Chiral Symmetry and the Isospin

We now proceed with the program outlined at the end of the previous section, namely, knowing from experiment that mesons exist we begin by introducing the relevant degrees of freedom. We also know that meson is a complicated solution of the QCD quark and gluon fields that involve a real quark-antiquark pair. However, without ever being able to find this solution, let us try to identify basic features of the meson that result from the underlying QCD structure.

Let us concentrate on a small piece of the QCD Lagrangian density (21),
i.e., on the up and down quark components of the middle term, i.e.,

To be specific, contains 24 components, i.e., two quarks, each in three colors, and each built as a four-component Dirac spinor. However, the Dirac and color structure is again not essential, so in the present section we may think about as two-component spinor. For a moment we have also disregarded the quark mass terms - we reinsert them slightly below.

What is essential now are the symmetry properties of .
This piece of the Lagrangian density looks like a scalar in the
two-component field , i.e., it is manifestly invariant with
respect to unitary mixing of up and down quarks. We formalize this
observation by introducing the isospin Pauli matrices,

What is slightly less obvious, but in fact trivial to anybody
acquainted with the relativistic Lorentz group, is the fact that
is also invariant with respect to multiplying the quark fields
by the Dirac matrix shown in Eq. (18). This property
results immediately from the commutation properties of the
matrices (remember that =). So in fact, we have
altogether six symmetry generators of , namely,

It is now easy to identify the symmetry group of . We
introduce the left-handed and right-handed
generators,

i.e., generates the SU(2) group, generates another SU(2) group, and since they commute with one another, the complete symmetry group is SU(2)SU(2). We call this group chiral.

This result is quite embarrassing, because it is in a flagrant disagreement with experiment. On the one hand, we know very well that particles appear in iso-multiplets. For example, there are two nucleons, a neutron and a proton, that can be considered as upper and lower components of an iso-spinor, and there are three pions, , , and , that can be grouped into an iso-vector. So there is no doubt that there is an isospin SU(2) symmetry in Nature, but, on the other hand, what about the second SU(2) group? In the Lorentz group, the Dirac matrix changes the parity of the field, so if was really a symmetry then particles should appear in pairs of species having opposite parities. This is not so in our world. Nucleons have positive intrinsic parity, and their negative-parity brothers or sisters are nowhere to be seen. Parity of pions is negative, and again, the positive-parity mirror particles do not exist any near the same mass.

So the Nature tells us that the SU(2)SU(2) symmetry of the QCD Lagrangian must be dynamically broken. It means that the Lagrangian has this symmetry, while the physical solutions do not. We already learned that these physical solutions are very complicated, and we are unable to find them and check what are their symmetries. But we do not really need that - experiment tells us that chiral symmetry must be broken, and hence, we can built theories that incorporate this feature on a higher level of description.

Before we construct a model in which the dynamical symmetry breaking
mechanism is explicitly built in (and before we show explicitly what
such a symmetry breaking really is), let us first reinsert the
quark-mass terms into the discussed piece of the Lagrangian:

That is about this far that we can move forward by using the QCD quark Lagrangian. We have identified basic symmetry properties of the QCD solutions, and now we have to go to the next level of description, namely, consider composite objects built of quarks. This way of proceeding is called the effective field theory (EFT). We do not build fields of composite objects from the lower-level fields. Instead, we consider the composite objects to be elementary, and we guess their properties from symmetry considerations of the lower-level fields; otherwise, it would have been too difficult a task. Before we arrive at sufficiently high energies, or small distances, at which the internal structure of composite objects becomes apparent, we can safely live without knowing exactly how the composite objects are constructed.