Subject: Physics 558 – Lecture 1 Winter 2003
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Date: Mon, 23 Jan 2006 14:22:49 -0600

Physics 558 – Lecture 1

Local Gauge Symmetries and
Gauge Bosons III – A (Quick) Review and some Group
Theory:

In
Lecture 14 last quarter we discussed the structure of a field theory with a
local gauge symmetry described by a Lie group.  The algebra of the group is specified by
the structure constants Cjkl
in

where the Tk are the n generators of the
group, k = 1 to n.  We
start with matter fields that are necessarily in a representation of the group,
typically the fundamental representation.
For the Standard Model the fundamental fields are fermions, the quarks
and leptons.  The general
local gauge transformation of the matter fields is then defined in terms
of an n-plet of scalar functions and is represented
by

where the tk are an appropriate (matrix)
representation, typically the fundamental representation, of the generators
Tk.   The gauge covariant derivative is
defined in terms of the n required vector gauge boson fields,
Bmk,

which also serves to define the
fundamental coupling constant g.
The corresponding gauge transformations of the vector gauge boson fields
take the form

The
associated field strength tensor is

The
Lagrangian that is invariant under such local gauge
transformations has the general form

This Lagrangian does not admit a mass term for the vector gauge
particle.  A term
like

is not gauge invariant due to
the term in the transformation of the vector field arising from the
derivative.  Under a gauge
transformation the gauge field does not simply “rotate”, which would leave this
term unchanged, but it is also “translated” by the space-time derivative term in
the gauge transformation.  It is the
derivative related terms in the transformation of the vector gauge particle that
cancel the terms that arise from the derivatives in the kinetic energy of the
matter fields and that ensure the overall invariance of the Lagrangian under
local gauge transformations.
It is these derivative terms that ensure that mass terms for the gauge
boson are not invariant.  Thus the
requirement of local gauge invariance and the presence of massless
gauge bosons are inextricably connected, whether the symmetry is Abelian or
non-Abelian.

We
also want to enlarge our command of the underlying group theory as applied to
such symmetric theories.  The matter
fields must appear in degenerate (i.e., all with the same mass)
irreducible (i.e., we cannot write the representation as a sum of smaller
representations – if we start with any member of the representation, we will get
to every other member of the representation by operating with some
element of the group) representations of the underlying symmetry group.  Thus it will be helpful to have a
language for characterizing these representations and we want to introduce some
useful notation.  As we have already
noted in the SU(2) case, the members of irreducible
representations are labeled by the eigenvalues of the generators (operators) in
the largest commuting (Abelian) subalgebra, called the Cartan subalgebra.  The dimension of the Cartan subalgebra
is called the rank of the algebra (and the group) and tells us how many quantum
mechanical operators can be simultaneously diagonalised.  Recall that for the special unitary
groups, SU(N), the dimension of the algebra (the number
of generators) is N2 – 1.
The rank is N – 1.  Thus for
the case of SU(2) (and, therefore, SO(3)) the rank is 1
and each member of each irreducible representations is characterized by the
eigenvalue of a single generator (operator).  This is typically taken to be Iz (or Jz in SO(3)).

The
irreducible representations themselves are characterized by the eigenvalues of
the so-called Casimir operators, the set of operators, typically nonlinear
functions of the generators, which commute with all of the generators.  (Note that the Casimir operators are not
themselves members of the algebra).
By Schur’s lemma the representation of a
Casimir operator is a multiple of the identity matrix, i.e., every
member of a given irreducible representation is an eigenvector of the Casimir
operator with the same eigenvalue.
For SU(2) of isospin there is a single Casimir
operator, which is just the total isospin

with eigenvalues i(i+1).
[For SO(3) the Casimir is the total angular
momentum J2.]
As we already know, the members of the irreducible representations of
isospin (or angular momentum) are represented by the 2 numbers i and m,

The individual states are
connected by the “ladder operators”

The possible irreducible
representations of SU(2) correspond to all the positive
real integers and half-integers (i.e., to these values for the isospin
i).
If we consider products of irreducible representations, we typically
obtain reducible representations, which can be expressed as sums of irreducible
representations.  For continuous
symmetries, the resulting quantum numbers (eigenvalues of the “total” Casimir
operator – the total isospin) are obtained from the appropriate “addition” of
the quantum numbers of the individual representations being added.  For example, the product of 2 isospin ½
representations (call them A and B) can be reduced into isospin 1
and isospin 0 representations,

As we have already
discussed, the specific details of this decomposition are provided by the
Clebsch-Gordan coefficients.  In
terms of the eigenvalues of the individual isospins,
IA and IB, and the total isospin,

and the corresponding z
components, we have

This formula allows us to
express any member of the resulting product representation as a linear
combination of the members of the original representations in the
product.

Now
we want to move on to the symmetry groups of rank 2.  Of particular interest is the group
SU(3).
This group has application to both the global SU(3) of quark flavors and the local gauge symmetry of
QCD.  In the former case the
fundamental representation is the triplet of (lowest mass) quarks flavors
u, d, s.  In the latter case the fundamental
representation is the triplet, a = 1, 2, 3 (or red, green, blue), with
one such triplet for each of the 6 quark flavors (f).  In the case of a local symmetry (QCD)
there are 8 gluon fields Gmk corresponding to the 8
generators of SU(3).  The structure constants are C jkl = f jkl, where these numbers can be found on our
SU(3) web page.
The generators are the Fk and
are represented by the lk matrices, tk = lk/2, which are also discussed
on the web page.  We will include
some of the results from the web page here.  The algebra is
given

where the rank
3 tensor f jkl is antisymmetric
in all indices and the non-zero values are given in the following table (plus
the antisymmetric permutations).

ijk

123

147

156

246

257

345

367

458

678

fijk

1

The
conventional forms of the fundamental representation of the generators (often
called the Gell-Mann matrices, the analogue of the Pauli matrices) are

These
matrices satisfy the following identities

and

where
dijk is a
symmetric rank 3 tensor with (nonzero) values (plus the symmetric
permutations) given in the following table.

ijk

111

146

157

228

247

256

338

344

dijk

ijk

355

366

377

448

558

668

778

888

dijk

We
know from our earlier discussion that individual members of an irreducible
representation of SU(3) will be labeled by 2 constants,
i.e., this is a group of rank 2.
These are conventionally chosen to be the eigenvalues of the generators
F3 and F8.   This point is already explicit in
the matrices above.  It is
l3 and l8 that are diagonal.

For
the global (approximate) SU(3) flavor symmetry, the
“physical” identification of these two generators is with isospin and
hypercharge

where the explicit form in terms
of the l matrices is for the
fundamental representation whereas the general identification applies to any
flavor SU(3) multiplet.  This is why
we plotted the representations of the baryons and mesons in a plane with axes
I3 and Y.
To connect to what we learned about the SU(3)
flavor symmetry we note that for the basic triplet of quarks, u, d, s, we
have

For the case of color SU(3) there is no corresponding special role for these two
generators, although we will continue to use the I notation.  We are typically only interested in the
singlet representation 1 (the hadrons), the triplet 3 (the quarks)
and the octet 8 (the gluons).

Whether we are considering
the irreducible representations of color SU(3) or of
flavor SU(3), the primary change from the representations of SU(2) is that the
representations are now 2-D structures rather than 1-D.  The two dimensions are required to
represent to the 2-quantum number labels that identify the individual members of
the representation.  The analogue of
the ladder operator of isospin is a set of 3 independent ladder
operators,

which can be thought of as
describing 3 SU(2) subalgebras and as generating steps along 3 axes in the 2-D
plane that are at 60° with respect to each other
as illustrated in the figure.  These
operators satisfy a set of suggestive commutation relations that follow from the
algebra of SU(3)

The
interested student should verify these relations and their interpretation in the
figure above.

With these operators we can
understand (and eventually decompose) representations of SU(3) much as we did with those of SU(2) (although the latter
exercise is much simpler).  Consider
a given irreducible SU(3) representation and start by
selecting the member of this representation with the largest value of
I3 (recall that this quantum number need not be the familiar
isospin).  This state is often
called the state of highest weight, Ymax.  It follows from the definitions that
this state is annihilated by those operators that step to larger
I3,

The
first result is familiar from SU(2).  As with the case of isospin, we can
identify the rest of the representation by operating with the lowering
operators.  The difference here is
that we have 3 such operators.  The
SU(3) representation will be labeled by two integers,
p and q (instead of the
single integer in the SU(2) case), that tell us when the “ladder of
states” terminates.  Thus we
have

The
3 states, Ymax,
V-pYmax and I-qV-pYmax, lie on the lower right
boundary of the representation as suggested in the figure.  Continuing in a similar manner utilizing
all of the SU(3) ladder operators, it is
straightforward to verify the following properties of the SU(3) representation
(p,q) as displayed in the “3-8” (or
I3-Y) plane.

·
The boundary is a convex
six-sided figure, which is symmetric under 120° rotations and reflections
in the vertical (8) axis.

·
There are no unfilled sites
on the boundary or inside.  There is
a single state at each point on the boundary, two states at each point on the
contour 1 step inside the boundary, three states at each point on the contour 2
steps inside the boundary and so on until the contour has a triangular shape,
when there are no more extra layers.

·
The total number of states
in the representation is

.  Thus we can identify

·
At the multi-occupied sites
the distinct members of the representation have different eigenvalues of the
SU(2) Casimir I2 (or
U2 or V2).

·
The simplest Casimir
operator for SU(3) has the form

The
other Casimir is cubic in the Fk and
it is simpler to label the representations in terms of (p,q).

For
the question of decomposing products of SU(3)
representations into irreducible representations, the most efficient notation is
that of Young diagrams.  These are
just left justified arrays of boxes with a specific set of (seemingly ad
hoc) rules for their manipulation and interpretation.  The rules include the
following.

1.
Each horizontal row of boxes
is at least as long as the horizontal row below it.

1.
We can think of the
horizontal direction as symmetrization (with respect to some internal index) and
the vertical direction as anti-symmetrization.  There are at most n rows for the case of
SU(n).

2.
For the SU(3) representation (p,q)
the first row has p more boxes than the second row and the second row has
q more boxes than the third row.   Thus we have

3.
The counting of states
within a given representation involves filling in the boxes starting with the
upper left hand corner.  For SU(N) you put N in that box and then increase the number when
moving to the left and decrease the number when moving down.  An example is
.  Next we must define a “hook”.  A hook is the set of boxes that form a
“right hook”, moving first up and then right.  For the previous example there are 3
possible hooks, 1 involving all three boxes, one involving only the right most
box and one involving only the bottom box, .  Without proof, we note that the number
of states in the representation represented by a Young diagram is given by the
product of all the boxes with numbers in them (i.e., the product of the
numbers in the boxes) divided by the product of the lengths (number of boxes) of
the hooks.  For the example above
for SU(3), we have  as expected.  Two other examples to test your
understanding are

To
actually combine multiplets, i.e., define a product of representations,
we need to carefully label things.
Here we use the notation of the PDG (see
http://pdg.lbl.gov/2000/youngrppbook.pdf).
Consider the product of 2 octets,

where we use boxes to represent
the first octet and letters for the second (with “a” for the first row, “b” for
the second, etc.).  Now we
proceed to “add the boxes” with the following rules.

1.
Start with the left-hand
Young diagram (the boxes)
.

2.
Add the “a’s” in all ways that produce a valid Young diagram, but
with no more than a single “a” in each column (initially symmetric labels cannot
be antisymmetrized)

.

3.
Starting in the second row
(where the “b’s” were initially) add the “b’s” subject to the constraint that, reading from right to
left starting at the end of the first row and moving on to the second row, the
number of “a’s” must be ³ the number of “b’s” (³ the number of “c’s”).  Thus the
allowed Young diagrams are

.

4.
Using the rules noted
earlier we can work out the multiplicity of each of these irreducible
representations, either with the “hooks” formula or with (p,q).

Thus
the final result (as we have already noted)
is

.

Looking ahead to the
application to the SU(3) of color we can reproduce some
other results that we have already used.
Consider the product of a quark and antiquark,

Next consider the product of
3 quarks, but begin by looking at 2 quarks,

With the third quark we
have

In
the context of color we are interested only in the color singlets for the mesons
and baryons respectively.  Applied
to the SU(3) of flavor, we see again that the mesons
should appear in octets and singlets while the baryons should form decuplets,
octets (of differing internal permutation symmetry) and singlets of flavor.  However, not all of these states can be
combined (with space, color and spin wave functions) to yield states with the
required overall asymmetry under permutations.  For example, the antisymmetric color
wave function requires net symmetry in the other quantum numbers.  For the ground state we expect the space
wave function to be symmetric.  The
spin wave function is either symmetric (S = 3/2) or mixed (S =
1/2).  Thus only the flavor
symmetric 10, with spin 3/2, and the appropriately mixed symmetry
8, with spin 1/2, can appear in the baryon ground
state.