Search for the Standard Model Higgs boson in the four leptons decay channel with multivariate analysis at the ATLAS experiment

Chapter1
The Standard Model and The
Higgs Boson
The Standard Model (SM) is a successful theory incorporating the present
understanding of fundamental particles and their interactions. The SM is per-
turbativeatsucientlyhighenergiesandrenormalizableduetoitsgaugeinvari-
ant formulation. It is able to accommodate basically all the known experimen-
tal facts and precise measurements performed in high energy particles colliders
over the last decades. However it remains \incomplete": the existence of dark
matter and the gravitational interaction are not described; the mechanism for
electroweak symmetry breaking that gives masses to the particles is not iden-
tied, and the associated particle, the Higgs boson, has not been observed yet.
ThereforeitiscrucialtoprovetheHiggsexistenceandthevalidityofthetheory
or completely exclude it over the entire allowed mass range.
This chapter briey describes the Standard Model and its key ingredients with
some attention to the mechanism which predicts the existence of the Higgs par-
ticle and describes the origin of the masses of the fundamental particles. Then
it will discuss the Higgs boson production, decay modes and the limits on its
mass.
1.1 The Standard Model (SM)
1.1.1 Particles and Interactions
In our current understanding, the physical world is composed by a few fun-
damental building blocks, which are collectively called matter, and is shaped
by their interactions, which are collectively called forces. In the Standard
Model particles can be divided in two categories: fermions and bosons [1].
6
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
The fermions are the bricks of the ordinary matter. They, by denition, have
semi-integer spins and the fundamental fermions in nature and therefore in SM
have all spin 1/2. Fermions obey Fermi-Dirac statistic and the Pauli exclusion
principle. Thefermionscanbegroupedintoleptons(l)andquarks(q),asshown
in Table 1.1. Quarks have an electric charge that is a fraction of the electron’s
one and carry a color charge. They do not exist as free particles but they are
combined to form particles called hadrons. Leptons are typically divided into
charged and neutral ones. The latter are referred to as neutrinos, interact only
weakly with the matter and can be transformed into each other when they pass
through matter. Moreover both leptons and quarks can be organized in three
symmetric sets of particles, called families, with increasing mass. Each of them
includes two quarks and two leptons.
There are four kinds of interactions between the fermions: the gravitational,
Fermions
Quarks
Generation
Q=+
2
3
, T
3;L
=+
1
2
, T
3;R
=0, Q= 1
3
, T
3;L
= 1
2
, T
3;R
=0,
Y
L
=+
1
3
, Y
R
=+
4
3
Y
L
=+
1
3
, Y
R
= 2
3
I u 1:7 3:3 MeV d 4:1 5:8 MeV
II c 1:18 1:34 GeV s 80 130 MeV
III t 173:1 0:6 1:1 GeV b 4:13 4:37 GeV
Leptons
Generation
Q= 1, T
3;L
= 1
2
, T
3;R
=0, Q=0, T
3;L
=+
1
2
, T
3;R
=0,
Y
L
= 1, Y
R
= 2 Y
L
= 1, Y
R
=0
I e 0:511 MeV  e <3 eV
II  106 MeV   <19 keV
III  1:78 GeV   <18:2 MeV
Table 1.1: Fermions of the SM. The properties of these particles are ex-
pressedintermsoftheirmass, theircharge(Q),theirweakisospin(I3,isthe
third component) and their hypercharge (Y). These quantum numbers are
relatedtoeachotherbythe relation: Q =T3+
Y
2
. Italsointroducesthedis-
tinction between left-handed and right-handed component of the fermionic
particle (indicated respectively by the subscript L and R) [1].
the electromagnetic, the weak and the strong. Each interaction is mediated by
one or more massive or massless spin-1 particles, summarized in Table 1.2 along
with the interaction’s range. The integer spin particles obey the Bose-Einstein
statisticsandarereferredtoasthebosons. Inparticulargravitationshouldalso
be mediated by a boson, the graviton, even if with spin 2 and not 1 as the oth-
ers, but there is still no evidence of its existence. The last boson which appears
in Table 1.2 is the Higgs boson. This particle has never been observed and its
search is the topic of this thesis. Here it is only mentioned that, on the contrary
of the other bosons, it is not a mediator of a force and it is expected by SM
predictions to be scalar, i.e. spin 0, and neutral. It appears as a consequence
7
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
of the Higgs mechanism after the spontaneous electroweak symmetry breaking
(see Section 1.1.3). Through this mechanism, as a consequence of the interac-
tion with the Higgs eld, the vector bosons and the fermions acquire mass.
It is noted that in the following the natural system of units ~ = c = 1 will be
used.
Force
Relative Range
Boson Spin
Mass
strength [m] [GeV]
Strong 1 10
 15
8 gluons (g) 1 0
Electromagnetic 10
 2
1 photon ( ) 1 0
Weak 10
 2
10
 13
W
 1 80:399 0:023
10
 2
10
 13
Z 1 91:1876 0:0021
Gravitational 10
 40
1 graviton (?) 2 0
Higgs (H) 0 ?
Table 1.2: Theparticleinteractionswiththeircarrierparticles,whosemass
and spin are reported. The relative strength and eective range of forces are
also shown [1]. The gravitational force is not described by the SM. The
graviton and the Higgs particle have not yet been observed experimentally.
1.1.2 The Gauge Field Theories
TheStandardModelhasbeenextendedfrommodelsdevelopedinthe1960’s
by Glashow, Winberg and Salam. The Standard Model description of particles
and forces in nature is based on the mathematical language of the Quantum
FieldTheory(QFT),whereparticlesareexcitationsoffundamentaleldswhich
are functions of, or extend in, space and time. Particle dynamics are described
by a Lagrangian density L, simply referred to as the Lagrangian hereafter
1
.
Fermions are represented mathematically by matter elds, interactions between
them are represented by gauge elds that operate on matter elds. For a given
Lagrangiandescriptionofasystem,gaugeinvarianceimpliesthatLisconserved
under local symmetry transformations
2
. Any such symmetry corresponds to a
conservation law and vice versa (Noether’s theorem), for instance if a temporal
translation invariance is required, a conservation of energy is obtained, whereas
spatial translation invariance implies conservation of momentum.
The SM is a renormalizable quantum eld theory which provides a unied ap-
proach for the description of the electromagnetic, weak and strong interactions.
TheStandardModelisbasedonthegaugesymmetrySU(3)
C
 SU(2)
L
 U(1)
Y
.
1
InthefollowingitwillnotusetheLagrangianLtodescribethesystem,buttheLagrangian
density L which is related to L by L =
R
Ld~ x. However for simplicity L will be called
Lagrangian as well.
2
These symmetries are said to be global if they are the same at any point in the Universe.
In the SM local symmetries are imposed as well as the global symmetries. These stricter
requirements imply that a certain local (position dependent) transformations should leave all
physical quantities conserved in the local space, apart from the globally preserved ones.
8
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
SU(3)
C
isthegroupofcolorsymmetry, describedwithintheframeofQuantum
Chromo Dynamics (QCD), SU(2)
L
the one of the weak isospin symmetry and
U(1)
Y
the one for the hypercharge symmetry. The symmetrySU(2)
L
 U(1)
Y
,
thatrepresentstheuniedweakandelectromagneticinteraction,isbrokenspon-
taneously by the Higgs mechanism (SU(2)
L
 U(1)
Y
!U(1)
Q
) [2, 3].
1.1.2.1 Quantum Electrodynamics (QED)
The Lagrangian of the massless electromagnetic eld A
 interacting with a
spin-1/2 eld of bare mass m is
L= 1
4
F
 F
 + (i
 D
  m) (1.1)
where = 
y
 0
and   are the 4 4 Dirac matrices satisfying the anti-
commutation relation: f  ;  g = 2g
 with g
 being the metric tensor. The
electromagnetic eld tensor is dened as
F
 =@
 A
  @
 A
 (1.2)
The Lagrangian in (1.1) is obtained by requiring that the Dirac Lagrangian of
the free spin-1/2 particle
L= (i
 @
  m) (1.3)
becomes symmetric under local U(1)
Q
3
transformations of the form:
U(x)=exp( ieQ (x)) (1.4)
where e is the unit electric charge and Q is the charge operator
4
.
TheLagrangian invariancecan bemaintainedwith theaddition of aspin-1 eld
A
 , called gauge boson, it is noted that under this transformation and A
 change as:
 (x)! 
0
(x)= (x)exp( ieQ (x)) (1.5)
A
 (x)!A
0
 (x)=A
 (x)+@
  (x) (1.6)
For (1.3) to be invariant under (1.4), the covariant derivative D
 D
 =@
 +ieA
 Q (1.7)
3
U(1) is the group of all complex numbers with module one.
4
Q =q
f
 , where q
f
=1 for the electrons.
9
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
hastobeintroduced. ThereforethenewLagrangiancanbewrittenas(1.1). Itis
characterizedbyatermrepresentingtheoriginalelectroneld( (i
 @
  m) ),
composedbythefermionickineticterm(i   @
 )andthefermionicmassterm
( m ). e   A
 Q is the interaction term between the vector eld A
 and
the electromagnetic current. The strength of the interaction is proportional to
the value of the constant e. The new eld A
 is thus the photon eld and
the interaction term appearing in the Lagrangian due to to the local gauge
invariancedescribestheelectromagneticinteractionsmediatedthroughphotons.
Finally the rst term is the kinetic term for the photon (  1
4
F
 F
 ).
The respective conserved current is the electric current:
j
 =e   Q (1.8)
The photon is massless, and a mass term of the form m
2
A
 A
 was not intro-
duced to preserve gauge invariance.
1.1.2.2 Quantum Chromodynamics (QCD)
The strong interactions are described by a local non-abelian gauge theory,
in which SU(3)
C
is the gauge group and gluons are the gauge bosons. The
corresponding Lagrangian is:
L= 1
4
G
 a
G
a
 +q
j
(i
 D
 jk
 M
jk
)q
k
(1.9)
where M
jk
is the quark mass matrix. The Latin indices refer to color and
assume values a = 1,2,:::8 for the eight gluons and j,k =1,2,3 for the three
quarks. The gluon eld tensor is dened as:
G
 a
=@
 G
 a
 @
 G
 a
 g
s
f
abc
G
 b
G
 c
(1.10)
hereG
 a
are the gluon elds, g
s
is the strong coupling andf
abc
are the structure
constants of the SU(3) group. The covariant derivative acting on the quark
elds is:
D
 jk
= jk
@
 +ig
s
(T
a
)
jk
G
 a
(1.11)
where T
a
are the generators of the SU(3) group dened by the commutation
relation [T
a
;T
b
]=if
abc
T
c
. In particularT
a
= a
=2, a
are the eight Gell-Mann
matrices. These are hermitian, 3 3 and traceless matrices.
The second term of (1.9) contains a quark-gluon interaction vertex, while the
rsttermcontainsthreeandfourgluoncouplings. Theseself-interactionsofthe
gluons, which have no analog in QED where the photon is electrically neutral,
are a consequence of the fact that gluons also carry color charge due to the
10
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
non-abelian nature of the group.
Gluons are required to be massless since the presence of a mass term for gauge
elds break the gauge invariance of Lagrangian.
1.1.2.3 Weak Interactions and Electroweak Unication
The fermions are grouped in left-handed and right-handed elds:
 
L
=P
L
 =
1
2
(1  5
) (1.12)
 
R
=P
R
 =
1
2
(1+ 5
) (1.13)
where P
L;R
are the chirality
5
operators and  5
=i
0
 1
 2
 3
[4].
Onlyleft-handedparticlesandright-handedantiparticlesparticipateintheweak
interaction. The left-handed fermions are SU(2)
6
doublets
7
,
 
L
=
 
 e
e
 !
L
;
 
    !
L
;
 
    !
L
;
 
u
d
0
!
L
;
 
c
s
0
!
L
;
 
t
b
0
!
L
(1.14)
while the right-handed fermions are singlets
 
R
=
    R
;
 e
  R
;:::
 u
 R
;
 d
 R
;::: (1.15)
AsaconsequenceofthesecouplingstheSU(2)symmetryisdenotedasSU(2)
L
.
Theelectromagneticandweakinteractionsareuniedintoasingletheorybythe
Glashow, Salam and Weinberg (GSW) theory, the Electroweak theory (EW).
The simplest unication of the parity violating weak force and the parity con-
serving electromagnetic force is the SU(2)
L
 U(1)
Y
gauge theory.
Local gauge invariance under SU(2) transformations requires introduction of
three massless spin 1 gauge bosons W
i
 , i = 1,2,3. The conserved quantity is
called weak isospin (T
a
with a = 1,2,3). An additional U(1) symmetry was
added to include the electromagnetic interaction in the EW theory. It is an
independent gauge symmetry of the weak hypercharge (Y), which is specied
according to the formulaQ=T
3
+
Y
2
, whereQ is the electric charge andT
3
the
third component of the weak isospin. This symmetry is denoted as U(1)
Y
and
it requires an additional gauge boson B
 with spin 1. The U(1)
Y
gauge boson
5
Chirality is a property of the eld dened by the operator  5
, which is formed, as shown
in the text, by the product of Dirac matrices so that it anti-commutes with all the others. In
case of massless particles the chirality corresponds to the helicity: fermions with right-handed
(left-handed)helicityaretheonesthathavethespinpointinginthesame(opposite)direction
of the momentum. For antifermions this convention is reversed.
6
SU(2) is the group of the special unitary 2 2 matrices. A unitary matrix satises: T
y
a
=
T
 1
a
, where T
y
a
is the hermitian conjugate matrix. The generators of SU(2) are Ta =  a=2,
where  a are the Pauli matrices dened in the weak isospin space.
7
d’,s’,b’ are the eigenstates of the weak interactions.
11
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
couples to both the right-handed and the left-handed components
8
.
The gauge invariant Lagrangian describing the electroweak interactions is
L= 1
4
W
a
 W
 a
 1
4
B
 B
 + i
 D
 (1.18)
where the eld tensor W
 and B
 are dened as:
W
i
 =@
 W
i
  @
 W
i
  g ijk
W
j
 W
k
 (1.19)
B
 =@
 B
  @
 B
 (1.20)
Additionally cubic and quartic self-couplings of the W
i
 elds have been intro-
duced. The covariant derivative is
D
 =@
 +igW
a
 T
a
+ig
0
1
2
B
 Y (1.21)
whereg is theSU(2) constant coupling andg
0
theU(1) constant coupling. The
interaction between the fermions and the gauge elds is
L
int
= 
L
   g
2
W
a
  a
+
g
0
2
B
 Y
 
L
 
R
   g
0
2
B
 Y
 
R
(1.22)
It should be noted that for the local gauge invariance to be conserved, no mass
terms for the fermions or the gauge bosons (m
2
B
 B
 andm
2
WW
 ) could be
introduced in the Lagrangian.
1.1.3 The Higgs Mechanism
The Standard Model, i.e. the SU(3)
C
 SU(2)
L
 U(1)
Y
theory, is the
combination of the electroweak theory and the QCD theory. The symmetry of
SU(2)
L
 U(1)
Y
, i.e. the invariance for a local gauge transformation, requires
the presence of massless gauge bosons in the EW theory. This conicts with
experimental measurements of W
 and Z gauge bosons, according which their
masses are large and can not be neglected (see Table 1.2). A solution has been
proposed by F. Englert, R. Brout, P. Higgs and independently G. Guralnik, C.
R. Hagen, and T. Kibble. They conjectured that the massless gauge bosons of
weak interactions acquire their mass through interaction with a scalar eld (the
8
TransformationsunderSU(2)
L
 U(1)
Y
oftheleft-handedandright-handedcomponents:
• Under SU(2)
L
transformation:
 
L
(x)! 
0
L
(x)=e
ig a
(x)
 a
2 
L
(x); 
R
(x)! 
0
R
(x)= 
R
(x) (1.16)
• Under U(1)
Y
transformation:
 
L
(x)! 
0
L
(x)=e
ig
0
 (x)
Y
2 
L
(x); 
R
(x)! 
0
R
(x)=e
ig
0
 (x)
Y
2 
R
(x) (1.17)
12
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
Higgs Field), resulting in a single massless gauge boson (the photon) and three
massive gauge bosons (W
 andZ). This is possible because the Higgs eld has
a potential function which allows degenerate vacuum solutions with a non-zero
vacuum expectation value [5].
In the context of the SU(2)
L
 U(1)
Y
symmetry, the Higgs mechanism is im-
plemented through an additionalSU(2)
L
doublet of complex scalar elds (four
real scalar elds)
 =
  +
 0
 =
r
1
2
  1
+i
2
 3
+i
4
 , (1.23)
the self-interaction of which leads to the spontaneous electroweak symmetry
breaking
9
. The quantum numbers of these elds are summarized in Table 1.3.
The Higgs sector of the Lagrangian is
T T
3
Y=2 Q
 +
1/2 1/2 1/2 1
 0
1/2 -1/2 1/2 0
Table1.3: Thequantumnumbersofthecomplexscalareldsofthe SU(2)L
doublet  .
L
H
=(D
  )
y
(D
  ) V( ) (1.24)
where the most general renormalizable form of the scalar potential is
V( )= 2
 y
 + ( y
 )
2
= 2
j j
2
+ j j
4
(1.25)
The potential is chosen such that it is an even function of the scalar eld, i.e.
V( ) = V(  ), so that the Lagrangian is invariant under the parity transfor-
mation  !   . The potential is parametrized by  and  .  , which is the
strength of the quartic self-coupling of the scalar eld (showed by  4
term), is
required to be positive so that the energy is bounded from below. This require-
ment ensures the existence of stable ground states. Two qualitatively dierent
cases, corresponding to manifest or spontaneously broken symmetry, may be
distinguished depending on the sign of the coecient  2
.
If  2
> 0, the potential has a unique minimum at  = 0 that corresponds to
the ground state, i.e. the vacuum. In terms of a quantum eld theory, where
9
It might break this symmetry simply introducing by hand a mass term for gauge bosons,
which violates the symmetry, however, such a procedure would destroy the renormalizability
of the theory. Then it uses a more elegant way to break the symmetry called \spontaneous
symmetry breaking". In this scenario, the gauge invariant Lagrangian is maintained, while
the state of lowest energy, which is interpreted as the vacuum state is not gauge invariant.
There is an innite number of states, each of which has the same ground state energy and
nature chooses one as a state of \true" vacuum.
13
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
(a)  2
>0.
(b)  2
<0.
Figure 1.1: IllustrationoftheHiggspotentialforascalareld  = 1+i 2
with  2
> 0 and  2
< 0.
 is an operator, the precise statement is that the operator  has zero vacuum
expectation value (vev), i.e. h i
0
= h0j j0i = 0. The vacuum obeys the re-
ection symmetry of the Lagrangian. In this case, aside from the  4
term the
Lagrangian is just the Lagrangian for a charged scalar particle of mass  and
massless gauge bosons.
If 2
<0, the Lagrangian has a mass term of the wrong sign for the eld  and
the minimum energy is not at  = 0. The potential adopts a shape known as
the \Mexican hat", with a maximum at =0, as can be seen in Figure 1.1(b).
The vacuum expectation value is obtained by looking at the stationary points
ofL:
@L
 @( y
 )
=0) 2
0
= y
  1
2
( 2
1
+ 2
2
+ 2
3
+ 2
4
)=  2
2  v
2
2
6=0 (1.26)
wherevisreferredtoasthevacuumexpectationvalueofthescalareld. Itmust
benotedthatitisnotzero. Thevaluesof(Re +
, Im +
, Re 0
, Im 0
)canrange
over the surface of a 4-dimensional sphere of radius v, such that v
2
=   2
=
and  y
 =j +
j
2
+j 0
j
2
. This implies that Lagrangian of  is invariant under
rotationsofthis4-dimensionalsphere. Theminimumofthispotentialnolonger
corresponds to a unique value of  but there is an innite number of states
14
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
with the same lowest energy, there is a degenerate vacuum. The solution for
the location of the minima,  0
, is satised by
 0
=e
i
r
  2
2 (1.27)
where 0   2 is the angle around the axis of the potential,V( ). Choosing
one of the non-zero ground states  0
for  2
< 0 spontaneously breaks the
SU(2)
L
 U(1)
Y
symmetry down to U(1)
Q
. Lagrangian is still invariant, but
thegroundstateisnomoresymmetricunderSU(2)
L
 U(1)
Y
. Thisisillustrated
in Figure 1.1: the potential in Figure 1.1(b) is symmetric under rotations but
any minimum chosen is not. Usually,  = 0 is used to x one vacuum state:
 0
( = 0)   vacuum
. The direction of the minimum in the SU(2)
L
 U(1)
Y
space is not determined since the potential depends only on the combination
 y
 . Withoutlossofgenerality,nowtheHiggseldisxedsuchthatthevacuum
expectation value of is dened to be a real parameter in the  0
direction, i.e.,
 1
= 2
= 4
=0,  2
3
=  2
= :
 0
=h0j j0i=
1
p
2
 0
v
 (1.28)
 +
is chosen to be zero because the vacuum state has to be neutral in order to
breakSU(2)
L
 U(1)
Y
symmetry saving theU(1)
Q
. Using the charge operator
Q on  0
and the properties of the Higgs eld, this leads to:
Q
0
=
 T
3
+
Y
2
  0
=( 1
2
+
1
2
) 0
=0 (1.29)
ThevevhastobeneutralbecauseifQ =0,asshownabove,performingU(1)
Q
transformation
 ! 0
=exp( ie (x)Q)  (1 ieQ ) = (1.30)
So that choice for the vacuum state breaks both SU(2)
L
and U(1)
Y
but not
U(1)
Q
. In this way the vacuum stays neutral but it carries a hypercharge and
an isospin so that it couples to weak bosons.
Thephysicalcontentofthetheoryisrevealedbyaperturbativeexpansionofthe
Lagrangianaroundthegroundstate, (x)canbeexpandedaboutthisparticular
vacuum, one can parametrize excitations from this ground state by
 =
1
p
2
e
i a(x) a
2v
 0
v+H(x)
 (1.31)
15
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
wheretherealelds  1
, 2
, 3
andH haveazerovacuumexpectationvalue. This
gives rise to a scalar eld H(x), the massive Higgs eld, which describes radial
excitations from the ground state changing the potential energy, and to three
massless scalar elds  a
(x), the Goldstone bosons, corresponding to angular
excitationswithoutpotentialenergychange. Thesethreemasslessscalarbosons
correspond to the three broken symmetry generators. These phase factors,
and thus the Goldstone bosons, can be eliminated by a local SU(2)
L
gauge
transformation with  (x) =  (x)=v, this gauge choice is referred to as unitary
gauge, leading to the following parametrization of the scalar eld:
 =
1
p
2
 0
v+H(x)
 (1.32)
Here the degrees of freedom represented by the Goldstone bosons are absorbed
(\eatenup")bythevectorparticlesW
 andZ,giventhemanadditionaldegree
of freedom: a longitudinal polarization, thus the vector bosons acquire mass.
Only massive particles with velocities below the speed of light can have longi-
tudinal degrees of freedom. The photon has only a transversal polarization
10
.
Thereforethe a
(x)disappearfromtheLagrangianandreappearasthelongitu-
dinal component of the massive gauge bosons. Since Q = 0, the ground state
is still symmetric under U(1)
Q
and the photon will remain massless.
1.1.3.1 Massive Gauge Bosons
The coupling of  to the gauge bosons takes place through the covariant
derivativeD
 . By expanding around the ground state of , i.e. introducing the
ansatz (1.32) into the Lagrangian of the electroweak theory (the Higgs sector is
expressed in (1.24)), it is now straightforward to see how the Higgs mechanism
generates masses for W
 and Z bosons. Evaluating the resulting kinetic term
(D
  )
y
(D
  ) at the vacuum expectation value  0
:
(D
  0
)
y
(D
  0
)=
     @
 +igW
a
  a
2
+ig
0
1
2
B
 Y
  0
    2
(1.33)
10
It is instructive to count the degrees of freedom after the spontaneous symmetry breaking
has occurred. The starting point is a Lagrangian with a complex scalar SU(2)
L
doublet  and four massless vector bosons. Counting degrees of freedom gives four from the scalars
and eight from the vector bosons, for a total of twelve. Through the Higgs mechanism the
Lagrangian is transformed into one real scalar, three massive vector bosons and one massless
vector boson. The massless vector boson is of course to be identied with the photon and the
single remaining scalar with the Higgs boson. Counting degrees of freedom again gives one
from Higgs, two from the photon and nine from the massive vector bosons, again adding up
to twelve.
16
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
The relevant terms are (it should be noted that  has hypercharge Y =1):
 L=
1
2
(0 v)
 1
2
gW
j
  j
+
1
2
g
0
B
  1
2
gW
k  k
+
1
2
g
0
B
  0
v
 =
=
1
8
(0 v)
 
gW
3
 +g
0
B
 g(W
1
  iW
2
 )
g(W
1
 +iW
2
 )  gW
3
 +g
0
B
 !
2 0
v
 =
=
1
8
v
2
 g
2
(W
1
  iW
2
 )(W
1
 +iW
2
 )+( gW
3
 +g
0
B
 )
2
 =
=
1
8
v
2
g
2
  W
1
  2
+
 W
2
  2
 +
1
8
v
2
 g
0
B
  gW
3
  g
0
B
  gW
3  =
=
 vg
2
 2
W
+
 W
  +
1
8
v
2
(W
3
 B
 )
 
g
2
 gg
0
 gg
0
g
02
!
 W
3 B
  (1.34)
wherethechargedgaugebosons,W
 , havealreadyacquiredmassandamixing
between W
3
 and B
 is observed. The corresponding mass eigenstates for the
neutral gauge bosons are obtained by diagonalizing the mass matrix:
W
  =
1
p
2
(W
1
  iW
2
 ) con m
W
=
1
2
gv (1.35)
Z
 =
gW
3
  g
0
B
 p
g
2
+g
02
con m
Z
=
1
2
v
p
g
2
+g
02
(1.36)
A
 =
g
0
W
3
 +gB
 p
g
2
+g
02
con m
A
=0 (1.37)
The spontaneous symmetry breaking rotates the four SU(2)
L
 U(1)
Y
gauge
bosons to their mass eigenstates by means of the gauge interaction term of the
Higgs elds: fW
1
 , W
2
 g!fW
+
 , W
  g efW
3
 , B
 g!fA
 , Z
 g.
W
 is associated to the charged current processes, A to the electromagnetic
currents and Z to the neutral currents. Once A
 is recognized as the photon,
the three couplings, previously described, are related to each others by
e=
gg
0
p
g
2
+g
02
(1.38)
Usually the ratio between g and g
0
is dened through the weak mixing angle
 W
, also known as the Weinberg’s angle:
tan W
=
g
0
g
(1.39)
It is important to notice that W
is a free parameter of the model since it is the
ratio of two coupling constants related to independent symmetry groups. Given
 W
, with sin W
=
g
0
p
g
2
+g
02
, all gauge couplings are determined by the electric
17
1. The SM and The Higgs Boson 1.1 The Standard Model (SM)
charge:
e=gsin W
=g
0
cos W
(1.40)
and thus the electroweak unication is achieved. In terms of  W
, the photon
and Z boson eld are
 Z
 A
  =
 
cos W
 sin W
sin W
cos W
!
 W
3
 B
  (1.41)
and the relation
m
W
m
Z
=cos W
(1.42)
for the masses of the gauge bosons is predicted.
By inserting (1.32) into the expression (1.25) for the Higgs potential V( ), the
mass term   2
H
2
for the Higgs eld H appears, implying the existence of a
new physical particle, the Higgs boson as already said, with mass
m
H
=
p
 2 2
=
p
2v (1.43)
It can be noticed that basically in this model there are two constants g and
g
0
related to the symmetry SU(2)
L
 U(1)
Y
and two parameters of the Higgs
potential  and  . Usually they are parametrized with the observables  , the
ne structure constant, G
F
, the Fermi constant, m
Z
, the Z boson mass, and
m
H
, the mass of the Higgs boson, for which the relations are summarized here:
 =
g
2
g
02
4  g
2
+g
02
 =
g
2
sin
2
 W
4 (1.44)
m
Z
=
1
2
v
p
g
2
+g
02
(1.45)
G
F
=
1
p
2v
2
(1.46)
m
H
=
p
 2 2
=
p
2v (1.47)
G
F
is the strength of the weak interaction in the eective and point-like de-
scription of weak interactions formulated by Fermi. The parameter v can be
determined from the measurement of the muon life time in the weak charged
current decay !e e
  . The interaction strength for muon decay is measured
very precisely to be G
F
= 1:16637(1) 10
 5
GeV
 2
[1] giving the value for the
vacuum expectation value
v =
 p
2G
F
  1=2
 246 GeV (1.48)
18
1. The SM and The Higgs Boson 1.2 The SM Higgs Boson
This value set the scale of the electroweak symmetry breaking
11
, but it is not
predicted by the SM. The relation between G
F
and the vev v comes from
G
F
p
2
=
g
2
8m
2
W
=
1
2v
2
(1.50)
which is a comparison between the Fermi theory and the charged current in the
limit of highly massive gauge bosons. In fact the muon decay at the leading
order can be described by the propagation of a W boson but Fermi showed
that this can be simplied to one vertex with the constant coupling G
F
. Once
the values of  , G
F
and m
Z
are known, it can be predicted from (1.35) and
(1.36) the mass of the W boson, at the lowest order m
W
= m
Z
cos W
 80
GeV, which has been conrmed experimentally by the Z’s and W’s discovery
at SppS (Super proton-antiproton Synchrotron) and by precise measurements
of m
W
and m
Z
at LEP (Large Electron Positron Collider). The v parameter
can be experimentally determined but there is no way to measure the value of
 before a discovery of the Higgs.
1.2 The SM Higgs Boson
1.2.1 Theoretical Constraints on the Higgs Mass
Despite the prediction of a Higgs boson, the Standard Model does not pro-
vide for its mass. Although the Higgs boson mass is a free parameter of the
theory, constraints on the possible mass values can be derived using theoretical
arguments regarding the energy regime in which the perturbative expansion of
the Standard Model is valid. Therefore these argumentations come from very
reasonable considerations, but they cannot provide stringent limits since they
dependontheabsenceofnewphysicsuptoacut-oenergyscale. Asitwillsee,
this means that it can be set a range of masses that is valid as long as virtual
eects of new physics enter in the calculation of the Higgs boson mass. These
arguments are: the unitarity in longitudinal scattering amplitudes; the pertur-
bativity of the Higgs self-coupling and the stability of the electroweak vacuum
[5, 6, 7, 8, 9].
1.2.1.1 Unitarity
A major decit of Fermi’s theory of weak interactions was the violation of
unitarity at the electroweak scale
p
s G
 1=2
F
, due to the assumption of point
11
By the relation
m
W
=
gv
2
=
g
2
p
2 m
H
(1.49)
it is showed that the Higgs mass sets the electroweak scale.
19