field theory - physics

1. Highlights From SU(2)XU(1) Basic Standard Model Construction
Roa, F. J. P., Bello, A., Urbiztondo, L.
Abstract
In this draft we present some important highlights taken from our study course in the subject of Standard
Model of particle physics although in this present 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:
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 draft, 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
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.

2. 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 𝑊⃗⃗⃗𝜈 − 𝜕𝜈 𝑊⃗⃗⃗ 𝜇 − 2𝑄′𝑊⃗⃗⃗𝜇 × 𝑊⃗⃗⃗𝜈
(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)
𝐴𝜇
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 𝐴𝜇
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
𝑄′𝜎 ∙ 𝜒 = 𝑄′∑ 𝜎𝑖 𝜒 𝑖
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−𝑖𝑌 𝐿 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−𝑖𝑌 𝐿 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 ∙ ( 𝑊⃗⃗⃗𝜇 − 𝜕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 transformin 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
𝑊⃗⃗⃗𝜇 → 𝑊⃗⃗⃗𝜇 + 𝜕𝜇 𝜒 + 2𝑄′
𝜒 × 𝑊⃗⃗⃗ 𝜇 (2.6.2)
Such transformations are needed to cancel the extra

3. 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−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗
∙ 𝜒⃗⃗
𝜓 𝐿
≈
𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ ( 𝜎 ∙ 𝑊⃗⃗⃗𝜇 +
𝑖𝑄′[( 𝜎 ∙ 𝜒), (𝜎 ∙ 𝑊⃗⃗⃗𝜇 )] ) 𝜓 𝐿
(2.7.1)
in which we note of the commutator
[( 𝜎 ∙ 𝜒), (𝜎 ∙ 𝑊⃗⃗⃗𝜇 )] = 𝑖2𝜎 ∙ ( 𝜒 × 𝑊⃗⃗⃗𝜇 )
(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+𝑌 𝐿)𝜒 𝑞
, 𝑒 𝑖(1−𝑌 𝐿)𝜒 𝑞)
(2.9.1)
This matrix utilizes the 𝜎3 Pauli matrix and the
Spinor doublet transforms as
𝜓 𝐿
→ 𝑒−𝑖𝑌 𝐿 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
𝜓𝑖
𝐿
=
1
2
(1 + 𝛾5) 𝜓𝑖 (2.9.4)
with Hermitian left-handed ad joint spinor given as
𝜓̅ 𝑖
𝐿
= (𝜓𝑖
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, 𝐴𝜇
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
) 𝐴𝜇
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−𝑖𝜎3 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 𝐴𝜇
𝑒𝑚
.
𝐴𝜇
𝑒𝑚
→ 𝐴𝜇
𝑒𝑚
+ 𝛿𝐴𝜇
𝑒𝑚

4. 𝛿𝐴𝜇
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
𝐴𝜇
𝑒𝑚
→ 𝐴𝜇
𝑒𝑚
+ 𝜕𝜇 Λ (2.18)
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4. Partial Higgs Boson theory
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5. Quantum Electrodynamics (QED)
pieces
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6. Concluding Remarks
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7. Acknowledgment
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8. References
[1]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.
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
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