vi Introduction
The branching ratio for these final states is of the order of 20% and they are the
second most likely final states for Higgs production at e
+
e
−
colliders in the LEP2
centre of mass energy. Their characteristics are a high value of missing energy and
the presence of two acoplanar jets. This is a very peculiar signature which places the
e
+
e
−
→ Hνν channel among the preferred ones for Higgs hunting.
This thesis is organised as follows. In the first chapter I will provide an exhaustive
description of the Higgs boson sector and its properties both in the Standard Model and
in the Minimal Supersymmetric extension of the Standard Model. Higgs production
mechanisms at LEP are examined.
The second chapter contains a brief description of the LEP accelerator and the DEL-
PHI detector. Since many publications exist in the literature I will give only a summary.
I dwell on the microvertex detector because I had the possibility to contribute to its
running during 1999 data taking.
Emphasis is placed in chapter 3 on the alignment of the DELPHI detector since
part of my PhD activities were dedicated to this task, which, during my PhD, has
constituted a challenge in the effort to improve the detector performances.
The Higgs search made in this thesis is pursued in two steps. The data collected
during 1998 at centre of mass energy of 189 GeV are analysed. First of all (chapter
4) a sequential series of cuts is applied in order to select a sample of events with an
increase in signal over background ratio. Then (chapter 5) the likelihood ratio method
is adopted to discriminate between signal and background events. There are two ir-
reducible background processes: e
+
e
−
→ Z
0
Z
0
with one Z
0
decaying into neutrinos,
and e
+
e
−
→ Z
0
γγ with the two photons in the beam pipe. The latter (which is called
the double radiative return to the Z
0
) represents a non-negligible background in the
e
+
e
−
→ Hνν channel. I was then motivated by a careful analysis of this process which
was not previously studied. As a result the first preliminary world measurement of the
cross section has been performed.
Since no evidence for a Higgs signal has been found, it is possible to set a lower
limit on the mass of the Higgs boson. In chapter 6 I will extensively described the
method used to define the limit and the limit itself will be computed. A detailed study
of systematic error sources has been done and the total systematic error has been
introduced in the limit derivation.
Finally, since at the moment of writing this thesis, 140 pb
−1
have been collected
during 1999 data taking, a preliminary analysis of these data has been performed
(chapter 7). Due to the Standard Model parameter fit indication of a light Higgs boson,
at each LEP energy step a great expectation is present and the data are immediately
analysed.
1Chapter 1
Phenomenology
In the framework of the Standard Model, it is foreseen the existence of a particle called
“Higgs boson”. The Standard Model without the Higgs boson is incomplete since it
predicts massless fermions and gauge bosons. Furthermore, the electroweak radiative
corrections to observables such as the W
±
and Z
0
boson masses would be infinite if
there were no Higgs boson. In this chapter I will describe why such particle has been
introduced, and I will review its general properties. The Standard Model has been
successful in predicting the properties of new particles and the structure of the basic
interactions. However there are some theoretical problems which cannot be solved
without the introduction of new physics. I will show the reasons why Supersimmatry
must be introduced and I will describe the properties of the Higgs boson in the frame-
work of Supersymmetry theory. The Higgs analysis, that is the subject of this thesys,
has been driven by properties here reviewed.
1.1 Standard Model
In this first section I briefly recall the properties of Standard Model. For a complete
review see for example [1].
Experimental observations of the weak interaction lead to classification of particles
into doublets and singlets of a representation of the symmetry group SU(2):
(
ν
e
e
)
L
(
ν
µ
µ
)
L
(
ν
τ
τ
)
L
e
R
µ
R
τ
R
(1.1)
for leptons, and
(
u
d
′
)
L
(
c
s
′
)
L
(
t
b
′
)
L
u
R
,d
′
R
c
R
,s
′
R
t
R
,b
′
R
(1.2)
2 1. Phenomenology
for quarks.
Indexes L and R stand for helicity Left and Right. The same classification is applied
to antiparticles with the exchange of L with R.
This classification reflects the existence of an underlying symmetry: the weak isospin
symmetry. The weak isospin is a conserved quantum number in weak interactions and
the SU(2) is a representation of isospin symmetry.
The index
′
(prime) indicates the eigenstates of the weak interaction: they are
connected with mass eigenstates by the Cabibbo-Kobayashi-Maskawa mixing matrix
(CKM):
d
′
s
′
b
′
=
V
ud
V
us
V
ub
V
cd
V
cs
V
cb
V
td
V
ts
V
tb
d
s
b
(1.3)
This matrix is described by complex elements, but unitarity constraint and the
arbitrarity in definition of quark fields leave only four free parameters: three real
angles and a complex phase.
Quantum numbers of elementary particles are summarised in tables 1.1 and 1.2
lepton T T
3
QY
ν
e
1/2 1/2 0 -1
e
−
L
1/2 -1/2 -1 -1
e
−
R
0 0 -1 -2
Table 1.1: Quantum numbers of leptons
quark T T
3
QY
u
L
1/2 1/2 2/3 1/3
d
L
1/2 -1/2 -1/3 1/3
u
R
0 0 2/3 4/3
d
R
0 0 -1/3 -2/3
Table 1.2: Quantum numbers of quarks
Hypercharge Y is related to electric charge Q and to weak isospin T by:
Q = T
3
+
Y
2
(1.4)
The hypercharge operator is the generator of U(1)
Y
symmetry group and isospin op-
erators are the generators of SU(2)
L
symmetry group. The ElectroWeak interaction
is a gauge theory based on SU(2)
L
× U(1)
Y
symmetry group. To have local gauge in-
variance, gauge fields are associated to these operators: a triplet W
a
a=1,2,3
is associated
to weak isospin, and a single field B is associated to hypercharge.
1.1. Standard Model 3
Quarks have an additional quantum number: the colour. It may have three different
values: red, yellow, blue. Colour charge cannot be observed: only colour singlet states
can be viewed
The Standard Model (SM) lagrangian is given by:
L
SM
= L
fermions
+ L
gauge
+ L
Higgs
+ L
mass
(1.5)
In the following sections each of the previous four contributions will be described
with particular interest for the Higgs field lagrangian.
1.1.1 Fermion fields lagrangian
L
fermions
= L
leptons
+ L
quarks
(1.6)
=
∑
families
[
(
ν
e
e
)
L
γ
µ
D
µL
(
ν
e
e
)
L
+ e
R
γ
µ
D
µR
e
R
]
(1.7)
+
∑
families
[
(
u d
)
L
γ
µ
D
µL
(
u
d
)
L
+ q
R
γ
µ
D
µR
q
R
]
where:
D
µL
= ∂
µ
− ig
$
T
$
W
µ
− ig
′
Y
2
B
µ
(1.8)
D
µR
= ∂
µ
− ig
′
Y
2
B
µ
are the covariant derivatives; f = f
†
γ
0
and f is the fermion field;
$
T = |T |$σ with
σ
a
a=1,2,3
Pauli matrices; γ
µ
are the Dirac matrices; g and g
′
are the coupling constants
of fermions to gauge fieldsW and B which appear in derivatives to maintain local gauge
invariance. W
a
a=1,2,3
and B are not the physical fields which are a linear combination
of them:
W
±
µ
=
1
√
2
(
W
1
µ
∓ iW
2
µ
)
(1.9)
(
A
µ
Z
µ
)
=
(
cos θ
W
sin θ
W
− sin θ
W
cos θ
W
)(
B
µ
W
3
µ
)
(1.10)
where
sin θ
W
=
g
′
√
g
2
+ g
′2
(1.11)
cos θ
W
=
g
√
g
2
+ g
′2
(1.12)
4 1. Phenomenology
Gauge bosons masses are:
m
A
= 0 (1.13)
m
W
=
1
2
vg (1.14)
m
Z
=
1
2
v
√
g
2
+ g
′2
(1.15)
From the previous equations:
ρ =
m
2
W
m
2
Z
cos
2
θ
W
= 1 (1.16)
which has been verified with great precision [5]:
ρ
exp
=1.0012± 0.0022 (1.17)
Once the electromagnetic coupling constant (electric charge) is defined by:
e = g sin θ
W
= g
′
cos θ
W
(1.18)
it is possible to rewrite the electroweak lagrangian for the interaction of fermions as:
L
fermions
= L
e.m.
+ L
c.c.
+ L
n.c.
(1.19)
where:
• Electromagnetic interaction:
L
e.m.
= −
∑
families
e
(
−eγ
µ
e+
2
3
uγ
µ
u−
1
3
dγ
µ
d
)
A
µ
(1.20)
It is possible to note the electric charge (−1;
2
3
;−
1
3
)offermions(e; u; d)asthe
coupling constants to photon field A
µ
• weak interaction by charge current:
L
c.c.
=
∑
families
g
′
√
2
(
ν
e
γ
µ
1− γ
5
2
eW
+
µ
+ eγ
µ
1− γ
5
2
ν
e
W
−
µ
)
(1.21)
+
∑
families
g
′
√
2
(
uγ
µ
1− γ
5
2
dW
+
µ
+ dγ
µ
1− γ
5
2
uW
−
µ
)
which couples left handed states with W
+
and W
−
bosons.
• weak interactions by neutral current:
L
n.c.
=
∑
fermions
−
1
2
1
√
g
2
+ g
′2
fγ
µ
(
g
V
f
+ g
A
f
γ
5
2
)
fZ
µ
(1.22)
which couples fermions to the Z
0
boson by a combination of axial and vector
components with strengths g
A
f
and g
V
f
whose values are summarised in table
1.3
1.2. Spontaneous symmetry breaking and Higgs mechanism 5
fermions g
V
f
g
A
f
leptons ν
e
,ν
µ
,ν
τ
g
2
+ g
′2
g
2
+ g
′2
e,µ,τ 3g
2
− g
′2
−g
2
− g
′2
quarks u,c,t −
5
3
g
2
+ g
′2
g
2
+ g
′2
d,s,b
1
3
g
2
− g
′2
−g
2
− g
′2
Table 1.3: Vector and axial coupling constants for leptons and quarks
1.1.2 Gauge fields lagrangian
Fields W and B are introduced on covariant derivatives to save local gauge invariance
under SU(2)
L
× U(1)
Y
. To complete the lagrangian for gauge fields it is necessary to
introduce the kinematic term:
L
gauge
= −
1
4
W
a
µν
W
µν
a
−
1
4
B
µν
B
µν
(1.23)
where field tensors are given by.
W
a
µν
= ∂
µ
W
a
ν
− ∂
ν
W
a
µ
+ g+
abc
W
bµ
W
cν
(1.24)
B
µν
= ∂
µ
B
ν
− ∂
ν
B
µ
(1.25)
In this way a self coupling term for W fields is present.
The lagrangian L = L
fermions
+ L
gauge
gives a description of a gauge-invariant and
renormalizable unified theory of weak and electromagnetic interactions. However all
leptons and gauge bosons till this point have zero mass. The addition of an ad hoc
mass term:
m
f
ff (1.26)
spoils the SU(2)× U(1) symmetry. This follows since expression 1.26 can be written
as:
m
f
f
(
1+γ
5
2
+
1− γ
5
2
)
f = m
f
[
f
L
f
R
+ f
R
f
L
]
(1.27)
but f
R
and f
R
are isoscalars under SU(2)× U(1), while f
L
and f
L
are isospinors and
so f
R
f
L
is not invariant under SU(2)
L
operations.
Similarly it is not possible to add a mass term for gauge bosons without spoiling the
symmetry. And, it is worthwhile to note, the gauge invariance is indispensable to have
a renormalizable theory.
Spontaneous symmetry breaking and Higgs mechanism are the instruments to over-
come the problems.
1.2 Spontaneous symmetry breaking and Higgs mech-
anism
The idea is to introduce a scalar field to add mass to fermions and gauge bosons. (in
the following the different possibilities to have one or more multiplets of a scalar field
6 1. Phenomenology
will be treated)
To understand how it works, let consider the simple case of a gauge massless field
A
µ
L(A)=−
1
4
F
µν
F
µν
(1.28)
where F
µν
= ∂
µ
A
ν
− ∂
ν
A
µ
To add the mass term for the gauge field without breaking the gauge invariance, a
term for a scalar field φ is added to the lagrangian:
L(φ)=(D
µ
φ)
∗
(D
µ
φ)− V (φ) (1.29)
where the potential is given by:
V (φ)=µ
2
φ
∗
φ+ λ(φ
∗
φ)
2
(1.30)
with µ
2
< 0andλ>0. Note that L = L(φ)+L(A) is invariant under gauge
transformation:
φ → (1 + igχ(x))φ
A
µ
→ A
µ
− ∂χ(x)
(1.31)
if covariant derivative is used: D
µ
= ∂
µ
+ igA
µ
.
The potential V (φ) has a minimum at φ = ve
iθ
/
√
2withv =
√
−µ
2
/λ at a value
which does not follow the potential symmetry. It is possible to compute the pertur-
bative excitations around this minimum (spontaneous symmetry breaking) with the
substitution
φ=(
v +H(x)
√
2
)e
iθ
(1.32)
(where H(x) is a real field). Since the lagrangian density is invariant under global
phase transformation φ(x)→ φ(x)e
iθ
,thevalueofθ is not significant and one can take
θ = 0. Then the lagrangian becomes:
L =
1
2
[(∂
µ
− igA
µ
)(v +H)][(∂
µ
+ igA
µ
)(v +H)] + (1.33)
1
2
µ
2
(v +H)
2
−
1
4
λ(v +H)
4
−
1
4
F
µν
F
µν
In the previous equation some terms are of particular interest:
• g
2
v
2
A
µ
A
µ
: mass term for gauge field
• λv
2
H : mass term for scalar field
• vg
2
HA
µ
A
µ
and g
2
H
2
A
µ
A
µ
: interactions of scalar field H with gauge field
• λH
4
and vλH
2
self-interactions of scalar field.
We have shown that by spontaneous symmetry breaking and Higgs mechanism, with-
out explicitly breaking the gauge invariance, it is possible to add a mass term for the
gauge field. As a “by product” a scalar massive field H appears: the Higgs boson.
1.2. Spontaneous symmetry breaking and Higgs mechanism 7
1.2.1 Higgs mechanism in Minimal Standard Model
The Higgs mechanism can be applied using as scalar field multiplets of SU(2)
L
×U(1)
Y
group (under some conditions specified later). The simplest choice is a doublet (T =
1/2) with hypercharge Y =1:
Φ=
(
φ
+
φ
0
)
=
(
(φ
1
+ iφ
2
)/
√
2
(φ
3
+ iφ
4
)/
√
2
)
(1.34)
where φ
i
are real fields.
Lagrangian for scalar field and potential are the same as 1.29 and 1.30, and the
covariant derivative has the form:
D
µ
= i∂
µ
− g
$
T ·
$
W
µ
− g
′
Y
2
B
µ
(1.35)
Perturbative expansion is computed around Φ
0
=
1
√
2
(
0
v
)
:
Φ(x)=
e
iσ(x)/v
√
2
(
0
v +H(x)
)
(1.36)
the σ(x) fields are the Goldstone bosons. At the end of the computation, mass terms
appear for W
µ
, Z
µ
and H fields (while A
µ
stays massless) and three degrees of freedom
are absorbed by longitudinal polarisation of W
±
and Z
0
.
Higgs boson mass is equal to v
√
2λ and it is not predicted by the model.
Apredictionforv can be obtained by the correspondence between this model and
the effective theory of charge currents at low energy:
v =
(
G
F
√
2
)
−
1
2
246GeV (1.37)
1.2.2 General scalar sector
The previous choice of a doublet is called minimal (MSM). A priori other multiplets
can be used to produce the Higgs mechanism, and moreover a more complex structure
of the scalar sector could explain the difference between fermion masses and can give a
new source of CP violation. Actually in the MSM the CP violation source is included in
the complex phase inside the CKM mixing matrix. However this source is not enough
to explain the asymmetry between matter and antimatter in the universe [2]. Models
in which the scalar sector has more than a doublet give new sources of CP asymmetry
and make possible the baryogenesis at the scale of the weak interaction.
So there are a lot of reasons to introduce a more complex structure in the scalar
sector of MSM. There are, of course, also some constraints to be explained later.
8 1. Phenomenology
Let us introduce the general possibility of N multiplets Φ
i
with isospin T
i
and hy-
percharge Y
i
:
Φ
i
=
φ
i,1
(T
i
)
φ
i,2
(T
i
− 1)
.
.
.
φ
i,j
(T
i,j
)
.
.
.
φ
i,2T
i
+1
(−T
i
)
(1.38)
where each φ
i,j
has electric charge: Q
i,j
= T
3
i,j
+
Y
2
(T
3
i,j
being the third component
of weak isospin of φ
i,j
).
Mass term for gauge bosons appears in developing the term in the lagrangian
|
∑
i
(
∂
µ
− g
$
T ·
$
W
µ
− g
′
Y
2
B
µ
)
Φ
i
|
2
(1.39)
if, and only if, at least one of the components of one multiplet Φ
i
has an expectation
value on vacuum not equal to zero:
< 0|Φ
i
|0 >=< Φ
i
>
0
=
0
.
.
.
0
1
0
.
.
.
0
(1.40)
In order that the vacuum state is invariant under local transformation of U(1)
em
group, it is necessary that non null components of < Φ
i
>
0
are electrically neutral.
Developing 1.39, the masses of gauge bosons have the following expressions:
m
2
W
=
g
2
2
∑
i
v
2
i
[
T
i
(T
i
+1)− (
Y
i
2
)
2
]
(1.41)
m
2
Z
=
g
2
cos θ
W
2
∑
i
v
2
i
(
Y
i
2
)
2
(1.42)
The number of massive scalar bosons (after absorption of three degree of freedom in
the longitudinal polarisation of massive gauge bosons ) is equal to 2
∑
N
i=1
(2T
i
+1)− 3.
Vacuum mean value is:
2
N
∑
i=1
v
2
i
[
T
i
(T
i
+1)− (
Y
i
2
)
2
]
(246GeV)
2
(1.43)
1.2. Spontaneous symmetry breaking and Higgs mechanism 9
1.2.3 Constraints on scalar sector
Constraints on the structure of scalar multiplets come from two different considerations:
• the gauge bosons mass relation ρ =
m
2
W
m
2
Z
cos
2
θ
W
= 1 must be verified at zero order
of quantic corrections;
• the generation mechanism of fermion masses where the neutral currents with
change of quark flavour (FCNC) at tree level must be avoided .
Let us examine both constraints.
1.2.3.1 Relation ρ=1
InthecaseofN multiplets (T
i
,Y
i
) the general expression for ρ is:
ρ =
∑
N
i=1
v
2
i
[
T
i
(T
i
+1)− (
Y
i
2
)
2
]
∑
N
i=1
v
2
i
(
Y
i
2
)
2
(1.44)
The relation ρ = 1 can be verified if each multiplet verifies:
T
i
(T
i
+1)=3(
Y
i
2
)
2
(1.45)
The simplest possibilities are the singlets (T =0,Y = 0) and doublets (T =1/2,Y =
±1). Singlets are usually used with other multiplets, in particular the doublets (it is
the case of some supersymmetric extension of MSM)
Also the representation (T =3,Y = 4) called septet,givesρ = 1 but its complex
structure makes it unlikely.
Another way to have ρ = 1 is by adjusting v
i
values. For example, if a triplet with
hypercharge Y
1
= 2 and a triplet with Y
2
=0areused:
ρ =
v
2
1
+2v
2
2
2v
2
1
(1.46)
using v
2
1
=2v
2
2
gives ρ = 1. However, this way of doing things seems artificial.
Unitarity of ρ implies a scalar sector with at least a doublet. It is possible to
add a triplet which will give a ρ value slightly different from one. In this case the
experimental value of ρ will constrain the vacuum mean value of the triplet depending
on its hypercharge [3]. A recent analysis gives [4]:
|v
1,Y
| < 0.03|v 1
2
,1
| (1.47)
where Y =0orY = ±2andv 1
2
,1
is the vacuum mean value of the doublet. But a low
value of v
1,Y
contradicts the role of scalar multiplets which needs vacuum mean value
10 1. Phenomenology
different from zero to break the electroweak symmetry. For this reason, this structure
is not likely.
Another possibility [4] is to add at the singlet of MSM a triplet of real fields (Y =0)
and a triplet of complex fields (Y = 2). Unitarity of ρ is assured if vacuum mean value
of the two triplets are equal. Also this possibility seems unlikely.
Consequently the constraint ρ = 1 (which is given by the experiments with great
precision), leads to a scalar sector made by doublet and singlet (MHDM: ”Multiple
Higgs Doublets Models”). This is the simplest and most likely solution which does not
require any other condition.
1.2.3.2 Mass lagrangian and suppression of FCNC
As was shown in 1.26 it is not possible to add a fermion mass term in the form m
f
ff
without explicitly breaking the symmetry.
Once the scalar sector is added, it can be used to overcome the problem. If the scalar
sector has a doublet Φ with hypercharge Y = ±1 the following term can be built (G
f
is a coupling constant):
G
f
[
(f
L
Φ)f
R
+ f
R
(Φ
†
f
L
)
]
(1.48)
which is invariant under SU(2) operations.
In the case of MSM there are three families of leptons and quarks and, moreover,
the possibility of quark flavour mixing must be taken into account. This mixing is
expressed in the frame of the MSM by the Cabibbo Kobayashi Maskawa matrix. As
was mentioned above this matrix is unitary: this reflects the absence at tree level of
transitions d ↔ s, s ↔ b, d ↔ b which implies the absence of neutral currents with
change of flavour (FCNC). Taking into account all these facts, the mass lagrangian
becomes:
L
mass
= L
leptons
mass
+ L
quarks
mass
(1.49)
=
∑
families
[G
e
(e
L
Φ)e
R
+G
e
(e
R
Φ
c
)e
L
]+h.c.
+
∑
i,j
[
G
′
i,j
(Q
iL
Φ)d
′
jR
+G
i,j
(Q
iL
Φ
c
)u
jR
]
+ h.c.
(1.50)
where
• Φ=
(
φ
+
φ
0
)
• Φ
c
=
(
φ
0∗
−φ
−
)
• Q
iL
=
(
u
u
)
iL
1.2. Spontaneous symmetry breaking and Higgs mechanism 11
• Gand G
′
are 3× 3 matrices
• i, j specify the families
• The
′
indicates the eigenstates of weak interaction (connected to mass eigenstates
as specified in 1.3)
After symmetry breaking and perturbative expansion of Φ near its minimum, the
quark masses lagrangian becomes:
(
1+
H(x)
v
)
∑
i,j
(
u
iL
M
ij
u
jR
+ d
′
iL
M
′
ij
d
′
jR
)
+ h.c. (1.51)
where M =
v
√
2
G and M
′
=
v
√
2
G
′
. Once these matrices are diagonalized, the la-
grangian for quark masses becomes:
(
1+
H(x)
v
)
(
m
u
uu +m
d
dd + ···
)
(1.52)
In the lagrangian there is no term of flavour change like Hds. This is due to the fact
that the diagonalisation of M and M
′
implies the diagonalization of Gand G
′
This is not the case for theory different from MSM. We can consider, for example,
the case of scalar sector made by two doublets Φ
1
and Φ
2
with hypercharge Y
1,2
=1.
The lagrangian for mass term of quarks is:
L
quarks
mass
=
∑
i,j
[
G
′(1)
ij
(Q
iL
Φ
1
)d
′
jR
+G
′(2)
ij
(Q
iL
Φ
2
)d
′
jR
]
+ h.c. (1.53)
After spontaneous symmetry breaking it gives the following terms
∑
ij
(
1+
H
1
(x)
v
1
)
(
d
′
iL
M
′(1)
ij
d
′
jR
)
+
∑
ij
(
1+
H
2
(x)
v
2
)
(
d
′
iL
M
′(2)
ij
d
′
jR
)
+ h.c. (1.54)
where v
i
and H
i
are respectively the vacuum mean value and the massive field
associated to doublet Φ
i
. The mixing matrix to be diagonalized is:
M
′
=M
′(1)
+M
′(2)
=
v
1
√
2
G
′(1)
+
v
2
√
2
G
′(2)
(1.55)
It is evident that diagonalization ofM
′
does not imply diagonalization of each ofG
′(1)
and G
′(2)
and so the interaction between scalar fields and quarks does not necessarily
preserve flavour. Since experimental measurements constrain the absence of FCNC at
tree level, the scalar sector should be built in such a way that FCNC are absent at
order zero of quantum corrections.
In the MHDM there are a lot of possibilities to do this, but the most elegant one
relies on a theorem proved by Glashow and Weinberg [6] [7] which states that: scalar
fields do not induce FCNC if and only if all quarks with the same electric charge:
12 1. Phenomenology
• have the same irreducible representation of SU(2)
L
(doublets or singlets in the
MSM)
• have the same quantum number T
3
• take mass from coupling to only one Higgs doublet
Actually, terms with FCNC disappear if, for quarks of same charge, mass matrix M
is proportional to interaction matrixG,thatisM is defined by only one scalar doublet.
This condition is equivalent to the request of an additional discrete symmetry, whose
definition is model dependent (it will be given later for models with two doublets).
In the MHDM there are five [8] different ways to couple Higgs doublets to fermions
respecting conditions of the above theorem. They are summarised in table 1.4 in which
it is specified also to which doublet each quark couples.
fermions models
I I
′
II II
′
III
u,c,t Φ
1
Φ
1
Φ
1
Φ
1
Φ
1
d,s,b Φ
1
Φ
1
Φ
2
Φ
2
Φ
2
e,µ,τ Φ
1
Φ
2
Φ
1
Φ
2
Φ
3
Table 1.4: Five ways to couple Higgs doublets to fermions respecting conditions for
suppression of FCNC defined by Glashow and Weinberg
If, for example, we consider the two doublets model II in which doublet Φ
1
(Φ
2
)
couples to quarks u (d), the lagrangian for quark mass becomes:
∑
i,j
[
G
′(1)
i,j
(Q
iL
Φ
1
)d
′
jR
+G
(2)
i,j
(Q
iL
Φ
2
)u
′
jR
]
+ h.c. (1.56)
and diagonalization of the mass matrix implies diagonalization of Gand G
′
.
We have shown that the request of unitarity of ρ and absence of FCNC at tree
level favour extensions of MSM made by at least two doublets of SU(2)
L
group with,
eventually, singlets; and that fermions couple to three different doublets (at maximum)
in five different ways. The desire for simplicity and a limited number of free parameters
from the scalar sector favour the case of two doublets.
1.2.4 Two doublets models
The interest to investigate these models, relays on the fact that the Minimal Super-
symmetric extension of Standard Model is made by a two doublets model.
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