content_image/425922.jpg content_image/425924.jpg content_image/425923.jpg content_image/425928.jpg content_image/425931.jpg content_image/425933.jpg content_image/425929.jpg content_image/425930.jpg content_image/425932.jpg content_image/425925.jpg content_image/425926.jpg content_image/425927.jpg
The first non-trivial question is to look for the inter- polating QCD operator which triggers the J P C = 0 ++ isoscalar states from the vacuum. On the lattice this has been done in a variety of ways, mainly involving quark operators (such as ¯ qq , see also [48]). A clear candidate, often used for the gluonium, is given by the trace of the energy-momentum tensor, Θ $^{µν}$, which satisfies the trace anomaly equation [49], 

<img src='content_image/425922.jpg'>

Here β ( α ) = µ $^{2}$dα/dµ2 denotes the beta function, α is the running coupling constant, γ$_{m}$ ( α ) = d log m/d log µ 2 is the anomalous dimension of the current quark mass m , the symbol D$_{µ}$ denotes the dilatation current, and G a µν is the field strength tensor of the gluon field. The operator Θ, besides having the J P C = 0 ++ quantum numbers, is renorm-invariant. In addition, in the chiral limit of massless quarks, m$_{q}$ → 0, it is SU$_{R}$ (2) ⊗ SU$_{L}$ (2) chirally invariant. This is different from the usual ¯ qq interpolating field, which is not chirally invariant and mixes under the chiral transformations with the ¯ qiγ$_{5}$⃗ τq operator. Actually, under the operation q → γ$_{5}$q , the operator Θ is chirally even while ¯ qq is chirally odd. 

In perturbation theory one has 

<img src='content_image/425924.jpg'>

where 

and N$_{f}$ denotes the number of active flavors. We recall that in the large- N$_{c}$ limit α ∼ 1 /N$_{c}$ and G 2 ∼ ( N 2 c − 1), i.e. , is proportional to the number of gluons. Hence, in Eq. (10) there is a contribution scaling as N 2 c and flavor- independent, as well as another contribution, subleading in N$_{c}$ and scaling as N$_{c}$N$_{f}$ (the higher orders in, for in- stance, the MS scheme, generate the N 2 f terms which are dependent on the renormalization scheme). In the chiral limit we only have the gluonic operator contribution to Eq. (10), however, still some information on the quark degrees of freedom remains via the β -function. Thus, in the chiral limit we may distinguish between the gluonic and quark contributions by the large- N$_{c}$ and N$_{f}$ scaling behavior, 

## A. Interpolating fields 

<img src='content_image/425923.jpg'>

<img src='content_image/425928.jpg'>

For a scalar-isoscalar particle | n 〉 we have a non- vanishing matrix element to the vacuum for the dilatation current, 

and hence 

for the on-shell particles. From now on the mass of the scalar-isoscalar states is denoted as m$_{n}$ . The two- point correlation function of the energy-momentum ten- sor reads 

<img src='content_image/425931.jpg'>

The energy momentum tensor is conserved, therefore 

where the factor of 1 / 3 complies with Eq. (15). After inserting a complete set of states we find 

<img src='content_image/425933.jpg'>

with 

where c.t. stands for the counterterms. According to the N$_{c}$ counting rules, the leading part of Π$_{ΘΘ}$ scales as N 2 $_{c}$, and the next-to-leading part as N$_{c}$N$_{f}$ . P C ++ 

The sum in Eq. (19) runs over all J = 0 isoscalar states, i.e. both mesons ( m ) and glueballs ( g ). The dif- ference is, however, in the scaling of the coupling con- stants f$_{n}$ with N$_{c}$ . If all states are contributing and no cancellations occur, then the glueball must have a larger coupling, f 2 n ∼ N 2 $_{c}$, whereas the ¯ qq -meson has f 2 n ∼ N$_{c}$N$_{f}$ : 

The Operator Product Expansion (OPE) in QCD yields [50–54] 

## B. Two-point correlations 

<img src='content_image/425929.jpg'>

<img src='content_image/425930.jpg'>

<img src='content_image/425932.jpg'>

<img src='content_image/425925.jpg'>

<img src='content_image/425926.jpg'>

<img src='content_image/425927.jpg'>

