() kk RRRRRR

()jj

GR()ijk = 0

(185)

H

L()ijk = 12(hE)jiRRR

(S0)k1 RLRLRL

() jj RRRRRR

()ii

HR()ijk =

(H

L()ijk

) (186)

HL(d)ijk = 12(hD)jiRRR

(S0)k1 RLRLRL

(d) jj RRRRRR

(d)ii

HR(d)ijk =

(H

L(d)ijk

) (187)10The order of the fields was chosen to be in agreement with SPheno[9] conventions.

33

H

L(u)ijk = 12(hU)jiRRR

(S0)k1 RLRLRL

(u) jj RRRRRR

(u)ii

HR(u)ijk =

(H

L(u)ijk

) (188)

IL()ijk = i 12(hE)jiRRR

(S0)k1 RLRLRL

() jj RRRRRR

()ii

IR()ijk =

(IL()ijk

) (189)

IL(d)ijk = i 12(hD)jiRRR

(S0)k1 RLRLRL

(d) jj RRRRRR

(d)ii

IR(d)ijk =

(IL(d)ijk

) (190)

IL(u)ijk = i 12(hU)jiRRR

(S0)k1 RLRLRL

(u) jj RRRRRR

(u)ii

IR(u)ijk =

(IL(u)ijk

) (191)

6.7 Trilinear scalar couplings with Higgs bosons

In this section we will show the trilinear scalar couplings envolving the Higgs boson and the

corresponding Goldstone bosons. The Higgs self interactions will be left for section 6.11.

6.7.1 Higgs Sfermion Sfermion

We write the lagrangian as

L =g(S+du)ijk S+i djuk + g(S+)

ijk S+i j

k + h.c.

+ g(S0dd)ijk S

0i djd

k + g

(S0uu)ijk S

0i uju

k + g

(S0)ijk S

0i j

k + g

(S0)ijk S

0i j

k

+ g(P 0dd)ijk P

0i djd

k + g

(P 0uu)ijk P

0i uju

k + g

(P 0)ijk P

0i j

k + g

(P 0)ijk P

0i j

k (192)

We list below these couplings.

g(S+du)ijk =

[vd2

((hDh

D)jk

g2

2jk

)RRR

(S) i1 +

vu2

((hUh

U)jk

g2

2jk

)RRR

(S) i2

]RRR

(d) jj RRR

(u)kk

+[(hU)jkRRR

(S) i1 + (AU)jkRRR

(S) i2

]RRR

(d) jj RRR

(u)k,k+3

+[(hD)kjRRR

(S) i2 + (A

D)kjRRR

(S) i1

]RRR

(d) j,j+3RRR

(u)kk

34

+[vu2RRR

(S) i1 +

vd2RRR

(S) i2

](hDhU)jk RRR

(d) j,j+3RRR

(u)k,k+3 (193)

g(S+)ijk =

[vd2

((hEh

E)jk

g2

2jk

)RRR

(S) i1

vu2

g2

2jk RRR

(S) i2

]RRR

() jj RRR

()kk

+[(hD)kjRRR

(S) i2 + (A

E)kjRRR

(S) i1

]RRR

() j,j+3RRR

()kk (194)

g(S0dd)ijk =

[vd

(1

12

(3g2+g2

)jk(hDhD)jk

)RRR

(S0)i1

vu12

(3g2+g2

)jkRRR

(S0)i2

]RRR

(d) jj RRR

(d)kk

+

[ 1

2(AD)jkRRR

(S0)i1 +

2(hD)jkRRR

(S0)i2

]RRR

(d) jj RRR

(u)k,k+3

+

[ 1

2(AD)kjRRR

(S0)i1 +

2(hD)kjRRR

(S0)i2

]RRR

(d) j,j+3RRR

(u)kk

+

[vd

(g2

6 (hDhD)j,k

)RRR

(S0)i1 vu

g2

6j,kRRR

(S0)i2

]RRR

(d) j,j+3RRR

(u)k,k+3 (195)

g(S0uu)ijk =

[ vd12

(3g2g2) jkRRR(S0)i1 +vu( 112 (3g2g2) jk(hUhU )jk

)RRR

(S0)i2

]RRR

(u) jj RRR

(u)kk

+

[2(hU)jkRRR

(S0)i1

12(AU)jkRRR

(S0)i2

]RRR

(d) jj RRR

(u)k,k+3

+

[2(hU)kjRRR

(S0)i1

12(AU)kjRRR

(S0)i2

]RRR

(d) j,j+3RRR

(u)kk

+

[vd g

2

3j,kRRR

(S0)i1 + vu

(g2

3 (hUhU)j,k

)RRR

(S0)i2

]RRR

(d) j,j+3RRR

(u)k,k+3 (196)

g(S0)ijk =

[vd

(1

4

(g2g2) jk(hEhE)jk)RRR(S0)i1 vu4 (g2g2) jkRRR(S0)i2

]RRR

(d) jj RRR

(d)kk

+

[ 1

2(AE)jkRRR

(S0)i1 +

2(hE)jkRRR

(S0)i2

]RRR

(d) jj RRR

(u)k,k+3

+

[ 1

2(AE)kjRRR

(S0)i1 +

2(hE)kjRRR

(S0)i2

]RRR

(d) j,j+3RRR

(u)kk

35

+[vd

(g2

2 (hEhE)j,k

)RRR

(S0)i1 vu

g2

2j,kRRR

(S0)i2

]RRR

(d) j,j+3RRR

(u)k,k+3 (197)

g(S0)ijk =

[vd4

(g2 + g2

)jkRRR

(S0)i1 +

vu4

(g2 + g2

)jkRRR

(S0)i2

]RRR

() jj RRR

()kk (198)

g(P 0dd)ijk = i

[12(AD)jkRRR

(P 0)i1 +

2(hD)jkRRR

(P 0)i2

]RRR

(d) jj RRR

(u)k,k+3

+ i

[12(AD)kjRRR

(P 0)i1 +

2(hD)kjRRR

(P 0)i2

]RRR

(d) j,j+3RRR

(u)kk (199)

g(P 0uu)ijk = i

[2(hU)jkRRR

(P 0)i1 +

12(AU)jkRRR

(P 0)i2

]RRR

(d) jj RRR

(u)k,k+3

+ i

[2(hU)kjRRR

(P 0)i1 +

12(AU)kjRRR

(P 0)i2

]RRR

(d) j,j+3RRR

(u)kk (200)

g(P 0)ijk = i

[12(AE)jkRRR

(P 0)i1 +

2(hE)jkRRR

(P 0)i2

]RRR

(d) jj RRR

(u)k,k+3

+ i

[12(AE)kjRRR

(P 0)i1 +

2(hE)kjRRR

(P 0)i2

]RRR

(d) j,j+3RRR

(u)kk (201)

g(P 0)ijk = 0 (202)

36

6.8 Quartic scalar couplings with Higgs bosons

In this section we will show the quartic scalar couplings envolving the Higgs boson and the

corresponding Goldstone bosons. The Higgs self interactions will be left for section 6.11.

6.9 Quartic sfermion interactions

In this section we will show the quartic scalar couplings envolving the sfermions among

themselves.

6.10 Higgs couplings with the gauge bosons

6.10.1 Higgs Gauge Boson Gauge Boson

6.10.2 Higgs Higgs Gauge Boson Gauge Boson

6.11 Higgs boson self interactions

In this section we will show the self couplings envolving the Higgs boson among themselves.

We give separately the 3-point and 4-point interactions.

6.11.1 Higgs Higgs Higgs

For the 3-point Higgs boson self interactions we write the Lagrangian as

L =16gS

0S0S0

i,j,k S0i S

0jS

0k +

1

2gS

0P 0P 0

i,j,k S0i P

0j P

0k + g

S0S+S

i,j,k S0i S

+j S

k (203)

where 1/6 and 1/2 are symmetry factors for the case of identical particles, chosen in such

a way that, for instance, the coupling gS0S0S0

i,j,k11 is really the Feynman rule for the vertice

(after multiplying by i as usual), that is,

gS0S0S0

i,j,k 3L

S0i S0j S

0k

(204)

and similarly for the other cases. To simplify the notation we write the values for the

unrotated couplings defined by the relations

gS0S0S0

i,j,k = gS0S0S0

i,j,k RRR(S0)i,i RRR

(S0)j,j RRR

(S0)k,k (205)

and similar relations. We get

gS0S0S0

1,1,1 =3g2vd4

3g2vd4

gS0S0S0

1,1,2 =g2vu4

+g2vu4

11We have that, obviously, gS0S0S0

i,j,k is completely symmetric and gS0P 0P 0

i,j,k is symmetric in the last two

entries.

37

gS0S0S0

1,2,1 =g2vu4

+g2vu4

gS0S0S0

1,2,2 =g2vd4

+g2vd4

gS0S0S0

2,1,1 =g2vu4

+g2vu4

gS0S0S0

2,1,2 =g2vd4

+g2vd4

gS0S0S0

2,2,1 =g2vd4

+g2vd4

gS0S0S0

2,2,2 =3g2vu4

3g2vu4

(206)

gS0P 0P 0

1,1,1 =g2vd4

g2vd4

gS0P 0P 0

1,1,2 =0

gS0P 0P 0

1,2,1 =0 gS0P 0P 0

1,2,2 =g2vd4

+g2vd4

gS0P 0P 0

2,1,1 =g2vu4

+g2vu4

gS0P 0P 0

2,1,2 =0

gS0P 0P 0

2,2,1 =0 gS0P 0P 0

2,2,2 =g2vu4

g2vu4

(207)

gS0S+S

1,1,1 =g2vd4

g2vd4

gS0S+S

1,1,2 =g2vu4

gS0S+S

1,2,1 =g2vu4

gS0S+S

1,2,2 =g2vd4

+g2vd4

gS0S+S

2,1,1 =g2vu4

+g2vu4

gS0S+S

2,1,2 =g2vd4

gS0S+S

2,2,1 =g2vd4

gS0S+S

2,2,2 =g2vu4

g2vu4

(208)

6.11.2 Higgs Higgs Higgs Higgs

For the 4-point Higgs boson self interactions we write the Lagrangian as

L = 14!gS

0S0S0S0

i,j,k,l S0i S

0jS

0kS

0l +

1

(2!)2gS

0S0P 0P 0

i,j,k,l S0i S

0jP

0kP

0l +

1

4!gP

0P 0P 0P 0

i,j,k,l P0i P

0j P

0kP

0l (209)

+1

2!gS

0S0S+S

i,j,k,l S0i S

0jS

+k S

l +

1

2!gP

0P 0S+0S

i,j,k,l P0i P

0j S

+k S

l +

1

(2!)2gS

+SS+S

i,j,k,l S+i S

j S

+k S

l

We get for the unrotated couplings

gS0S0S0S0

1,1,1,1 =3g2

4 3g

2

4, gS

0S0S0S0

1,1,1,2 = gS0S0S0S0

1,1,2,1 =0, gS0S0S0S0

1,1,2,2 =g2

4+g2

4

gS0S0S0S0

1,2,1,1 =0, gS0S0S0S0

1,2,1,2 = gS0S0S0S0

1,2,2,1 =g2

4+g2

4, gS

0S0S0S0

1,2,2,2 =0

gS0S0S0S0

2,1,1,1 =0, gS0S0S0S0

2,1,1,2 = gS0S0S0S0

2,1,2,1 =g2

4+g2

4, gS

0S0S0S0

2,1,2,2 =0 (210)

38

gS0S0S0S0

2,2,1,1 =g2

4+g2

4, gS

0S0S0S0

2,2,1,2 = gS0S0S0S0

2,2,2,1 =0, gS0S0S0S0

2,2,2,2 =3g2

4 3g

2

4

gS0S0P 0P 0

1,1,1,1 =g2

4 g

2

4, gS

0S0P 0P 0

1,1,1,2 = gS0S0P 0P 0

1,1,2,1 =0, gS0S0P 0P 0

1,1,2,2 =g2

4+g2

4,

gS0S0P 0P 0

1,2,1,1 =0, gS0S0P 0P 0

1,2,1,2 = gS0S0P 0P 0

1,2,2,1 =0, gS0S0P 0P 0

1,2,2,2 =0,

gS0S0P 0P 0

2,1,1,1 =0, gS0S0P 0P 0

2,1,1,2 = gS0S0P 0P 0

2,1,2,1 =0, gS0S0P 0P 0

2,1,2,2 =0,

gS0S0P 0P 0

2,2,1,1 =g2

4+g2

4, gS

0S0P 0P 0

2,2,1,2 = gS0S0P 0P 0

2,2,2,1 =0, gS0S0P 0P 0

2,2,2,2 =g2

4 g

2

4(211)

gP0P 0P 0P 0

1,1,1,1 =3g2

4 3g

2

4, gP

0P 0P 0P 0

1,1,1,2 = gP 0P 0P 0P 0

1,1,2,1 =0, gP 0P 0P 0P 0

1,1,2,2 =g2

4+g2

4,

gP0P 0P 0P 0

1,2,1,1 =0, gP 0P 0P 0P 0

1,2,1,2 = gP 0P 0P 0P 0

1,2,2,1 =g2

4+g2

4, gP

0P 0P 0P 0

1,2,2,2 =0,

gP0P 0P 0P 0

2,1,1,1 =0, gP 0P 0P 0P 0

2,1,1,2 = gP 0P 0P 0P 0

2,1,2,1 =g2

4+g2

4, gP

0P 0P 0P 0

2,1,2,2 =0, (212)

gP0P 0P 0P 0

2,2,1,1 =g2

4+g2

4, gP

0P 0P 0P 0

2,2,1,2 = gP 0P 0P 0P 0

2,2,2,1 =0, gP 0P 0P 0P 0

2,2,2,2 =3g2

4 3g

2

4

gS0S0S+S

1,1,1,1 =g2

4 g

2

4, gS

0S0S+S

1,1,1,2 = gS0S0S+S

1,1,2,1 =0, gS0S0S+S

1,1,2,2 =g2

4+g2

4,

gS0S0S+S

1,2,1,1 =0, gS0S0S+S

1,2,1,2 = gS0S0S+S

1,2,2,1 =g2

4, gS

0S0S+S

1,2,2,2 =0,

gS0S0S+S

2,1,1,1 =0, gS0S0S+S

2,1,1,2 = gS0S0S+S

2,1,2,1 =g2

4, gS

0S0S+S

2,1,2,2 =0,

gS0S0S+S

2,2,1,1 =g2

4+g2

4, gS

0S0S+S

2,2,1,2 = gS0S0S+S

2,2,2,1 =0, gS0S0S+S

2,2,2,2 =g2

4 g

2

4(213)

gP0P 0S+S

1,1,1,1 =g2

4 g

2

4, gP

0P 0S+S

1,1,1,2 = gP 0P 0S+S

1,1,2,1 =0, gP 0P 0S+S

1,1,2,2 =g2

4+g2

4,

gP0P 0S+S

1,2,1,1 =0, gP 0P 0S+S

1,2,1,2 = gP 0P 0S+S

1,2,2,1 =g2

4, gP

0P 0S+S

1,2,2,2 =0,

gP0P 0S+S

2,1,1,1 =0, gP 0P 0S+S

2,1,1,2 = gP 0P 0S+S

2,1,2,1 =g2

4, gP

0P 0S+S

2,1,2,2 =0,

gP0P 0S+S

2,2,1,1 =g2

4+g2

4, gP

0P 0S+S

2,2,1,2 = gP 0P 0S+S

2,2,2,1 =0, gP 0P 0S+S

2,2,2,2 =g2

4 g

2

4(214)

gS+SS+S

1,1,1,1 = gS+SS+S

2,2,2,2 = g2

2 g

2

2

39

gS+SS+S

1,1,2,2 = gS+SS+S

1,2,2,1 = gS+SS+S

2,1,1,2 = gS+SS+S

2,2,1,1 =g2

4+g2

4

gS+SS+S

1,1,1,2 = gS+SS+S

1,1,2,1 = gS+SS+S

1,2,1,1 = gS+SS+S

1,2,1,2 = gS+SS+S

1,2,2,2 = 0

gS+SS+S

2,1,1,1 = gS+SS+S

2,1,2,1 = gS+SS+S

2,1,2,2 = gS+SS+S

2,2,1,2 = gS+SS+S

2,2,2,1 = 0 (215)

6.12 Ghost interactions

6.12.1 Ghost Ghost Gauge Boson

6.12.2 Ghost Ghost Higgs

40

A Changelog

24/6/2011

Corrected misprints in Eq. 161, Eq. 162, Eq. 168, Eq. 169 and Eq. 170. We

thank R. Fonseca for pointing out these.

23/11/2010

Added the 3-point and 4-point Higgs self-interaction in sections 6.11.1 and

6.11.2.

18/11/2010

Added the HiggsSfermionSfermion couplings of section 6.7.1

16/11/2010

Changed the convention in Eq. (161) to write the couplings for Hi instead of

H+i . This leads to Eq. (177) and Eq. (178) in agreement with SPheno.

15/11/2010

Changed the convention of the left-handed rotation matrices for leptons, Eq. (90).

Corrected a misprint in Eq. (161).

Changed notation in Eq. (168) for CL()iAX and C

R()iAX to be uniform with the

other terms.

Changed some of the couplings in Eqs. (169)-(176) to be consistent with the

previous changes.

All the couplings in Eqs (169)-(176) were checked against SPheno[9] with com-

plete agreement.

16/2/2005

sin2 and cos2 were exchanged in Eq. 78.

28/11/2003

Corrected misprints in Eq. 96, Eq. 170, Eq. 171 and Eq. 172.

27/10/2002

Changed Version Number to 1.6.xxx corresponding to the year 2002.

Changed minor points in notation in Eq. 62 and related expressions.

41

31/8/2001

Corrected Q = T3 + Y in Eq. 43.

17/6/2001

Introduced the V V V and V V V V couplings.

27/5/2001

Introduced the version numbering convention.

42

References

[1] OPAL Collaboration, L3 Collaboration, DELPHI Collaboration, ALEPH Collabo-

ration, LEP Electroweak Working Group, and SLD Heavy Flavor Group, preprint

CERN-PPE/97-154.

[2] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in Elementary Particle

Theory, ed. N. Svartholm, p. 367, Stockholm (1968); S.L. Glashow, J. Iliopoulos,

and L. Maiani, Phys. Rev. D2 (1970) 1285.

[3] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 75 (1995) 1028; D0 Collabora-

tion, S. Abachi et al., Phys. Rev. Lett. 77 (1997) 3303; Phys. Rev. Lett. 78 (1997)

3634.

[4] ALEPH Collaboration, R. Barate et al., Phys. Lett. B422 (1998)) 369; DELPHI

Collaboration, P. Abreu it al., Phys. Lett. B397 (1997) 158; Phys. Lett. B423

(1998) 194; L3 Collaboration, M. Acciari et al. Phys. Lett. B403 (1997) 168; Phys.

Lett. B413 (1997) 176; OPAL Collaboration, K. Ackerstaff et al. Phys. Lett. B397

(1997) 147; Eur. Phys. J. C2 (1998) 597.

[5] H.E. Haber and G.L. Kane, Phys. Rep. 117, 75 (1985).

[6] J.F. Gunion and H.E. Haber, Nucl. Phys. B 272, 1 (1986); erratum-ibid. B 402, 567

(1993).

[7] H. Haber et al., Phys. Rev. Lett. 66 (1991) 1815; R. Barbieri et al., Phys. Lett.

B258 (1991) 395; J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B262 (1991) 477.

[8] M. A. Daz, J. C. Romao, and J. W. F. Valle, Nucl. Phys. B524 (1998) 23.

[9] W. Porod, Comput. Phys. Commun. 153, 275 (2003) [arXiv:hep-ph/0301101].

43

Introduction and MotivationSUSY Algebra, Representations and Particle ContentSUSY AlgebraSimple Results from the AlgebraSUSY Representations

How to Build a SUSY ModelKinetic TermsSelf Interactions of the Gauge MultipletInteractions of the Gauge and Matter MultipletsSelf Interactions of the Matter MultipletSupersymmetry Breaking LagrangianRParity

The Minimal Supersymmetric Standard ModelThe Gauge Group and Particle ContentField Strengths and Covariant DerivativesThe Superpotential and SUSY Breaking LagrangianSymmetry BreakingThe Constrained Minimal Supersymmetric Standard Model

Mass Matrices in the MSSMThe Chargino Mass MatricesThe Neutralino Mass MatricesNeutral Higgs Mass MatricesCharged Higgs Mass MatricesLepton Mass MatricesQuark Mass MatricesSlepton Mass MatricesSneutrino Mass MatricesSquark Mass Matrices

Couplings in the MSSMGauge Self-InteractionsVVVVVVV

Charged Current CouplingsNeutral Current Couplings3-point gauge boson couplings to scalarsGauge boson couplings to sfermions (V)Gauge boson couplings to Higgs (VHH)

4-point gauge boson couplings to scalarsGauge boson couplings to sfermions (VV)Gauge boson couplings to Higgs (VVHH)

Scalar couplings to fermionsCharged scalars couplings to fermionsNeutral scalars couplings to fermionsFinal formulas for the scalar couplings to fermionsFinal formulas for the Higgs couplings to SUSY fermionsFinal formulas for the Higgs couplings to SM fermions

Trilinear scalar couplings with Higgs bosonsHiggs Sfermion Sfermion

Quartic scalar couplings with Higgs bosonsQuartic sfermion interactionsHiggs couplings with the gauge bosonsHiggs Gauge Boson Gauge BosonHiggs Higgs Gauge Boson Gauge Boson

Higgs boson self interactionsHiggs Higgs HiggsHiggs Higgs Higgs Higgs

Ghost interactionsGhost Ghost Gauge BosonGhost Ghost Higgs

Changelog

Documents

Mssm Model