1. R.R. Silva, M.E. Bracco, S.H. Lee, M. Nielsen
The Z (4430) as a Tetraquark State +
What is the state ?
R.R. Silva, M.E. Bracco, S.H. Lee, M. Nielsen
Introduction:
The Z [1] is an exotic meson observed in the pi+psi’ invariant mass spectrum in the B meson decay into K+(pi+psi’). Therefore the simplest quark content of Z is (udcc) and it can not be made of two quarks. Many conjectures have been made about how these quarks are organized, Fig.1.
Xiang Liu et al. [2] calculated the mass of Z assuming it is a molecular state D*D1 in the meson exchange effective theory. Z + + +
(tetraquark state)
diquark-anti-diquark
molecular state +
Fig.1
VIII International Workshop Relativistic Aspects of Nuclear Physics (RANP08) CBPF-RJ (11/2008)

2. 2. QCDSR
The QCDSR are an approximate phenomenological treatment of nonperturbative effects in QCD.The QCDSR calculation of hadron masses centers around the two-point correlation function:
The fundamental assumption of the QCDSR is the principle of quark-hadron duality. Specifically, we assume that there is an interval over which a hadron may be equivalently described at both the quark level and at the hadron level.
Lee et al. [3] calculated the mass of Z in the framework of QCD sum rules (QCDSR) [4] using the current likes the molecular type, but there is not meson exchange. The interaction between the D* and D1 is due the QCD vacuum.
In this work we explore another possibility for Z in QCDSR. We consider Z meson as a diquark-anti-diquark state with IG(JP)=1+(0-). A possible current describing such state is given by:
where a,b,... are color indices and C is the charge conjugation matrix. + +
. (2)
j(x) =
, (1)

3. Inserting Eq.1 into Eq.2 we obtain the correlation the correlation function at the quark level, the right hand side of the sum rule (RHS). The non-perturbative contributions are considered in full quark propagator [5]. We have used the factorization approximation for multi-quark condensates, and have use the fixed-point gauge. The correlation function at the phenomenological level is calculated by defining,
Inserting in Eq.2 the completed states of Z+ we obtain the left hand side of the sum rule (LHS).
where is the spectral density. The spectral density is described, as usual, as a sharp pole representing the lowest resonance plus a smooth continuum representing higher mass states:
On the RHS, we consider the contributions of condensates up to dimension eight. To keep the charm quark mass finite, we use the momentum-space-expression for the charm quark propagator. The light quark part of the correlation function is calculated in the coordinate-space.Then, the resulting light-quark part is Fourier transformed to the momentum space.
. (5)
. (3)
, (4)

4. . (6) . (7) After equating the Eq.4 and Eq.6,
The correlation function in the RHS can be written as:
where
with
After equating the Eq.4 and Eq.6,
we making a Borel transform to both sides, and transferring the continuum contribution to the RHS
assuming quark-hadron duality. The sum rule, is given by:
where the dimension-eight condensate, is given by:
The integration limits are:
. (6)
. (7)

5. 3. Results
In the complete theory, the mass extracted from the sum rule should be independent of the Borel mass M. To determine the allowed Borel window, we analyse the OPE convergence and the pole contribution. To fix the continuum threshold range we extract the mass from the sum rule Eq.7 and accept such value of So if the obtained mass is around 0.5 GeV smaller than sqrt(So).
In the numerical analysis, the input values are:
Fig.2: The OPE convergence in the region M >2.2 GeV
for sqrt(So)=4.9 GeV. Perturbative contribution (dotted line),
contribution (dashed line), contribution (long-
dashed line), (dot-dashed line) and the total
contribution (solid line). 2
Fig.3: The dashed line shows the relative pole continuum
contributions for sqrt(So)=4.9 GeV.

6. Acknowledgements References In [6] we also calculated Z with JP=1-,
but we get a bigger mass for this state, thus we conclude that the Z is
a JP = 0− state.
Acknowledgements
This work has been partly supported
by FAPESP and CNPq-Brazil, and
By the Korea Research Foundation.
References
[1] BELLE Coll, PRL (2008)
[2] Xiang Liu et al., PRD (2008)
[3] S. H. Lee at al., PLB66128(2008)
[4] M.A. Shifman, A.I. and Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979).
[5] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127, 1 (1985).
[6] Bracco et al., PLB671240(2009)
Δ=sqrt(So) -m Z
Δ>0.5 GeV
Δ<0.4 GeV
Fig.4: The Z mass as a function of the Borel mass for different values of sqrt(So): sqrt(So)=4.8 GeV dashed line, sqrt(So)=4.9 GeV solid line and sqrt(So)=5 GeV dot-dashed line. The crosses indicate the Borel window.
Taking into account the variations
on M , s0 and m in the regions
indicated above we get:
which is a little bigger than the
experimental value [1]. 2 c