452Paper

On SU(2)L × U(1)Y , Weak Isospin, and Weak Hypercharge in the
GSW Model of Electroweak Interactions
A study of continuous groups∗
Isaac Mooney†
University of Michigan Physics Department
(Dated: May 1, 2015)
Abstract
Abstract: The group SU(2), the special unitary group of degree 2, is a non-Abelian group under
matrix multiplication consisting of all two dimensional complex unitary matrices having a determi-
nant of one. Physically, this group corresponds to rotations of complex spinors in real space. This
group is perhaps most important in the theory of the electroweak interaction. This interaction,
described by the gauge group SU(2) x U(1), where U(1) is the circle group, has a symmetry with
weak isospin, the analogue to isospin in the strong interaction, which gives rise to the three gauge
bosons, W+, W−, and W0. In this paper, we will delve further into the relation between the SU(2)
x U(1) gauge group and weak isospin, after ﬁrst covering the requisite group theoretic concepts.
∗
A hearty thanks to Professor Deegan for a wonderful semester
†
imoo@umich.edu
1

I. INTRODUCTION
As we will rely heavily on group theory for this introduction to the electroweak theory, we
will cover some of the key concepts first. Next, via representation theory, we will use them
to understand symmetries which underly physical processes, and finally we will examine the
consequences of broken symmetry in a system.
II. GROUP THEORY
A group is simply a set which equipped with a binary operation, which satisfies four
axioms: (1) Closure under said operation, (2) Associativity, (3) Existence of an identity
element in the group, and (4) Every element has an inverse which is also in the group[5].
These groups can be either discrete or continuous. Although the study of finite groups is a
rich field of mathematics, we will focus on that of infinite groups for their greater application
to modern theoretical physics (we will not quite get to CPT symmetry).
A. Lie Groups
We now would like to consider groups whose elements vary analytically with respect to
a set of parameters. A Lie group, on the most basic level, is a group which satisfies this
condition, as well as the usual axioms of a group, altered slightly because of the group’s
continuous nature. [13]. We define the dimension of the Lie group as the number of linearly
independent parameters.
Another useful concept is that of the Lie algebra which we associate with our Lie groups.
In essence, the Lie algebra is the part of a Lie group within a small interval of the identity,
which corresponds to infinitesimal transformations[10]. More technically, it is a vector space,
V , over a field, F, with an associated binary operation which takes V × V → V with V
(where V is called the Lie bracket) satisfying the axioms: bilinearity, asymmetry, and the
existence of the Jacobi identity.
2

1) Bilinearity:
[ax + by = a[x, z] + b[y, z] (1)
[z, ax + by] = a[z, x] + b[z, y] (2)
for all a, b ∈ F and x, y, z ∈ V
2) Asymmetry:
[x, y] = −[y, x] (3)
for all x, y ∈ V .
3) Jacobi Identity:
[x, y], z + [z, x], y + [y, z], x = 0 (4)
for all x, y, z ∈ V .
B. Generators
Recall that we considered the Lie algebra for the fact that the elements correspond to
infinitesimal transformations. This leads to another definition: the generators of a Lie group.
These are the set of group elements which cannot be contained within a subgroup of the
group unless that subgroup is the entire group. It turns out that these elements correspond
to infinitesimal vectors which generate the group locally, by the exponential map, which
we now show. Take some generator, X, and a parameter λ, by which the elements vary
analytically. When λ is infinitesimal, as desired, we denote it dλ. We know for a group
element, G, G(λ) |λ=0= 1, so for dλ close to the identity, we can Taylor expand this, and
keep only the first order term:
G(dλ) = 1 + idλX + ... (5)
For this to apply to general λ, we can express it as λ
N
repeated N times, where N is large
enough that this expression is small. So, [1]
G(λ) = lim
N→∞
1 +
iλX
N
N
(6)
3

But this is the definition of the exponential map, so we can rewrite:
G(λ) = exp(iλX) (7)
Differentiating this equation:
−i
dG(λ)
dλ λ=0
= X (8)
And for a general set of parameters, λi, we have the general form:
Xi = −i
∂
∂λi
G(λ)
λ=0
, i = 1, ..., N (9)
This is a wonderful result because the behavior of an element in the group is given entirely
by its behavior in some infinitesimally small parameter range. We have left to check that the
product of two Lie group elements expressed in terms of these generators is again within the
group, as it must be by closure. Take two elements: Gl = eiλX
, Gm = eiµX
. Their product
is given by:
eiλX
eiµX
= 1 + iλlXl + O(λ2
) 1 + iµmXm + O(µ2
) (10)
= 1 + i(λ + µ)lXl − λlµmXlXm + O(λ2
) + O(µ2
) (11)
= 1 + i(λ + µ)lXl −
1
2
(λ + µ)l(λ + µ)mXlXm (12)
−
1
2
λlµm[Xl, Xm] + O(λ2
) + O(µ2
) (13)
One might recognize a semblance of a power series for exp(i(λ+µ)X) in the first three terms
of this last expression, and we want this to be equal to exp(iνX) for ν = λ + µ, in order for
this closure to be satisfied. However, the commutator of the generators is non-zero, so we
require that it be a linear combination of generators:
[Xl, Xm] = i
n
flmnXn = fn
lmXn (14)
where the coefficients of this linear combination are called structure constants, and charac-
terize the Lie group. One last definition will be useful before continuing on, which is that
of rank - the maximum number of mutually commuting, independent generators.
4

III. REPRESENTATION
A. Representation of Discrete Groups
Now that we have covered these concepts, it will be illuminating to discover the resulting
physics. There is a rich interplay between the worlds of physics and mathematics in this
particular area. To make this possible, we use representations of group elements which act
the same way, but have more physical signiﬁcance. A representation of a group G takes its
elements to a set of linear operators. There are two constraints on these operators:
1) P(e) = 1, where 1 is now an operator (named the identity operator).
2) P(a)P(b) = P(ab), for a, b ∈ G. So the group operation carries over to the new repre-
sentation, and this representation mapping takes the group identity element to the identity
operator in the vector space of the linear operators. We also have that every representation
of a discrete group is a unitary representation.
As an example of a representation of a discrete group, take the dihedral group D3[14], which
is the symmetry group of order 6 of a 3-sided regular polygon. A unitary representation is
given by[1]:
U(I) =


1 0
0 1

 , U(C3) =


−1
2
1
2
√
3
−1
2
√
3 −1
2


U(C2
3 ) =


−1
2
−1
2
√
3
1
2
√
3 −1
2

 U(C2) =


1 0
0 −1


U(C2) =


−1
2
1
2
√
3
1
2
√
3 1
2

 , U(C2 ) =


−1
2
−1
2
√
3
−1
2
√
3 1
2


Clearly we have that the identity element gives the identity operator as desired, and using
the fact that C2C2 = C2
3 ,
U(C2)U(C2) =


1 0
0 −1




−1
2
1
2
√
3
1
2
√
3 1
2

 , (15)
U(C2C2) = U(C2
3 ) =


−1
2
−1
2
√
3
1
2
√
3 −1
2

 (16)
So this has been satisﬁed for two of the group elements, and it would be easily checked
that the rest satisfy this requirement, as well. We would now like to know whether or not
5

this representation can be broken into direct sums of smaller representations. If so, we call
it reducible. If not, it is an irreducible representation. In the former case, there must be
a unitary similarity transformation by which the representation is a single block diagonal
matrix with each term in the direct product as each block.
B. Representation of Continuous Groups
We consider the group of rotations in three-dimensional space, SO(3) - the special orthog-
onal group in 3 dimensions, so called because rotation of a set of vectors preserves angle, so
the matrix of the representation we deﬁne must be orthogonal, and because the determinant
of these rotation matrices will be 1 (hence the special moniker). For rotation about the third
axis, our representation will be:
R(λ) =





cos(λ) sin(λ) 0
−sin(λ) cos(λ) 0
0 0 1





(17)
and this matrix is already in block diagonal form.
IV. SYMMETRY AND INVARIANCE
Consider, momentarily, an equilateral triangle with vertices labeled A, B, and C. If one
were to permute any of the two vertices, the appearance of the original triangle would be
recovered. This is due to an inherent symmetry of the equilateral triangle under rotations
by 120◦
. Obviously, these rotations come with certain properties: if we perform the 120◦
rotation three times, we should regain the original triangle with the vertices in the initial
position; if we perform two of these rotations, it is the same as performing one in the
opposite direction on the initial position. The set of these permutations, with composition
as its binary operation, forms what is called the (rotation) symmetry group of the equilateral
triangle, S3 [2]. The elements of this group are the permutations of the triangle which leave
it unaltered, i.e. the group of all isometries (distance-preserving maps) under which the
triangle is invariant, which is to say that it remains unchanged under transformation. S3 is
a non-Abelian group, i.e. its operation - composition - is non-commutative; performing two
6

successive permutations, is not in general the same as performing the latter first, and the
first, subsequently.
This is an example of a discrete symmetry group. More specifically, it is a finite point
group, meaning that we have no translational symmetry. However, there are also continuous
symmetry groups. For a simple example, consider rotations of a sphere. In general, these
continuous symmetry groups are Lie groups.
Noether’s theorem illustrates an extremely useful property. Informally, a continuous
symmetry in a system corresponds to conserved quantities. Conservation of current is given
by the local continuity equation ∂µjµ
= 0. As an example of a conserved current, take the
Klein-Gordon equation, for which the Lagrangian density (a locally defined Lorentz scalar
field which, integrated over all spacetime, gives the action of the system) is
L =
∂ψ
∂xν
∂ψ∗
∂xµ
+ m2
ψψ∗
(18)
which is invariant under complex rotations, so Noether’s theorem gives us that the conserved
current is:
jν
= i
∂ψ
∂xµ
ψ∗
−
∂ψ∗
∂xµ
ψ ηνµ
(19)
which, upon multiplication by charge, will yield the electric current density. This is a good
starting point for gauge theory.
V. GAUGE THEORY
Much of the work we have already done was in preparation for gauge theory, defined
as a field theory in which the Lagrangian is invariant under a continuous group of local
transformations (also called gauge transformations[8]). These transformations make up the
theory’s symmetry group (a Lie group as mentioned previously). As before, this Lie group
has generators forming an associated Lie algebra. Again, as we discussed previously, the
gauge group generators will have an associated vector field which is coined a gauge field. In
quantum field theory, where the fields are quantized, the quanta manifest in the gauge fields
by what are called gauge bosons. One gauge group we are interested in is SU(2)L × U(1)Y .
First, we will define each of these groups.
7

VI. SU(2)
SU(2) =


µ −ν∗
µ ν∗

 | µ, ν ∈ C, |µ|2
+ |ν|2
= 1 The corresponding Lie algebra is:
su(2) =


ir −z∗
z −ir

 | r ∈ R, z ∈ C (20)
with generators
u1 =


0 i
i 0

 , u2 =


0 −1
1 0

 , u3 =


i 0
0 −i

 (21)
These are familiar to the old Pauli matrices, and in fact, u1 = iσ1, u2 = −iσ2, u3 = iσ3,
so the obvious representation for this group is one analogous to spin. We are now able to
use our familiar angular momentum kets: |j, m and raising and lowering operators (the
process is too involved for the length of this paper) to deﬁne the spin j representation of
su(2) via the elements of the corresponding matrix: [Jj
a]kl = j, j +1−k|Ja|j, j +1−l . The
representation of SU(2) gives rise to what is called weak isospin, which is a concatenation
of isobaric spin. It is a property of a particle relating to the weak force. The subscript on
su(2)L is indicative of the fact that the chirality of the particle will determine whether the
particle is a member of a singlet or doublet. A doublet is a two-particle system of particles
with the same quantum number (weak isospin is denoted T). As with angular momentum,
the third component rivals the total in importance, and it is more common to see T3 than T.
T3 is conserved under weak interactions, because of the underlying SU(2) symmetry. The
doublets are:


νe
eL

 ,


νµ
µL

 ,


ντ
τL

 ,


uL
dL

 ,


cL
sL

 ,


tL
bL

 (22)
and the matrix representation of SU(2) acts on them as Ta
= σa
2
for a = 1, 2, 3, whereas
fermions with positive chirality have T = 0, and do not interact weakly. They are the
singlets:
eR, µR, τR, uR, cR, tR, dR, sR, bR (23)
And the gauge bosons are named W+
, W−
, and W0
.
8

VII. U(1)
U(1) ∼= T = {z ∈ C | |z| = 1}, and the exponential map for the group is θ → eiθ
since
these are 1 × 1 unitary matrices. The irreducible representations of this group are just the
automorphisms of the circle group, ρn(eiθ
) =


cos(nθ) −sin(nθ)
sin(nθ) cos(nθ)

 for n ∈ Z>0. This
representation gives rise to weak hypercharge and has a corresponding vector boson Bµ.
Weak hypercharge is a quantum number which relates weak isospin to electric charge:
Qf = Tf,3 + Yf (24)
where Tf,3 is the third component of the weak isospin, Yf is the hypercharge, and Qf is the
electric charge of the field. We take the gauge group of our unified Glashow-Salem-Weinberg
electroweak theory to be SU(2)L × U(1)Y .
VIII. SYMMETRY BREAKING
Consider the simple example of a ball sitting on the peak of a hill. Any slight fluctuation
of the surrounding system could push the ball and cause it to roll down the hill. When an
observer sees the initial state, there is a symmetry about the vertical axis of the ball + hill
system, but when the ball rolls to the bottom, although the ball still retained its obligatory
spherical symmetry, and the hill retained it’s reflection symmetry (when viewed from the
side), the system is no longer symmetric from that vantage point. This is spontaneous
symmetry breaking – the surrounding system, i.e. the vacuum, is not invariant, so the
entire system including the vacuum will not be invariant, despite the fact that the equations
of motion of the system remain so under the process.
The particular symmetry breaking which is of importance for our gauge group is the
Higgs mechanism. The Higgs field spontaneously breaks the symmetry and gives particles
mass. The specifics of the Higgs mechanism are beyond the scope of this paper, and in fact
there are many possibilities from which to choose (e.g. Abelian Higgs mechanism, Affine
Higgs mechanism, etc.) but the result is:
SU(2)L × U(1)Y → U(1)EM (25)
9

where the generator of this new group is Q = Y
2
+ T3 and the gauge transformations are a
combination of the previous transformations[11]:
Φ → Φ = exp iθ


1 0
0 0

 Φ (26)
The product of this symmetry breaking consists of the morphing of the W0
and B0
bosons
into the Z0
boson, and the photon, denoted γ, via the Weinberg angle, θW as follows:


γ
Z0

 =


cos(θW ) sin(θW )
−sin(θW ) cos(θW )




B0
W0

 (27)
We still have redundant 4 degrees of freedom with the production of a Higgs boson, and the
W+
, W−
, and Z bosons.
IX. CONCLUSIONS
We have merely scratched the surface of an extremely interesting and complicated inter-
weaving of physical and mathematical ﬁelds, in which exciting research is still being done.
The aim was to familiarize ourselves with the underlying mathematics and apply our under-
standing to the many intricate formulations within the Standard Model electroweak theory
in order to better conceptualize the various phenomena.
[1] Harris Arfken, Weber. Mathematical Methods for Physicists. Academic Press, 7th edition,
2012.
[2] Fred Richman Bernard Johnston. Numbers and Symmetry. CRC Press, 1 edition, 1997.
[3] G. Moore C. Burgess. The Standard Model: A Primer. Cambridge University Press, revised
edition, 2007.
[4] J.F. Cornwell. Group Theory in Physics, volume 1 and 2. Academic Press, 1984.
[5] Howard Georgi. Lie Algebras in Particle Physics. Westview Press, 2nd edition, 1999.
[6] Howard Georgi. Weak Interactions and Modern Particle Theory. Dover Publications, 2009.
[7] Volker Heine. Group Theory in Quantum Mechanics. Pergamon Press, 1960.
[8] Wolfgang Pauli. Relativistic ﬁeld theories of elementary particles. Rev. Mod. Phys. 13: 203–32,
1941.
10

[9] P. Renton. Electroweak Interactions. Cambridge University Press, 1990.
[10] P.A. Rowlatt. Group Theory and Elementary Particles. Longmans, Green and Co. Ltd., 1966.
[11] J.D. Wells S.P. Martin. Lecture notes on elementary particle physics, part i. 2015.
[12] L.-F. Li T.-P. Cheng. Gauge Theory of Elementary Particle Physics. Oxford University Press,
1988.
[13] M.J.G. Veltman. Lie groups in physics. English version - G. ’t Hooft.
[14] Eric Weisstein. Dihedral group.
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