Particle Data Group entries for the eta and eta'.

would not only be invariant under chiral $SU(3)$-flavor transformations

where the are the SU(3) Gell-Mann matrices acting in flavor space and the phi_a (a=1,...,8) are some arbitrary numbers, but also invariant under chiral U(1) transformations

where the phase is the same for all flavors. As a consequence of these symmetries, the classical Noether currents

In QCD (i.e. in the full quantum field theory), these two symmetries are realized in different ways:

Chiral SU(3) is not only a symmetry of the classical QCD Lagrangian, but also of the full quantum field theory. However, while the QCD Hamiltonian is invariant under chiral SU(3), this symmetry is spontaneously broken in the ground state. As a consequence, eight light (they are not massless since the quark masses are only small, but not zero) Goldstone bosons appear: the pseudoscalar meson octet.

While chiral U(1) is a symmetry of the classical QCD Lagrangean,
in the quantum formulation it is no longer a symmetry.
The reason is the famous Adler-Bell-Jackiw anomaly^{1}

In other words, the chiral U(1) symmetry is broken already by the the quantum Hamiltonian. Thus there is no symmetry left to be spontaneously broken by the QCD vacuum and hence there should be no Goldstone boson. In fact, the lightest meson with the right quantum numbers is the .

Up to this point, everything is quite rigorous and theorists generally agree with each other. However, there are many interesting issues left regarding the .

- While the anomaly clearly suggests that one should not expect a light SU(3) flavor singlet pseudoscalar meson, it is not entirely clear what physical mechanism (in terms of quark and gluon degrees of freedom) is reponsible for the high mass of the .
- For the analysis as well as interpretation of many experiments involving the meson, it is necessary to know certain transition matrix elements. Lacking enough detailled knowledge about the microscopic structure of hadrons, one is often dependent on symmetries. Without the anomaly, the would be essentially a pion with different flavor quantum numbers, so that symmetry arguments would be applicable. In order to estimate the deviations of physical matrix elements from predictions based on symmetry arguments, it would be very helpful to understand the large mass of the microscopically.

From the quark model point of view, the main distinction between the and other pseudoscalar mesons arises from the fact that the is a flavor singlet.

The can thus mix with states that are purely gluonic, i.e.
contain no quarks or antiquarks. If one includes such mixing perturbatively,
one obtains a surprising result: Since all glueball states are expected to be
heavier than mesons in the pseudoscalar octet, simple quantum mechanics
tells us that the mixing of such states with the unperturbed
*lowers* the mass of the because increasing the Hilbert space
in general lowers the energy of the ground state. This is just the opposite
of what we want.

Fortunately, the larger Hilbert space is not the only difference between the Hamiltonian in the pion sector versus the Hamiltonian in the sector: Consider for example the QCD Hamiltonian in Coulomb gauge It is necessary to pick a gauge that does not have negative norm states - otherwise the above variational arguments are invalid! Consider the Fock space component where the meson consists of a quark an antiquark and a gluon. The pair is in a color octet state and thus an annihilation interaction (via the instantaneous Coulomb interaction) is possible for the meson. Such an interaction can be repulsive, and since it proceeds via the instantaneous interaction, there is no two gluon intermediate state and this contribution was not included in the above variational argument. Note that, even in the g Fock sector, an annihilation of the into a gluon is not possible for the pion because of its flavor quantum numbers.

Summarizing the discussion so far, we have discovered two effects which work in opposite directions:

- The can mix with ``pure glue'' states, which always lowers its mass
- There is an instantaneous annihilation interaction acting in higher Fock components for the , which can act repulsively

Note that while the annihilation interaction can increase the mass,
there is no general argument that it has to. Furthermore, without doing a
calculation it is not clear which of the two effects (mixing vs. annihilation)
is more important.
In order to answer this question, Donoghue and Gomm performed a
calculation in the bag model^{2}, and thier calculation
not only gave a repulsive effect due to the annihilation but also showed
that the repulsive effect is stronger than the mixing effect. In fact, they
were able to fit the mass with a reasonable set of mixing
parameters.

Unfortunately, the Donoghue-Gomm calculation is the only serious work in this direction and it is not clear whether the result is an artefact of the bag model or of the non-gauge invariant calculation scheme.

When one adds a term

to the QCD Lagrangean then the energy density in the vacuum (ground state) does in general depend on - this is similar to the fact that the energy density in a solid depends on the magnitude of an external magnetic or electric field.

However, for massless quarks, one can show that the dependence
of the vacuum energy density disappears. From this fact, Witten derived
a relation between the mass of the and the * topological
susceptibility*

where is the energy density of the vacuum in the presence of a angle and in
the absence of quarks.
This result is exact in the chiral limit to leading order in 1/N_{C}.

The correlator of vanishes in perturbation theory. In fact, it is believed that instantons play a leading, if not dominant, role for the r.h.s. of the last Eq. Obviously, this means that is nonperturbative as well - but this is of course true for all bound states!

Certainly, there are instantons in QCD and they contribute largely
(or dominantly) to the topological susceptibility. Also QCD has an
meson which is heavy (we know that from experiment).
Furthermore, there is an equation relating the two. But it would be
premature to conclude on the basis of Eq. (\ref{d2e}) alone that
instantons *cause* the mass. An interesting observation
was made by 't Hooft, who showed that integrating out instanton
degrees of freedom leads to an effective interaction which involves
a determinant of the quark fields. Practically speaking, that
determinant leads to a point-like quark-quark interaction, which
acts repulsively in the U_{A}(1) channel and thus provides a qualitative
understanding of how instantons could perhaps lead to modifications
in quark models.

There is no strict definition of the term would-be Goldstone boson. However, in the context of QCD, the would be considered a would-be Goldstone boson if it behaved (coupling constants, decay constants, form factor) like a ``heavy pion'' with the only difference being its flavor quantum numbers. Since many experiments involving are indirect measurements, extraction of its properties from the data is often model dependent. Therefore, it is currently impossible to tell from the currently available experimental data whether or not the is just a heavy partner of the pion.

Theoretically, all we can tell is that with the anomaly turned off
(e.g. by taking N_{C} to infinity the does indeed
become a Goldstone boson. But does this justify the term
``would-be Goldstone boson''? Clearly, if the mass shift were small
(for example, for ``QCD'' with 100 colors) one would expect from
that the wave function of the still looks almost like
the one for the pion. However, unfortunately the mass shift for
the is rather large. and therefore there is no a priori
reason to expect that the really is a ``would-be Goldstone boson''.
On the other hand, there is no evidence to the contrary either and
it would be very important to design experiments (or lattice calculations)
that can help answer this question. This issue was in part the motivation for
holding this workshop, but it turs out to be rather difficult to come
up with a feasable experiment to address this issue.
A gedanken experiment can be easily formulated: for example, one could look
at the deep inelastic structure function of both and .
This way one could see if for example the gluon content differs when comparing
these two mesons. Unfortunately, the lifetime of the is much to
short for performing a Drell-Yan experiment, but the above example at least
shows one way how one could perform such a would-be-Goldstone-boson test in
principle. A more practical way to do this would perhaps be to study the
fragmentation at e^{+}e^{-} colliders.

^{1}S.L. Adler, Phys. Rev. **177**, 2426 (1969);

J.S. Bell and R. Jackiw, Nuovo Cimento **60**, 47 (1969);

W. Bardeen, Phys. Rev. **184**, 1848 (1969).

^{2}E. Witten, Nucl. Phys. B **156**, 269 (1979).

^{3}J.F. Donoghue and H. Gomm, Phys. Rev. D **28**, 2800 (1983).

^{4}G. 't Hooft, Phys. Rept. **142**, 357 (1986).

Particle Data Group entries for the eta and eta'.