For the $U(1)$ problem, is the Kugo and Ojima Goldstone quartet wrong?

Question

On page 96 in "Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem", Prog. Theor. Phys. Suppl. 66 (1979) 1, KO state the following:

Finally we should comment on the current belief that the $U(1)$ problem was resolved by instanton. It is not correct; namely:, the instanton by itself cannot assure the unphysicalness of the $U(1)$ Goldstone boson $\chi$ which is contained in $J^{\mu}_5 $ because of the chiral $U(1)$ Ward identity ($ 7 \cdot 4$ ).No one has ever proved in the framework of "instanton physics" in a satisfactory manner that the $\chi$ really contained in the gauge-variant current $J^{\mu}_5$is not contained in the gauge-invariant one, $j^{\mu}_5$, although this problem has been discussed by many authors.

Are Kugo and Ojima correct? I thought 't Hooft proved that the instantons solved the $U(1)$ problem.

Answer 1

The expectation of the axial current divergence in a $\theta$ shifted $QCD$ vacuum is given by

$\partial_{\mu} \langle J^{\mu5}_{\mathrm{inv}} \rangle_{\theta} = 2m_q \langle \bar{q}i\gamma^5q \rangle_{\theta} + \langle \Xi \rangle_\theta,$

where the first term on the right hand side is the explicit breaking term due to the quark masses and the second term,

$ \Xi = \frac{2 g^2N_f}{32\pi^2}\ F_{\mu\nu }^a F^{\mu\nu a},$

comes from the contribution added to the current to make it gauge invariant. The volume density of this term is called the topological susceptibility.

According to 't Hooft, the expectation value in the second term is nonvanishing, since it is proportional to the expectation of the fluctuation of the index in the presence of instantons:

$\langle \Xi \rangle_\theta = \langle(N_+-N_-)^2\rangle_{\theta}.$

There is strong evidence from lattice QCD that this contribution is nonvanishing. This means that the $U(1)$ axial symmetry is explicitly broken even in the absence of quark masses, therefore the $\eta^{\prime}$ is not a Goldstone which is a possible explanation of its large mass compared to the “Goldstone Octet”

Kugo argues that if we use the gauge variant current which is the true Noether current instead of the gauge invariant one we get the usual current conservation broken only due to the quark masses, thus naively $\eta^{\prime}$ could be a Goldstone. However, he argues that this gauge variant Goldstone field belongs to a quarted obtained by the action on it by the BRST operator thus experiences quartet decoupling from the physical spectrum just as the temporal, longitudinal gluon polarizations and ghosts in the Kugo-Ojima theory.

Answer 2

I think 't Hooft and Kugo are solving different problems.

't Hooft addresses the issue that the anomaly involves a topological term. As a result, in perturbation theory there is no theta dependence and the anomaly equation by itself does not solve the $U(1)$ problem. He shows that topological objects (semi-classically, instantons) generate theta dependence. He also shows that instanton zero modes provide a microscopic mechanism for chirality non-conservation, and that instantons generate an effective vertex that can give a mass to the $\eta'$.

Kugo addresses the issue that the anomaly involves a total divergence. As a result I can define a gauge dependent current which is conserved, and therefore has a Goldstone pole. This means that I should show that this pole does not couple to any gauge invariant currents. Kugo provides a mechanism (BRST quartet), by which this can happen.