Using the method of QCD sum rules, we estimate the energy of the lowest hybrid charmonium
state, and find it to be at the energy of the

A hybrid meson is a state composed of a quark and antiquark color octet, along with a valence gluon, giving a color zero
particle. Hybrids are of great interest in studying the nature of QCD. The
nonperturbative method of QCD sum rules [

The

In Section

We now will use
the method of QCD sum rules to attempt to find the lowest hybrid charmonium
state, assuming that such a pure hybrid charmonium meson with quantum numbers

The starting point of the method of QCD sum rules is
the correlator, which for a hybrid meson is

QCD sum rule
study of a state

Next,

After a Borel transform,

This sum rule is used to estimate the heavy-hybrid
mass,

Since mesons with certain quantum numbers, such as

For heavy-quark hybrids, the higher-order terms in the OPE are quite small, so the QCD sum rule method is more accurate. Since we are not trying to predict exotic hybrids, however, there is the serious complication of nonhybrid meson-hybrid meson mixing, which will be discussed in detail in what follows. This is an important aspect of our present work.

There have been many lattice QCD calculations of
glueballs and light-quark hybrids [

For a hybrid meson with quantum numbers

It is important to recognize that the correlator used in the QCD sum rule method is similar to the correlator used in the lattice gauge approach, with the same objective of finding the mass of a heavy-hybrid meson.

As described in the preceding subsection, the correlator is evaluated in the method of QCD sum rules via an OPE and a dispersion relation.

The QCD sum rule method uses an OPE in dimension (or
inverse momentum) (

Lowest-order term in sum rule.

Using the standard quark and gluon propagators (see
Appendix

Extracting the scalar correlator

The second term in the OPE for the heavy-quark hybrid
correlator includes the gluon condensate, illustrated in Figure

Gluon condensate term in sum rule.

For this
process, the correlator

To ensure convergence of the OPE of the correlator,
one performs a Borel transform

Using the equations in Appendix

In a similar way, taking the Borel transform of

The method of QCD sum rules uses a dispersion relation
for the correlator, which it equates to the correlator's operator product.
Following the usual convention, we call the dispersion relation the LHS and the
OPE the RHS:

For convenience in carrying out the sum rule, we have
fit the RHS of the correlator to a polynomial in the Borel mass,

We find that for all values of

Taking the ratio of

The result of the QCD sum rule fit is shown in Figure

QCD sum rule for heavy-hybrid charm meson.

The parameters

In our treatment of hadronic decays of the

In energy regions where there are both scalar mesons
and scalar glueballs, it is expected that

The QCD sum rule calculation makes use of the correlator

Scalar glueball-meson coupling theorem.

The results of the QCD sum rule calculations [

80% scalar glueball at 1500 MeV

80% scalar meson at 1350 MeV

Light Scalar Glueball 400–600 MeV

The Sigma/Glueball model follows from the existence of
the

The solution to the

Lowest-order PQCD diagram for a

Decay of a

The corresponding hybrid decay involves the matrix
element

Assuming that the 2 s state is a

Using the method of QCD sum rules, we have shown that
the

There have been many lattice calculations of exotic
hybrid mesons. There is experimental evidence for an exotic light-quark

It is interesting that the energy difference between
the

The current to create a heavy-quark hybrid meson with

The first term in the OPE, shown in Figure

The correlator for the process shown in Figure

The next term in the OPE for this heavy-quark system,
where quark condensates are negligible, is the gluon condensate term, shown in
Figure

From this, one finds

A key method that enables one to use the operator
expansion to get accurate sum rules is the use of the Borel transform [

Two key equations which we need are (with

Transforms used in the body of the paper are

This work was supported in part by the NSF/INT Grant no. 0529828. The authors thank Professors Pengnian Shen, Wei-xing Ma, and other IHEP, Beijing colleagues for helpful discussions. They thank Professor Y. Chen for discussions of lattice QCD in comparison to QCD sum rules for hybrid states.