theoretical particle physics

1. Highlights From SU(2)XU(1) Basic Standard Model Construction
Ferdinand Joseph P. Roaa
, Alwielland Q. Bello b
, Engr. Leo Cipriano L. Urbiztondo Jr.c
a
Independent Physics Researcher, 9005 Balingasag, Misamis Oriental
b
Natural Sciences Dept., Bukidnon State University
8700 Malaybalay City, Bukidnon
c
IECEP, Sound Technology Institute of the Philippines
Currently connected as technical consultant/expert for St. Michael College of Caraga (SMCC)
8600 Butuan City, Agusan del Norte
Abstract
In this paper we present some important highlights taken from our study course in the subject of
Standard Model of particle physics although in this current draft we are limited only to discuss the basics of
SU(2)XU(1) construction. The highlights exclude the necessary additional neutrinos aside from the left-
handed ones which are presented here as massless.
Keywords: Standard model, gauge group, Lagrangian, doublet, singlet
1. Introduction
This paper serves as an exposition on an
initial and partial construction of SU(2)XU(1)
model in Quantum Field Theory whose complete
SU(2)XU(1) structure represents the Electro-
Weak Standard model. The discussions center on
Lagrangian that must be invariant or symmetric
under the SU(2)XU(1) gauge group. It must be
noted that the whole of The Standard Model has
the mathematical symmetry of the
SU(3)XSU(2)XU(1) gauge group to include the
Strong interaction that goes by the name of
Chromodynamics. Such is ofcourse beyond the
scope of this present draft.
In its present form, this paper is mainly
based on our group’s study notes that include our
answers to some basic exercises and workouts
required for progression. So we might have used
some notations by our own convenient choice
though as we understand these contain the same
notational significance as that used in our main
references.
The initial and partial SU(2)XU(1)
construction presented here is intended primarily
to illustrate gauge transformation of fields and
how such fields must transform so as to observe
invariance or symmetry of the given Lagrangian.
Concerning neutrinos, the Dirac left-
handed spinor doublet discussed here aside from
the left-handed electron it contains, it also has a
left-handed neutrino that is rendered massless in
the Yukawa coupling terms. In addition to these,
the other Fermion is the right-handed electron. As
there is only one left-handed spinor doublet and
one right-handed spinor singlet no other type of
fermions such as additional neutrinos are present
in this initial and partial SU(2)XU(1)
construction. In a later section, it will be shown
how this left-handed neutrino is made massless in
the mentioned Yukawa coupling terms.
2. Partially Unified Lagrangian

2. Let us start our highlights say with a
partially unified Lagrangian,
ℒ( 𝑆𝑈(2) × 𝑈(1)) 𝑃𝑎𝑟𝑡 = ℒ( 𝜓 𝐿, 𝜓2
𝑅
, 𝜙 ) +
ℒ( 𝑊, 𝐵 )
(1.1)
This is for fields under the 𝑆𝑈(2) × 𝑈(1) gauge
symmetry group [1]. In this, the necessary
additional fermions in the complete Electro-Weak
theory [2] are not yet included. The basic fermions
present here are contained in the component
Lagrangian
ℒ( 𝜓 𝐿,𝜓2
𝑅
, 𝜙 ) = 𝑖𝜓̅ 𝐿 𝛾 𝜇 𝐷𝜇(𝐿) 𝜓 𝐿 +
𝑖𝜓̅2
𝑅
𝛾 𝜇 𝐷𝜇(𝑅) 𝜓2
𝑅
−
𝑦( 𝜓̅2
𝑅
𝜙 † 𝜓 𝐿 + 𝜓̅ 𝐿 𝜙𝜓2
𝑅 ) +
1
2
| 𝐷𝜇 𝜙|
2
− 𝑉(𝜙)
(1.2)
This component Lagrangian incorporates a
Left-handed spinor doublet, 𝜓 𝐿, Right-handed
spinor singlet 𝜓2
𝑅
and scalar doublet 𝜙. The Left-
handed spinor doublet consists of initial Left-
handed Fermions – the left-handed neutrino 𝜓1
𝐿
and the left-handed electron, 𝜓2
𝐿
. The right-handed
spinor singlet represents for the right-handed
electron, while the scalar doublet represents for the
Higgs field, which consists of a vacuum
expectation value (vev) and a scalar component
called the Higgs Boson, then three Goldstone
bosons.
As a partially unified Lagrangian under
the cited gauge symmetry group, Lagrangian (1.1)
also consists of a component part ℒ( 𝑊, 𝐵 ) that
contains the three components of 𝑆𝑈(2) vector
gauge boson field 𝑊⃗⃗⃗ and one 𝑈(1) vector gauge
boson field, 𝐵𝜇. Such component Lagrangian is
given by[3, 4]
ℒ( 𝑊, 𝐵 ) = ℒ 𝑊 + ℒ 𝐵
(1.3)
where one subcomponent goes for the boson field
𝑊⃗⃗⃗
ℒ 𝑊 = −
1
4
𝐹𝜇𝜈 ∙ 𝐹 𝜇𝜈 = −
1
4
∑𝐹𝜇𝜈
(𝑖)
𝐹(𝑖)
𝜇𝜈
3
𝑖=1
(1.4)
(We note: Greek index as space index, while Latin
index as particle index.)
The anti-symmetric tensor 𝐹𝜇𝜈 in (1.4) is
given by
𝐹𝜇𝜈 = 𝜕𝜇 𝑊⃗⃗⃗ 𝜈 − 𝜕𝜈 𝑊⃗⃗⃗ 𝜇 − 2𝑄′𝑊⃗⃗⃗ 𝜇 × 𝑊⃗⃗⃗ 𝜈
(1.5)
The 𝑆𝑈(2) vector gauge boson takes three
components, 𝑊⃗⃗⃗ = (𝑊𝜇
(1)
, 𝑊𝜇
(2)
𝑊𝜇
(3)
), where
Latin indices take parameter values 1, 2, 3. In
short hand, we write for a component in the cross
product as [5]
𝐴 × 𝐵⃗ | 𝑎
= 𝜀 𝑎𝑏𝑐 𝐴 𝑏 𝐵 𝑐 (1.6)
This is written in terms of the components 𝜀 𝑎𝑏𝑐 of
Levi-Civita symbol.
The remaining subcomponent of (1.3) is
for the solely U(1) gauge boson 𝐵𝜇 whose
Lagrangian in turn is given by
ℒ 𝐵 = −
1
4
( 𝜕𝜇 𝐵 𝜈 − 𝜕𝜈 𝐵𝜇)
2
(1.7)
We must also take note the complex linear
combinations that give out the W-plus and W-
minus gauge bosons
𝑊𝜇
(±)
=
1
√2
(𝑊𝜇
(1)
± 𝑖 𝑊𝜇
(2)
) (1.8)
and the SO(2)-like rotations
𝑍 𝜇 = 𝐵𝜇 𝑠𝑖𝑛𝛼 − 𝑊(3)𝜇 𝑐𝑜𝑠𝛼 (1.9.1)
𝐴 𝜇
𝑒𝑚 = 𝐵𝜇 𝑐𝑜𝑠𝛼+ 𝑊(3)𝜇 𝑠𝑖𝑛𝛼 (1.9.2)
with respect to the mixing angle alpha, which
mixing (rotation-like) gives out one massive Z
field and one massless gauge boson that represents
the electromagnetic field 𝐴 𝜇
𝑒𝑚.
3. Transformations Under The
SU(2)XU(1) Subgroups
In this section, we highlight the left-
handed spinor doublet as the specific illustration
whose 𝑆𝑈(2) × 𝑈(1) 𝐿 subgroup is characterized
by the hypercharge 𝑌𝐿, a label we choose by our
own convenient notation. Such subgroup is
represented by the matrix
𝑒−𝑖𝑌 𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ (2.1)
This is in exponentiated form, where 𝜎𝑖 (𝑖 =
1, 2,3) are the Pauli matrices. We must make the
identifications

3. 𝑄′𝜎 ∙ 𝜒 = 𝑄′∑ 𝜎𝑖 𝜒𝑖
3
𝑖=1
𝜒 𝑞 = 𝑄′𝜒3
(2.2)
Associated with this particular subgroup is
the covariant derivative operator for the left-
handed spinor doublet as characterized also by the
hypercharge, 𝑌𝐿.
𝐷𝜇(𝐿) = 𝜕𝜇 + 𝑖𝑄𝑌𝐿 𝐵𝜇 + 𝑖 𝑄′∑ 𝜎𝑖 𝑊(𝑖)𝜇
3
𝑖=1
(2.3)
We see in this that the hypercharge goes along
with the U(1) gauge field.
We note in the matrix (2.1) the U(1) part
as given by 𝑒−𝑖𝑌𝐿 𝜒 𝑞, while the SU(2) part by the
2X2 matrix 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ . Under this subgroup, the
left-handed spinor doublet transforms as
𝜓 𝐿 → 𝑒−𝑖𝑌𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ 𝜓 𝐿 (2.4)
So to first order in 𝑄′ this will result in the
transformation of covariant derivative operation
𝐷𝜇(𝐿) 𝑒−𝑖𝑌𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ 𝜓 𝐿 =
𝑒−𝑖𝑌𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ ( 𝜕𝜇 + 𝑖𝑄𝑌𝐿( 𝐵𝜇 −
𝑄−1 𝜕𝜇 𝜒 𝑞)+ 𝑖𝑄′𝜎 ∙ ( 𝑊⃗⃗⃗ 𝜇 − 𝜕𝜇 𝜒 − 2𝑄′ 𝜒 ×
𝑊⃗⃗⃗ 𝜇) ) 𝜓 𝐿
(2.5)
For our present purposes let us take the
invariance of Lagrangian (1.1) with respect to the
transformation of the left-handed spinor doublet
that is given in (2.4) under the 𝑆𝑈(2) × 𝑈(1) 𝐿
gauge group. This invariance requires that the
gauge vector bosons must also transform in the
following ways
𝐵𝜇 → 𝐵𝜇 + 𝑄−1 𝜕𝜇 𝜒 𝑞 (2.6.1)
for the U(1) gauge field, while to first order in 𝑄′,
the 𝑆𝑈(2) vector boson transforms as
𝑊⃗⃗⃗ 𝜇 → 𝑊⃗⃗⃗ 𝜇 + 𝜕𝜇 𝜒 + 2𝑄′ 𝜒 × 𝑊⃗⃗⃗ 𝜇 (2.6.2)
Such transformations are needed to cancel the
extra terms picked up in (2.5) when the left-
handed spinor doublet transforms under its own
gauge subgroup.
For these results, it is fairly
straightforward exercise to obtain the following
approximated identity
𝜎 ∙ 𝑊⃗⃗⃗ 𝜇 𝑒−𝑖𝑌𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗
∙ 𝜒⃗⃗ 𝜓 𝐿 ≈
𝑒−𝑖𝑌 𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ ( 𝜎 ∙ 𝑊⃗⃗⃗ 𝜇 + 𝑖𝑄′[( 𝜎 0 ∙
𝜒),(𝜎 ∙ 𝑊⃗⃗⃗ 𝜇)] ) 𝜓 𝐿
(2.7.1)
in which we note of the commutator
[( 𝜎 ∙ 𝜒),(𝜎 ∙ 𝑊⃗⃗⃗ 𝜇)] = 𝑖2𝜎 ∙ ( 𝜒 × 𝑊⃗⃗⃗ 𝜇)
(2.7.2)
which is also a straightforward exercise to prove.
Given the SU(2) gauge transformation
(2.6.2), the W-gauge boson Lagrangian ℒ 𝑊 also
transforms as
−4ℒ 𝑊 = 𝐹𝜇𝜈 ∙ 𝐹 𝜇𝜈 → 𝐹𝜇𝜈 ∙ 𝐹 𝜇𝜈 +
2(2)𝑄′𝐹𝜇𝜈 ∙ (𝜒 × 𝐹 𝜇𝜈)
(2.8.1)
This is also taken to first order in 𝑄′. By cyclic
permutation we note that
𝐹𝜇𝜈 ∙ ( 𝜒 × 𝐹 𝜇𝜈) = 𝜒 ∙ ( 𝐹 𝜇𝜈 × 𝐹𝜇𝜈 ) = 0
(2.8.2)
This drops the second major term of (2.8.1) off,
proving the invariance of ℒ 𝑊 under gauge
transformation.
We can proceed considering the given
Spinor doublet under the 𝑆𝑈(2) × 𝑈(1) 𝐿 diagonal
subgroup whose matrix is given by
𝑒−𝑖𝑌 𝐿 𝜒 𝑞 𝑒−𝑖𝜎3 𝜒 𝑞 = 𝑑𝑖𝑎𝑔( 𝑒−𝑖(1+𝑌𝐿 )𝜒 𝑞, 𝑒 𝑖(1−𝑌𝐿 )𝜒 𝑞)
(2.9.1)
This matrix utilizes the 𝜎3 Pauli matrix and the
Spinor doublet transforms as
𝜓 𝐿 → 𝑒−𝑖𝑌𝐿 𝜒 𝑞 𝑒−𝑖𝜎3 𝜒 𝑞 𝜓 𝐿 (2.9.2)
It is to be noted that as a doublet this Spinor
doublet is a 2X1 column vector wherein each
element in a row is a left-handed Dirac spinor in
itself.
𝜓 𝐿 = (
𝜓1
𝐿
𝜓2
𝐿
) (2.9.3)
In this draft the authors’ convenient
notation for each of these left-handed Dirac
spinors is given by

4. 𝜓 𝑖
𝐿
=
1
2
(1 + 𝛾5) 𝜓 𝑖 (2.9.4)
with Hermitian left-handed ad joint spinor given
as
𝜓̅ 𝑖
𝐿
= (𝜓 𝑖
𝐿
)† 𝛾0 =
1
2
𝜓̅ 𝑖(1− 𝛾5)
(2.9.5)
In our notations, our fifth Dirac gamma matrix 𝛾5
has the immediate property
𝛾5 = −𝛾5 (2.9.6)
Alternatively, under this diagonal
subgroup and given (1.9.1) and (1.9.2), we can
write the covariant left-handed derivative operator
in terms of the 𝑍 𝜇 field and the electromagnetic
field, 𝐴 𝜇
𝑒𝑚.
𝐷𝜇(𝐿) = 𝜕𝜇 + 𝑖𝑄′( 𝜎1 𝑊(1)𝜇 + 𝜎2 𝑊(2)𝜇) +
𝑖𝑄′
𝑐𝑜𝑠𝛼
( 𝑌𝐿 𝑠𝑖𝑛2 𝛼 − 𝜎3 𝑐𝑜𝑠2 𝛼) 𝑍 𝜇 +
𝑖𝑄′( 𝜎3 + 𝑌𝐿 ) 𝐴 𝜇
𝑒𝑚 𝑠𝑖𝑛𝛼
(2.10)
It is to be noted that 𝑆𝑈(2) × 𝑈(1) 𝐿 is
non-Abelian gauge group whose generators (the
Pauli matrices) do not commute so that we can
have the following results
𝜎1 𝑒−𝑖𝜎3 𝜒 𝑞 = 𝑒−𝑖𝜎3 𝜒 𝑞( 𝜎1 𝑐𝑜𝑠2𝜒 𝑞 − 𝜎2 𝑠𝑖𝑛2𝜒 𝑞)
(2.11.1)
and
𝜎2 𝑒−𝑖 𝜎3 𝜒 𝑞 = 𝑒−𝑖𝜎3 𝜒 𝑞( 𝜎1 𝑠𝑖𝑛2𝜒 𝑞 + 𝜎2 𝑐𝑜𝑠2𝜒 𝑞)
(2.11.2)
As the Left-handed spinor doublet
transforms under (2.9.2) the covariant
differentiation with (2.10) also takes the
corresponding transformation
𝐷𝜇(𝐿) 𝑒−𝑖𝑌𝐿 𝜒 𝑞 𝑒−𝑖𝜎3 𝜒 𝑞 𝜓 𝐿 = 𝑒−𝑖𝑌 𝐿 𝜒 𝑞 𝑒−𝑖𝜎3 𝜒 𝑞( 𝜕𝜇 −
𝑖( 𝑌𝐿 + 𝜎3 ) 𝜕𝜇 𝜒 𝑞 + 𝑖𝑄𝑌𝐿 𝐵𝜇 +
𝑖 𝑄′( 𝜎1 𝑊′
(1) 𝜇 + 𝜎2 𝑊′
(2) 𝜇) +
𝑖 𝑄′ 𝜎3 𝑊(3)𝜇 ) 𝜓 𝐿 (2.12)
where we take note of the SO(2) like rotations
𝑊(1)𝜇 → 𝑊′
(1) 𝜇 = 𝑊(1)𝜇 𝑐𝑜𝑠2𝜒 𝑞 +
𝑊(2)𝜇 𝑠𝑖𝑛2𝜒 𝑞
𝑊(2)𝜇 → 𝑊′
(2) 𝜇 = −𝑊(1)𝜇 𝑠𝑖𝑛2𝜒 𝑞 +
𝑊(2)𝜇 𝑐𝑜𝑠2𝜒 𝑞 (2.13)
A quick drill would show the invariance
∑ 𝑊′
( 𝑖) 𝜇 𝑊′(𝑖)
𝜇
2
𝑖=1
= ∑ 𝑊( 𝑖) 𝜇 𝑊(𝑖)
𝜇
2
𝑖=1
(2.14)
Corresponding to the transformation
(2.12) of covariant differentiation is the U(1) like
gauge transformation of 𝑊(3)𝜇.
𝑊(3)𝜇 → 𝑊(3)𝜇 + 𝑄′−1 𝜕𝜇 𝜒 𝑞
(2.15.1)
These transformations consequently lead
to U(1) gauge transformation of 𝐴 𝜇
𝑒𝑚.
𝐴 𝜇
𝑒𝑚 → 𝐴 𝜇
𝑒𝑚 + 𝛿𝐴 𝜇
𝑒𝑚
𝛿𝐴 𝜇
𝑒𝑚 = ( 𝑄−1 𝑐𝑜𝑠𝛼 + 𝑄′−1
𝑠𝑖𝑛𝛼) 𝜕𝜇 𝜒 𝑞 =
2𝑒−1 𝜕𝜇 𝜒 𝑞
(2.15.2)
where
𝑄′ 𝑠𝑖𝑛𝛼 = 𝑄 𝑐𝑜𝑠𝛼 = 𝑒/2 (2.15.3)
The massive 𝑍 𝜇 field stays gauge invariant
𝑍 𝜇 → 𝑍 𝜇 + 𝛿𝑍 𝜇 = 𝑍 𝜇 (2.16.1)
since
𝛿𝑍 𝜇 = ( 𝑄−1 𝑠𝑖𝑛𝛼 − 𝑄′−1
𝑐𝑜𝑠𝛼 ) 𝜕𝜇 𝜒 𝑞 = 0
(2.16.2)
In order to conform with conventional or
that is standard notations, we may have to identify
the spacetime-dependent parameter 𝜒 𝑞 in terms of
Λ(𝑥 𝜇).
𝜒 𝑞 =
1
2
𝑒Λ (2.17)
so that the U(1) gauge transformation of the
electromagnetic field can be written as
𝐴 𝜇
𝑒𝑚 → 𝐴 𝜇
𝑒𝑚 + 𝜕𝜇 Λ (2.18)
4. The Yukawa Coupling
From (1.2) let us proceed with the
Yukawa coupling.
ℒ 𝑦 = −𝑦( 𝜓̅2
𝑅
𝜙 † 𝜓 𝐿 + 𝜓̅ 𝐿 𝜙𝜓2
𝑅 )
(3.1.1)
Under all (diagonal) subgroups of
SU(2)XU(1), the transformations lead to the
following end result
𝜓̅ 𝐿 𝜙𝜓2
𝑅
→ 𝜓̅ 𝐿 𝜙𝜓2
𝑅
𝑒−𝑖(1− 𝑌𝐿 )𝜒 𝑞 𝑒−𝑖 𝑌𝑅 𝜒 𝑞
(3.1.2)
or

5. 𝜓̅2
𝑅
𝜙 † 𝜓 𝐿 → 𝜓̅2
𝑅
𝜙 † 𝜓 𝐿 𝑒 𝑖𝑌𝑅 𝜒 𝑞 𝑒 𝑖(1− 𝑌𝐿 )𝜒 𝑞
(3.1.3)
We take note in here that to the right-
handed spinor singlet we attribute the hypercharge
𝑌𝑅. SU(2)XU(1) symmetry also requires the
Yukawa term to remain invariant under
SU(2)XU(1) gauge transformations. This
invariance requires a relation between
hypercharges that is given by
𝑌𝑅 = 𝑌𝐿 − 1 (3.2)
Under U(1) gauge subgroup the right-
handed spinor singlet transforms as
𝜓2
𝑅
→ 𝑒−𝑖𝑌 𝑅 𝜒 𝑞 𝜓2
𝑅
(3.3.1)
while under the SU(2)XU(1) the scalar doublet
transforms as
𝜙 → 𝑒−𝑖𝜒 𝑞 𝑒−𝑖𝜎3 𝜒 𝑞 𝜙 (3.3.2)
The values of the mentioned hypercharges
play important roles in the coupling or decoupling
of the fields involved in the Yukawa terms. For the
left-handed spinor doublet its hypercharge has the
value 𝑌𝐿 = − 1. This value decouples the left-
handed neutrino from the electromagnetic field so
that only the left-handed electron interacts with the
electromagnetic field. This can be seen in the
matrix
( 𝜎3 + 𝑌𝐿) 𝜓 𝐿 = (
0
−2𝜓2
𝐿) (3.4.1)
(As noted.)
( 𝜎3 − 1 ) 𝜓 𝐿 𝐴 𝜇
𝑒𝑚 = (
0
−2𝜓2
𝐿) 𝐴 𝜇
𝑒𝑚
(3.4.2)
In (3.2) we consider 1 as the hypercharge
given to the scalar doublet and with this value we
see in the following matrix
(1 + 𝜎3 ) 𝜙0 𝐴 𝜇
𝑒𝑚 = (
0
0
) 𝐴 𝜇
𝑒𝑚 (3.4.3)
that the electromagnetic field decouples from the
vacuum expectation value (vev) 𝜙0 of the Higgs
field thus, rendering this electromagnetic field
massless.
Conveniently, we can re-group the terms
in (3.1.1) so as to separate out a mass term and an
interaction term.
ℒ 𝑦 = ℒ 𝑦(𝑚𝑎𝑠𝑠) + ℒ 𝑦(𝑖𝑛𝑡) (3.5)
The mass term gives masses to the
electrons and the interaction term gives the
interaction of the Higgs boson with fermions that
have mass. This mass term basically gives the
interactions of the left-handed and right-handed
electrons with the constant real component 𝛽 of
the scalar doublet. (This constant real component
is the vacuum expectation value (vev) of the Higgs
field.) In these said interactions the mentioned
fermions acquire their masses in the process.
ℒ 𝑦(𝑚𝑎𝑠𝑠) = −𝑦𝛽( 𝜓̅2
𝑅
𝜓2
𝐿
+ 𝜓̅2
𝐿
𝜓2
𝑅 ) = −𝑦𝛽𝜓̅2 𝜓2
(3.6.1)
(Noted)
𝜓̅2
𝑅
𝜓2
𝐿
=
1
2
𝜓̅2(1 + 𝛾5) 𝜓2 (3.6.2)
𝜓̅2
𝐿
𝜓2
𝑅
=
1
2
𝜓̅2(1 − 𝛾5) 𝜓2 (3.6.3)
The left-handed neutrino is ultimately not
included in the mass term and the absence of this
fermion in this term signifies that the said fermion
does not interact with the constant real component
of the scalar doublet so it does not acquire mass.
The masses of the other fermions that do interact
with the constant real component of the scalar
doublet are directly proportional to that vev,
𝑚 𝜓 ∝ 𝛽 with y as the constant of proportionality.
In the other Yukawa interaction term, the
real scalar component (the Higgs boson 𝜂) of the
scalar doublet can be seen to interact with both the
left-handed and right-handed electrons.
ℒ 𝑦( 𝑖𝑛𝑡) = −𝑦𝜂( 𝜓̅2
𝑅
𝜓2
𝐿
+ 𝜓̅2
𝐿
𝜓2
𝑅 ) − 𝑦( 𝜑1 𝜓̅1
𝐿
+
𝑖𝜀 𝜓̅2
𝐿 ) 𝜓2
𝑅
− 𝑦𝜓̅2
𝑅( 𝜑1
∗
𝜓1
𝐿
− 𝑖𝜀𝜓2
𝐿)
(3.7)
Although in (3.7) we see that the massless
left-handed neutrino seems to interact with the
right-handed electron any such interaction will just
be removed by a gauge choice
𝑅𝑒[ 𝜑1] = 𝐼𝑚[ 𝜑1] = 𝐼𝑚[ 𝜑2] = 0 (3.8.1)
𝜑2 = 𝜂 + 𝑖𝜀
𝑅𝑒[ 𝜑2] = 𝜂
that sets the Goldstone bosons to vanish. After this
gauge choice is imposed, the interaction term (3.7)
will just contain the interaction of the Higgs boson

6. with those fermions that gain masses, the
electrons.
ℒ 𝑦( 𝑖𝑛𝑡) = −𝑦𝜂𝜓̅2 𝜓2 (3.8.2)
While the first generation fermions (with
the exception of the massless left-handed neutrino)
such as the left-handed and right-handed electrons
acquire their masses from the Yukawa coupling,
the massive vector gauge bosons such as the W
plus/minus and Z fields gain their masses from the
constant part of the kinetic term of the scalar
doublet.
1
2
| 𝐷𝜇 𝜙0|
2
= 𝑄′2 𝛽2 𝑊𝜇
(+)
𝑊(−)
𝜇
+
𝑄′2 𝛽2
2𝑐𝑜𝑠2 𝛼
𝑍 𝜇 𝑍 𝜇
(3.9.1)
This part contains the couplings of 𝑊𝜇
(±)
vector
gauge boson and 𝑍 𝜇 fields with the real non-zero
constant value 𝜙0 of the scalar doublet. It is in
these interactions that the named vector gauge
bosons and Z field acquire their masses. From
(3.9.1) we read off the mentioned masses
(squared)
1
2
𝑀 𝑊
2
= 𝑄′2 𝛽2, 𝑀 𝑍
2
=
1
2𝑐𝑜𝑠2 𝛼
𝑀 𝑊
2
(3.9.2)
For the case of the scalar doublet, we have
the diagonal subgroup from (3.3.2), which we
write explicitly in matrix form
𝑒−𝑖𝜒 𝑞 𝑒−𝑖𝜎3 𝜒 𝑞 = 𝑑𝑖𝑎𝑔( 𝑒−𝑖2𝜒 𝑞, 1)
(3.10.1)
This considering the hypercharge of the scalar
doublet as 𝑌 = 1.
Given the transformation (3.3.2) for the
scalar doublet, under the diagonal subgroup
(3.10.1), the covariant derivative operator can be
expressed as
𝐷𝜇 = 𝜕𝜇 + 𝑖𝑄′( 𝜎1 𝑊(1)𝜇 + 𝜎2 𝑊(2)𝜇) +
𝑖𝑄′
𝑐𝑜𝑠𝛼
( 𝑠𝑖𝑛2 𝛼 − 𝜎3 𝑐𝑜𝑠2 𝛼 ) 𝑍 𝜇 +
𝑖𝑄′( 𝜎3 + 1 ) 𝐴 𝜇
𝑒𝑚 𝑠𝑖𝑛𝛼
(3.10.2)
𝜙 = 𝜙0 + 𝜑 = (
0
𝛽
) + (
𝜑1
𝜑2
) (3.10.3)
𝜑2 = 𝜂 + 𝑖𝜀
𝑅𝑒[ 𝜑2] = 𝜂 (Higgs boson)
Goldstone bosons:
𝑅𝑒[ 𝜑1], 𝐼𝑚[ 𝜑1], 𝐼𝑚[ 𝜑2]
(3.10.4)
As already mentioned earlier in (3.8.1), these
Goldstone bosons must vanish.
Under the diagonal subgroup (3.10.1) as
the scalar doublet transforms as (3.3.2), the real
constant part 𝜙0 stays invariant and the Higgs
boson remains invariant as well since 𝜑2 is
invariant, while 𝜑1 transforms under a U(1) phase
transformation.
𝜑1 → 𝜑′1 = 𝑒−𝑖2𝜒 𝑞 𝜑1 (3.10.5)
In this initial construction, there remains
the Higgs boson to give mass to.
5. References
[1]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcour
se.html
[2] W. Hollik, Quantum field theory and the
Standard Model, arXiv:1012.3883v1 [hep-ph]
[3]Siegel, W., FIELDS, arXiv:hep-th/9912205 v2
[4]Griffiths, D. J., Introduction To Elementary
Particles, John Wiley & Sons, Inc., USA, 1987
[5]Arfken, G. B., Weber, H. J., Mathematical
Methods For Physicists, Academic Press, Inc., U.
K., 1995