Seminar
Quantum Field Theory
Institut für Theoretische Physik III
Universität Stuttgart
2013
Contents
1 Motivtion
1.1 Where’s the mass? . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2 Spontaneous Symmetry Breaking
3
2.1
2.2
Symmetry Breaking: Basic principles . . . . . . . . . . . . . . . .
Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . .
2.2.1 Discrete Symmetry . . . . . . . . . . . . . . . . . . . . . .
3
5
5
2.2.2
2.2.3
Continuous Symmetry . . . . . . . . . . . . . . . . . . . .
Goldstone’s Theorem . . . . . . . . . . . . . . . . . . . . .
6
9
3 Higgs mechanism
3.1 Basic points of gauge theory . . . . . . . . . . . . . . . . . . . . .
11
11
3.2
The Anderson-Higgs mechanism . . . . . . . . . . . . . . . . . . .
3.2.1 Abelian . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
12
3.2.2
14
Bibliography
Non Abelian . . . . . . . . . . . . . . . . . . . . . . . . . .
19
iii
Chapter 1
Motivtion
1.1 Where’s the mass?
To give a slight motivation to the consecutively discussed analysis we start with
the fact that experimental results predict the existence of massive gauge fields.
Based on this observation we are interested to find an appropriate description
for massive gauge fields and the observed masses. Recall that examining the
Lagrangian of a model that involves N scalar fields
1 ~ 2
~ 2 − λ (φ
~ 2 )4 ,
− µ2 φ
L = (∂ φ)
2
4
(1.1)
the masses of the scalar fields are generally to find in the term with the scalar fields
squared (φ~2 ), whereupon the mass value corresponds to the constant in front of
φ~2 . Following this example we could straightforwardly construct a massive gauge
1
1 Motivtion
field theory by adding an A2µ term to the Lagrangian
1 2
~ 2 − µ2 φ
~ 2 − λ(φ
~ 2 )2 − m2 Aµ Aµ
L = − Fµν
[Aµ ] + (Dµ [Aµ ]φ)
4
(1.2)
to characterize our massive gauge fields. Unfortunately this easy way of construction suffers due to
U (1)
Aµ −−→ Aµ −∂µ α(x) : m2 Aµ Aµ → m2 (Aµ Aµ −Aµ ∂ µ α−∂µ αAµ +∂µ α∂ µ α), (1.3)
here in simplest case of U(1) gauge theory, loss of gauge freedom and thus fails
for further analysis. Anyway there is a way out to construct massive gauge fields
and retaining gauge freedom, namely by the concept of spontaneous symmetry
breaking that will be introduced in the following chapter.
2
Chapter 2
Spontaneous Symmetry Breaking
2.1 Symmetry Breaking: Basic principles
Discussing symmetry breaking demands a precise definition of symmetry. Consider a system satisfying a symmetry denoted by the symmetry group G. Then the
Lagrangian L of the system is invariant under symmetry transformations U ∈ G,
i.e.
φ(x) → Uφ :
L(φ) = L(Uφ(x)).
(2.1)
Concerning a theory of N scalar fields φi invariant under the special orthogonal
symmetry group SO(N) the N scalar fields can be chosen to transform as an
N-component vector
φi → [R]ij φj
R ∈ SO(N).
(2.2)
Orthogonal groups O(N) > SO(N) are the most popular symmetry groups of
physical systems. SO(N) and O(N), respectively, is a continuous symmetry group
what means that the symmetry operations R ∈ SO(N) can be expressed in dependence of continuous parameters θa :
R = eiθT
, θT =
X
θa T a .
(2.3)
a
3
2 Spontaneous Symmetry Breaking
The N-dimensional rotation group has N (N2−1) generators T a 1 . An infinitesimal
transformation under SO(N) can be written in the expanded form
Rij φj = φi + δφi = (1 + θa T a )ij φj .
(2.6)
Consider the Lagrangian of N scalar fields
L=
i
1h ~ 2
~ 2 − λ (φ
~ 2 )2
(∂ φ) − µ2 φ
2
4
(2.7)
~ = (φ1 , . . . , φN ) exhibiting O(N) symmetry. An intuitive way to break the
with φ
O(N) symmetry would be to perform it explicitly by hand and add terms such as
~ 2 to (2.7) that do not respect the symmetry. The symmetry is then
φ21 , φ41 and φ21 φ
~ transforms just as a (N − 1)-component
broken down to O(N − 1) under which φ
vector. Actually breaking symmetry by hand is not very interesting as we attain
not that much new. Instead of breaking the symmetry explicitly by hand there
is another way, where we can let the system break the symmetry itself. This
phenomenon is known as spontaneous symmetry breaking and is discussed in the
following.
1
As an example for N = 2, assume the Lagrangian of 2 scalar fields φ1 and φ2 involved:
L=
1
1
λ
((∂φ1 )2 + (∂φ2 )2 ) − µ2 (φ21 + φ22 ) − (φ21 + φ22 )2 .
2
2
4
This Lagrangian is invariant under the transformation
cos(θ) − sin(θ)
φ1
~
Uφ =
,
sin(θ) cos(θ)
φ2
(2.4)
(2.5)
that is an element of SO(2). This group has one continuous parameter θ and one generator
respectively.
4
2.2 Spontaneous Symmetry Breaking
2.2 Spontaneous Symmetry Breaking
2.2.1 Discrete Symmetry
To give the simplest example for spontaneous symmetry breaking consider the
1D-Lagrangian of the classical ϕ4 -field theory
1
m2
λ
L = (∂µ φ)2 − φ2 − φ4
2
| 2 {z 4! }
(2.8)
−V (φ)
with V (φ) the potential of the system. Let us now replace the constant m2 of
the mass term by a negative parameter −µ2 , i.e. simply flip the sign of the mass
term, we get
µ2
λ
1
(2.9)
L = (∂µ φ)2 + φ2 − φ4 ,
2
2
4!
2
and the potential energy V (φ) = − m2 φ2 + 4!λ φ4 turns to a double-well potential
(see figure 2.1). This Lagrangian exhibits discrete reflection symmetry φ → −φ.
The Hamiltonian and hence the energy of the system is given by
H=
Z
dx
1 2 1
π + (∇φ)2 + V (φ)
2
2
>
Z
dx V (φ).
(2.10)
Since any spatial variation only increases the energy, we can estimate (2.10) down
to the minimum and look for the minimum of V (φ). The value to minimize the
potential V (φ) is
s
6
φ0 = ±v = ±
µ,
(2.11)
λ
called the vacuum expectation value. Before starting to deal with the interpretation of the theory, take a brief look at the tunneling barrier. Calculating
the tunneling barrier reveals the crucial difference between QFT and quantum
R
mechanics, that the barrier is now [V (0) − V (±v)] dx and hence infinite. The
tunneling is shut down and the system once seated in one of the ground state
values ± v remains there. For interpretation assume the system being near one
of the minima, i.e.
φ = v + φ′ (x).
(2.12)
5
2 Spontaneous Symmetry Breaking
(b) N=2
(a) N=1
Figure 2.1: (a) Potential energy V (φ) before and after symmetry breaking
for reflection symmetry: The potential energy becomes a doublewell potential
with the vacuum expectation value (potential minima)
q
φ0 = ± 6/λµ. (b) Potential energy of spontaneously broken
SO(2) symmetry: Mexican hat potential with an infinite number of
vacuum states.
Inserting (2.12) into (2.9) and rewriting L after some calculations in terms of φ′
one gets the expression
1
1 2 ′2
2
L = (∂µ φ′ ) −
2µ φ −
2
2
s
6 ′3 λ ′4
µφ − φ .
λ
4!
(2.13)
The linear term of φ′ disappears and the shifted field φ′ creates a particle with
√
mass 2µ, while the reflection symmetry is not evident any more. The symmetry
φ → −φ is spontaneously broken.
2.2.2 Continuous Symmetry
A physically more interesting case than breaking discrete symmetry occurs by
studying spontaneous breaking of a continuous symmetry, that would be for dimensions N ≥ 2. This study involves N scalar fields φi (x) retained in the Lagrangian
m2 i 2 λ h i 2 i2
1
(φ ) .
(2.14)
(φ ) −
L = (∂µ φi )2 +
2
2
4
6
2.2 Spontaneous Symmetry Breaking
The Lagrangian density (2.14) is invariant under rotations
φi → Rij φj ,
(2.15)
where the N scalar fields transforms as a N component vector under the N × N
rotation matrices R of the N-dimensional orthogonal group O(N). The potential
energy
1
λ h i 2 i2
V (φi ) = − µ2 (φi )2 +
(φ )
(2.16)
2
4
~ 0 that fulfills
and hence the energy of the system minimizes at the constant field φ
the condition
~ 0|2 = (φi )2 = v 2 =
|φ
0
µ2
.
λ
(2.17)
~ 0 , whilst the direcNote, that condition (2.17) fixes the magnitude of the vector φ
tion of the vector is arbitrary. For practical application (and conventionally) it’s
~ 0 to point in the Nth direction, i.e.
wise to choose the vector φ
~ 0 = (0, 0, ..., 0, v)T .
φ
(2.18)
Again we examine the case of the N-salar fields being near the vacuum expectation
value characterized by the shifted fields
~
φ(x)
= (π k (x), v + φ′ (x))
k = 1, ..., N − 1.
(2.19)
With (2.19) we immediately gain the Lagrangian
√
√
1
1
1
L = (∂µ π k )2 + (∂µ φ′ )2 − (2µ2 )φ′2 − λµφ′ − λµ(π k )2 φ′
2
2
2
λ ′4 λ k 2 ′2 λ ′k 2 2
− φ − (π ) φ − [(π ) ]
4
2
4
(2.20)
(2.21)
in terms of the shifted fields π k and φ′ . Lagrangian (2.21) describes N − 1 massles
fields π k and a massive field φ′ as in (2.13). The O(N) symmetry is not apparent any more, i.e. spontaneously broken down to the subgroup O(N − 1)2 . The
~ along the trough of the pomassless fields describe oscillations of the vector φ
tential (2.16), which is an (N − 1)-dimensional surface and holds the unbroken
2
Corresponds to rotations of the π k fields among themselves.
7
2 Spontaneous Symmetry Breaking
~ is chosen to point in the 1
Figure 2.2: Topview on the 2D-potential: Vector φ
direction. The left figure shows the Cartesian representation of the 2
scalar fields, while the right displays the analogy in polar coordinates.
O(N − 1) symmetry. Figure 2.1 shows the potential for the N = 2 case that has
the shape of a Mexican hat or the bottom of a punted wine bottle. As there is
~ = (φ1 , φ2)T satisfying the minimizing conan infinite number of directions of φ
dition (2.17) the potential holds an infinite number of vacua being all physically
equivalent. Choosing the field vector to point in the 1 direction (figure 2.2) induces a Lagrangian given by the above mentioned form (2.21) with one massless
shifted field along the 2 direction and a massive shifted field along the 1 direction.
Besides the Cartesian expression one is free to parametrize the shifted fields in
polar coordinates
φ(x) = ρ(x)eiθ(x) = (v + χ(x))eiθ(x) ,
v=
s
µ2
,
λ
(2.22)
that yields the Lagrangian

s

s

µ2 λ 3
2µ2
χ − λχ4  + 
χ + χ2  (∂θ)2 ,
L = v 2 (∂θ)2 + (∂χ)2 − 2µ2 χ2 − 4
2
λ
(2.23)
θ(x) can be identified as the massless field, while χ(x) appears to be massive.
We gained the same outcome as in the Cartesian representation. On the basis of
the complex representation though its easier to recognize the massless field θ(x)
describing oscillations along the trough of the potential, as was doing φ2 before.
8
2.2 Spontaneous Symmetry Breaking
This motion does not cost energy, in contrast to radial oscillations producing a
massive field.
2.2.3 Goldstone’s Theorem
As a matter of fact appears it, that the emergence of massless fields in consequence
of spontaneous symmetry breaking is a general result, known as Goldstone’s theorem. The O(N) symmetry exhibits n(O(N)) = N (N2−1) continuous symmetries,
i.e. has n(O(N)) generators T a . Consider H|0i = 0, the vacuum state having zero
energy, and the system being invariant under the symmetry group G composed
by n(G) generators T a . Then it follows
eiθ
aT a
|0i = |0i → T a |0i = 0,
(2.24)
i.e. the generator T a annihilates the vacuum. Upon symmetry breaking some
symmetry operations turn up that do not leave the vacuum invariant any more.
Consequently the corresponding generators do not annihilate the vacuum state
and exhibit zero energy:
∃T a : T a |0i =
6 0 → H(T a |0i) = [H, T a ]|0i = 0.
(2.25)
For example, we had in 2 dimensions O(2) symmetry with a single direction of
rotation corresponding to the single generator of the group. After spontaneous
symmetry breaking we obtained an infinite number of states with minimum energy, that can all be created by applying the single generator to a vacuum state.
Therefore, if the system appears to be invariant under the symmetry group G
with n(G) generators and is after symmetry breaking left invariant under a subgroup H < G, n(G) − n(H) generators emerge not annihilating the vacuum state.
This number of generators corresponds to the number of massless fields appearing
in the Lagrangian, called Nambu Goldstone bosons. One can simply show
that these Goldstone bosons have mass zero. For that purpose we start with the
Lagrangian of a theory of several fields φa (x)
L = (terms with derivatives) − V (φ)
(2.26)
9
2 Spontaneous Symmetry Breaking
~ 0 being the constant that minimizes V (φ):
and φ
∂
V
(φ)
= 0.
a
∂φa
φ (x)=φa
(2.27)
0
An Expansion of V about the minimum constant gives
1
∂2
V (φ) = v(φ0 ) + (φ − φ0 )a (φ − φ0 )b
V
2
∂φa ∂φb
with the coefficient
∂2
V
∂φa ∂φb
!
!
(2.28)
φ0
= m2ab
(2.29)
φ0
in front of the quadratic term, which is a symmetric matrix. The eigenvalues of
this matrix give the masses of the fields. Consider a general continuous symmetry
transformation of the form
φa → φa + α∆a (φ)
(2.30)
with α being an infinitesimal transformation ∆a (φ) a function of the φ’s. For the
constant field the derivative terms of L vanish and we get
V (φa ) = V (φa + α∆a (φ)) → ∆a (φ)
∂
V (φ) = 0.
∂φa
(2.31)
Differentiating (2.31) with respect to φb at φ = φ0 gives
0=
∂∆a )
∂φb
!
∂V (φ)
∂φa
φ0
|
{z
=0
!
∂2
+∆a (φ0 )
V
∂φa ∂φb
φ0
!
.
(2.32)
φ0
}
In case of unbroken symmetry the infinitesimal transformation ∆a (φ0 ) vanishes
and condition (2.32) is obvious. After symmetry breaking it holds that ∆a (φ0 ) 6= 0
and ∆a (φ0 ) is the eigenvector to the eigenvalue zero. So the Nambu Goldstone
boson has mass zero.
10
Chapter 3
Higgs mechanism
So far spontaneous symmetry breaking was discussed only for global symmetry.
This chapter shall go a step further and discuss the consequences of spontaneous
symmetry breaking in presence of local symmetry. Due to the fact that then
conventionally local gauge invariance is demanded, one has to include local gauge
theories next to the concept of spontaneous symmetry breaking in the same theory.
The combination of the two offers new prospects for constructing quantum field
theory models, i.e. a scheme for create massive bosons.
3.1 Basic points of gauge theory
Having local symmetry implies that the symmetry operations are dependent on
position in space U = U(x) ∈ G, i.e. varies from place to place. The elements of a
continuous symmetry group can be expressed in terms of the exponential function
a (x)T a
U(x) = eiα
,
(3.1)
where αa (x) are position dependent continuous parameters and {T a } the set of
generators of the symmetry group. In the case of an Abelian symmetry group
(3.1) reduces to U(x) = eiα(x) . To acquire a Lagrangian invariant under local
symmetry operations, the derivative
∂µ → Dµ = ∂µ − igAµ (x)a T a
Abelian
−→
∂µ + ieAµ
(3.2)
has to be replaced by the general covariant derivative Dµ , with Aaµ being the gauge
potentials and g a coupling constant. The gauge potentials are determined by the
11
3 Higgs mechanism
Lie algebra structure constants f abc of the underlying symmetry group:
Aaµ → Aaµ − f abc θb Acµ + ∂µ θa
Abelian
−→
1
Aµ + ∂µ α.
e
(3.3)
Furthermore the general field strength is given by
Fµν [A] =
i
h
1
[Dµ , Dν ] = ∂µ Aaν T a − ∂µ Aaν T a − ig Aaµ T a , Aaν T a
−ig
(3.4)
and reduces in the Abelian case to the familiar form
∂µ Aν − ∂ν Aµ .
(3.5)
Lastly the Maxwell Lagrangian has the form
1
L = − Fµν F µν .
4
(3.6)
3.2 The Anderson-Higgs mechanism
This section will introduce the analysis of gauge theories with spontaneous symmetry breaking, known as the Anderson-Higgs mechanism. The first part
will start with a simple case of an Abelian symmetry group and proceed in the
following part to generalize the analysis for the non-Abelian case.
3.2.1 Abelian
We start with the Abelian U(1) gauge theory, where the Lagrangian of a complex
scalar field coupled to an electromagnetic field and being invariant under the local
transformation
φ(x) → eiα(x) φ(x)
1
Aµ (x) → Aµ (x) − ∂µ α(x)
e
(3.7)
has the form
1
L = − Fµν [A]F µν [A] + (Dµ [A]φ)† Dµ [A]φ + µ2 φ† φ − λ(φ† φ)2
|
{z
}
4
−V (φ)
12
(3.8)
3.2 The Anderson-Higgs mechanism
with the covariant derivative Dµ = ∂µ + ieAµ . For the complex scalar field expressed in polar coordinates φ = ρeiθ L transforms into
1
L = − Fµν F µν + ρ2 (∂µ θ − eAµ )2 + (∂ρ)2 + µ2 ρ2 − λρ4 .
4
(3.9)
Note that comprising local symmetry, i.e. presence of gauge fields, changes the
term (∂µ θ)2 of the Lagrangian into (∂µ θ − eAµ )2 . We now define a field Bµ to be
the combination
1
(3.10)
Bµ = Aµ − ∂µ .
e
As Bµ is invariant under the gauge transformation (3.7) and
Fµν [A] = ∂µ Aν − ∂ν Aµ = ∂µ Bν − ∂ν Bµ = Fµν [B],
(3.11)
the Lagrangian can be expressed in terms of the field Bµ :
1
L = − Fµν [B]F µν [B] + ρ2 e2 Bµ2 + restterms.
4
(3.12)
(3.12) contains (second term) a mass expression for the field Bµ with the mass
value given by the expectation value of the radial field ρ(x). Actually the mass
of the field Bµ would be zero solely because the vacuum expectation value of
ρ(x) vanishes, unless the symmetry is spontaneously broken. For spontaneous
q
symmetry breaking induces a non zero vacuum expectation value v =
one gets
1
1
1
L = − Fµν [B]F µν [B] + M 2 Bµ2 + e2 vχBµ+ e2 χ2 Bµ2
4
2
2
|
{z
}
µ2 /λ and
interaction
√
λ
1
µ4
+ (∂χ)2 − µ2 χ2 − λµχ3 − χ4 +
4
4λ}
|2
{z
√
field χ with mass
(3.13)
2µ
when the radial field is shifted by χ(x) with respect to the vacuum expectation
value v. The Lagrangian (3.13) depicts the theory of a massive field Bµ with mass
M = ev,
(3.14)
13
3 Higgs mechanism
√
a field χ(x) with mass 2µ and the interaction between the two. The attendance
of the phase field θ(x) in the Lagrangian, namely the Nambu Goldstone in the
ungauged U(1) theory, is no longer obvious. It has disappeared though not completely. Furthermore due to the fact that the involvement of the Nambu Goldstone
boson, i.e. the consequence of spontaneous symmetry breaking, is necessary for
the gauge boson to acquire mass, one says that the gauge field Aµ has gained
mass by eating the Nambu-Goldstone boson. The mechanism of a massless gauge
field becoming massive upon spontaneous symmetry breaking by extinction of a
Nambu-Goldstone boson is known as the Higgs mechanism.
3.2.2 Non Abelian
Generalization
Consider a system of scalar fields {φi } in a Lagrangian invariant under the symmetry group G with the transformations
φi → (1 + iθa T a )ij φj .
(3.15)
Enlarging the symmetry group G to a local gauge symmetry, i.e. with the covariant
derivative (3.2), the kinetic term in the Lagrangian for the φi ’s is
1
1
1
(Dµ φi ) = (∂µ φi )2 + gAaµ (∂µ φi Tija φj ) + g 2 Aaµ Abµ (T a φ)i (T b φ)i.
2
2
2
(3.16)
Upon spontaneous symmetry breaking the φi ’s get vacuum expectation value (φ0 )i
and are expanded about these values. (3.16) then contains a term
∆L =
1 2 a
g (T φ0 )i (T b φ0 )i Aaµ Abµ
{z
}
2|
(3.17)
m2ab
giving the structure of the gauge boson mass. m2ab is a mass matrix, which is
positive semidefinite, whereby all gauge bosons receive positive masses. To get
the masses one has to diagonalize the mass matrix to find the mass eigenstates
with the corresponding mass eigenvalues. Considering the symmetry of a system
is spontaneously broken down from G to a subgroup H, n(G) − n(H) Nambu-
Goldstone bosons are eaten and just as much massive gauge bosons appear in
14
3.2 The Anderson-Higgs mechanism
case of a gauged theory. The corresponding generators T a generate the non zero
mass eigenvalues of the mass matrix. But the mass matrix also contains zero mass
eigenvalues. The remaining n(H) generators, precisely the generators still leaving
the vacuum invariant
T a φ0 = 0
(3.18)
(cf. (2.24)) are responsible for these zero eigenvalues.
Example: Glashow-Weinberg-Salam Theory of Weak Interactions
This subsection will give an example of application of the general formalism proposed in the last subsection. It will start with the Non-Abelian group SU(2) and
proceed with the spontaneously broken U(1) × SU(2) gauge theory, which is the
central point a model introduced by Glashow, Weinberg and Salam (GWS). This
model gives an appropriate unified description of weak and electromagnetic interactions. If we choose the spinor representation of SU(2), the covariant derivative
of a model with an SU(2) field coupled to a scalar field φ is
Dµ φ = (∂µ − igAaµ τ a )φ
(3.19)
with τ a = σ a /2 and σ a being the Pauli matrices






0 1 2 0 −i 3  1 0
σ1 = 
,σ =
,σ =
.
1 0
i 0
−1 0
(3.20)
Upon spontaneous symmetry breaking φ acquires the vacuum expectation value
 
1 0
hφi = √  
2 v
(3.21)
and the kinetic term
 
0
1 |Dµ φ|2 = g 2 0 v τ a τ b   Aaµ Abµ + . . .
2
v
|
{z
∆L
(3.22)
}
15
3 Higgs mechanism
exposes the gauge boson masses. As the first term of (3.22) is a sum over a and
P
b ( a,b ) it contains the expressions
1
τ a τ b + τ a τ b = {τ a , τ b } = δab
2
(3.23)
the mass term turns into
g 2 v 2 a aµ
A A .
8 µ
It comes up that all three gauge bosons equally gain the mass
∆L =
mA =
gv
.
2
(3.24)
(3.25)
It arises that whether all three generators of SU(2) are broken equally depends
on the representation of SU(2). If one chooses φ to transform after the vector
representation of SU(2) it shows up that only the gauge bosons corresponding to
the generators 1 and 2 gain mass while the third gauge boson remains massless.
The explicit analysis to the vector representation of SU(2) and an example for
SU(3) gauged theory can be found in [2]. With the yet achieved results we can
turn to the GWS theory. As the GWS model contains a massless photon but
the spinor representation of SU(2) exhibits no massless gauge bosons we add the
U(1) gauge symmetry, i.e. set up the U(1) × SU(2) gauge symmetry. The gauge
transformation is then
a a
φ → eiα τ eiβ/2 φ
(3.26)
in which the scalar field is assigned a +1/2 charge under U(1) transformation.
Using the formalism of the previous section the covariant derivative is given by
1
Dµ φ = (∂µ − igAaµ τ a − i g ′Bµ )φ
2
(3.27)
with Aaµ and Bµ being the to the U(1) and SU(2) symmetries corresponding gauge
bosons and g, g ′ different coupling constants. To get the mass term we evaluate
the square of (3.27) at the vacuum expectation value (3.21) and receive
1
1
∆L =
0 v gAaµ τ a + g ′ Bµ
2
2
=
16
 
0
1
gAbµ τ b + g ′ B µ  
2
v
i
v2 h 2 1 2
g (Aµ ) + g 2 (A2µ )2 + (−gA3µ + g ′Bµ )2 .
8
(3.28)
(3.29)
3.2 The Anderson-Higgs mechanism
The mass matrix holds three non zero mass eigenvalues and thus three massive
vector bosons, namely the W - and Z-bosons
1 v
(3.30)
Wµ± = √ A1µ ∓ iA2µ with mass eigenvalue mW = g
2
2
q
v
1
3
′
2 + g ′2 . (3.31)
g
Zµ0 = √ 2
gA
−
g
B
with
mass
eigenvalue
m
=
µ
Z
µ
2
g + g ′2
The forth mass eigenvector, orthogonal to Zµ0
Aµ = √
1
3
′
gA
+
g
B
µ
µ
g 2 + g ′2
(3.32)
corresponds to the zero mass eigenvalue mA =0.
17
Bibliography
[1] A. Zee, Quantum Field Theory in a Nutshell: (Second Edition). Princeton
University Press, 2010. 1
[2] M. E. Peskin and D. Schroeder, An Introduction to Quantum Field Theory.
Addison Wesley, 1996. 16
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