Pseudoscalar gluonia

We shall be concerned with the correlator:

where QR(x) is the renormalized gluon topological density which mixes under renormalization with the divergence of the flavour singlet axial current mix as follows [129] (see Section 10.3.3 in Part III):

where:

and we have quoted the formulae for nf flavours. The correlation function X(k2) obeys the inhomogeneous RGE [260]:

The anomalous dimension is:

The extra RG function β(L) (so called because it appears in the longitudinal part of the Green function of two axial currents) is given by

The RGE is solved in the standard way, giving

where αs(t) is the running coupling. The different QCD diagrams contributing to the correlator are shown in Fig. 37.1.

The perturbative expression for the two-point correlation function in the scheme is [455]:

The non-perturbative contribution from the gluon condensates (coming from the next lowest dimension operators in the OPE) is [382]:

Pseudoscalar meson-gluonium mixing

Let us consider the off-diagonal two-point correlator:

shown in Fig. 37.2.

Its QCD expression reads [458]:

where one can notice that the mixing from the OPE vanishes in the chiral limit. However, one should notice that this mixing acts on the gluonium propagator, that is, it affects the mass splitting but not its decay width which is governed by a three-point function. Unfortunately, several authors mix these two features in the literature.