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.