# The Possible $J^{PC}=0^{--}$ Exotic State

Chun-Kun Jiao<EMAIL_ADDRESS>Wei Chen<EMAIL_ADDRESS>Hua-Xing Chen
<EMAIL_ADDRESS>Shi-Lin Zhu<EMAIL_ADDRESS>Department of Physics and
State Key Laboratory of Nuclear Physics and Technology
Peking University, Beijing 100871, China

###### Abstract

In order to explore the possible existence of the exotic $0^{--}$ state, we
have constructed the tetraquark interpolating operators systematically. As a
byproduct, we notice the $0^{+-}$ tetraquark operators without derivatives do
not exist. The special Lorentz structure of the $0^{--}$ currents forbids the
four-quark type of corrections to the spectral density. Now the gluon
condensates are the dominant power corrections. None of the seven
interpolating currents supports a resonant signal. Therefore we conclude that
the exotic $0^{--}$ state does not exist below 2 GeV, which is consistent with
the current experimental observations.

exotic state, QCD sum rule

###### pacs:

12.39.Mk, 11.40.-q, 12.38.Lg

##  I Introduction

Most of the experimentally observed hadrons can be interpreted as
$q\bar{q}$/$qqq$ states and accommodated in the quark model Amsler:2008zzb ;
Klempt:2007cp . Up to now there has accumulated some evidence of the exotic
state with $J^{PC}=1^{-+}$ Adams:2006sa ; Abele:1999tf ; Thompson:1997bs .
Such a quantum number is not accessible for a pair of quark and anti-quark. It
is sometimes labelled as an exotic hybrid meson with the particle contents
$\bar{q}g_{s}G_{\mu\nu}\gamma^{\nu}q$. Recently we have investigated the
$1^{-+}$ state using the tetraquark currents Chen:2008qw . The extracted mass
and characteristic decay pattern are quite similar to those expected for the
exotic hybrid meson. Such a result is expected. Since the gluon field creates
a pair of $q\bar{q}$ easily, the hybrid operator
$\bar{q}g_{s}G_{\mu\nu}\gamma^{\nu}q$ transforms into a tetraquark
interpolating operator with the same exotic quantum number. In quantum field
theory different operators with the same quantum number mix and tend to couple
to the same physical state.

Using the same tetraquark formalism developed in the study of the low-lying
scalar mesons Chen:2006hy  and the exotic $1^{-+}$ mesons Chen:2008qw , we
study the possible $J^{PC}=0^{--}$ states composed of light quarks. For a
neutral quark model state $q\bar{q}$, we know that $J=0$ ensures $L=S$ hence
$C=(-)^{L+S}=+1$. In other words, states with $J^{PC}=0^{--},0^{+-}$ are
strictly forbidden. On the other hand, the gauge invariant scalar and
pseudoscalar operators composed of a pair of the gluon field are
$g^{2}_{s}G_{\mu\nu}^{a}G^{a\mu\nu}$ and
$\epsilon^{\mu\nu\alpha\beta}g^{2}_{s}G_{\mu\nu}^{a}G^{a}_{\alpha\beta}$, both
of which carry the even C-parity.

We construct all the local tetraquark currents with $J^{PC}=0^{--}$. There are
two kinds of constructions: $(qq)(\bar{q}\bar{q})$ and $(\bar{q}q)(\bar{q}q)$.
They can be related to each other by using the Fierz transformation. As usual,
we use the first set Chen:2006hy . Their flavor structure can be
$\mathbf{\bar{3}}_{f}\otimes\mathbf{3}_{f}$,
$\mathbf{6}_{f}\otimes\mathbf{\bar{6}}_{f}$, and
$\mathbf{\bar{3}}_{f}\otimes\mathbf{\bar{6}}_{f}\oplus\mathbf{6}_{f}\otimes\mathbf{3}_{f}$
($(qq)(\bar{q}\bar{q})$). With all these independent currents, we perform the
QCD sum rule analysis. As a byproduct, we notice that there does not exist any
tetraquark interpolating operator without derivative for the $J^{PC}=0^{+-}$
case.

This paper is organized as follows. In Sec. II, we construct the tetraquark
currents with $J^{PC}=0^{--}$ using the diquark ($qq$) and antidiquark
($\bar{q}\bar{q}$) fields. The tetraquark currents constructed with the quark-
antiquark ($\bar{q}q$) pairs are shown in Appendix.A. We present the spectral
density in Sec. III and perform the numerical analysis in Sec. IV. For
comparison, we present the finite energy sum rule analysis in the Appendix.B.
The last section is a short summary.

##  II tetraquark interpolating currents

###  II.1 The $J^{PC}=0^{--}$ Tetraquark Interpolating Currents

In this section, we construct the tetraquark interpolating currents with
$J^{PC}=0^{--}$ using diquark and antidiquark fields. Such a quantum number
can not be accessed by a $q\bar{q}$ pair. The currents can be similarly
constructed by using the quark-antiquark pairs. However, as shown in
Appendix.A, these two constructions are equivalent as we have shown several
times in our previous studies Chen:2006hy ; Chen:2008qw .

The pseudoscalar tetraquark currents can be constructed using five independent
diquark fields, which are constructed by five independent $\gamma$-matrices

$\displaystyle
S_{abcd}=(q_{1a}^{T}Cq_{2b})(\bar{q}_{3c}\gamma_{5}C\bar{q}^{T}_{4d})\,,$
$\displaystyle
V_{abcd}=(q_{1a}^{T}C\gamma_{5}q_{2b})(\bar{q}_{3c}C\bar{q}^{T}_{4d})\,,$
$\displaystyle
T_{abcd}=(q_{1a}^{T}C\sigma_{\mu\nu}q_{2b})(\bar{q}_{3c}\sigma^{\mu\nu}\gamma_{5}C\bar{q}^{T}_{4d})\,,$
(1) $\displaystyle
A_{abcd}=(q_{1a}^{T}C\gamma_{\mu}q_{2b})(\bar{q}_{3c}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4d})\,,$
$\displaystyle
P_{abcd}=(q_{1a}^{T}C\gamma_{\mu}\gamma_{5}q_{2b})(\bar{q}_{3c}\gamma^{\mu}C\bar{q}^{T}_{4d}).$

where $q_{1-4}$ represents the $up$, $down$ and $strange$ quarks, and $a-d$
are the color indices.

To compose a color singlet pseudoscalar tetraquark current, the diquark and
antidiquark should have the same color and spin symmetries. So the color
structure of the tetraquark is either $\mathbf{6}\otimes\mathbf{\bar{6}}$ or
$\mathbf{\bar{3}}\otimes\mathbf{3}$, which is denoted by labels $\mathbf{6}$
and $\mathbf{3}$ respectively. Therefore, considering both the color and
Lorentz structures, there are altogether ten terms of products

$\displaystyle\\{S\oplus V\oplus T\oplus A\oplus
P\\}_{Lorentz}\otimes\\{3\oplus 6\\}_{Color}.$ (2)

We list them as follows

$\displaystyle 6_{F}\otimes\bar{6}_{F}~{}(S)$
$\displaystyle\left\\{\begin{array}[]{l}S_{6}=q_{1a}^{T}Cq_{2b}(\bar{q}_{3a}\gamma_{5}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\gamma_{5}C\bar{q}^{T}_{4a})\,,\\\
V_{6}=q_{1a}^{T}C\gamma_{5}q_{2b}(\bar{q}_{3a}C\bar{q}^{T}_{4b}+\bar{q}_{3b}C\bar{q}^{T}_{4a})\,,\\\
T_{3}=q_{1a}^{T}C\sigma_{\mu\nu}q_{2b}(\bar{q}_{3a}\sigma^{\mu\nu}\gamma_{5}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\sigma^{\mu\nu}\gamma_{5}C\bar{q}^{T}_{4a})\,,\end{array}\right.$
(6) $\displaystyle\bar{3}_{F}\otimes 3_{F}~{}(A)$
$\displaystyle\left\\{\begin{array}[]{l}S_{3}=q_{1a}^{T}Cq_{2b}(\bar{q}_{3a}\gamma_{5}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\gamma_{5}C\bar{q}^{T}_{4a})\,,\\\
V_{3}=q_{1a}^{T}C\gamma_{5}q_{2b}(\bar{q}_{3a}C\bar{q}^{T}_{4b}-\bar{q}_{3b}C\bar{q}^{T}_{4a})\,,\\\
T_{6}=q_{1a}^{T}C\sigma_{\mu\nu}q_{2b}(\bar{q}_{3a}\sigma^{\mu\nu}\gamma_{5}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\sigma^{\mu\nu}\gamma_{5}C\bar{q}^{T}_{4a})\,,\end{array}\right.$
(10) $\displaystyle\bar{3}_{F}\otimes\bar{6}_{F}~{}(M)$
$\displaystyle\left\\{\begin{array}[]{l}A_{6}=q_{1a}^{T}C\gamma_{\mu}q_{2b}(\bar{q}_{3a}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4a})\,,\\\
P_{3}=q_{1a}^{T}C\gamma_{\mu}\gamma_{5}q_{2b}(\bar{q}_{3a}\gamma^{\mu}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\gamma^{\mu}C\bar{q}^{T}_{4a})\,.\end{array}\right.$
(13) $\displaystyle 6_{F}\otimes 3_{F}~{}(M)$
$\displaystyle\left\\{\begin{array}[]{l}P_{6}=q_{1a}^{T}C\gamma_{\mu}\gamma_{5}q_{2b}(\bar{q}_{3a}\gamma^{\mu}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\gamma^{\mu}C\bar{q}^{T}_{4a})\,,\\\
A_{3}=q_{1a}^{T}C\gamma_{\mu}q_{2b}(\bar{q}_{3a}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4a})\,.\end{array}\right.$
(16)

In the above expressions, the flavor structure is fixed at the same time due
to the Pauli principle. The currents $S_{6}$, $V_{6}$, $T_{3}$ belong to the
symmetric flavor representation $\mathbf{6_{F}}\otimes\mathbf{\bar{6}_{F}}(S)$
where both diquark and antidiquark fields have the symmetric flavor structure.
The currents $S_{3}$, $V_{3}$, $T_{6}$ belong to the antisymmetric flavor
representation $\mathbf{\bar{3}_{F}}\otimes\mathbf{3_{F}}(A)$, where both
diquark and antidiquark fields have the antisymmetric flavor structure.
$A_{6}$, $P_{3}$ for $\mathbf{\bar{3}_{F}}\otimes\mathbf{\bar{6}_{F}}(M)$ and
$A_{3}$, $P_{6}$ for $\mathbf{6_{F}}\otimes\mathbf{3_{F}}(M)$, where $M$
represents the mixed flavor symmetry. The isovector states with charges can be
observed in the experiments more easily, therefore in this paper we
concentrate on the isovector currents which was shown in the $SU(3)$
tetraquark weight diagram in Fig 1 Chen:2008qw . We have:

$\displaystyle qq\bar{q}\bar{q}(S),~{}qs\bar{q}\bar{s}(S)~{}~{}$
$\displaystyle\in$ $\displaystyle~{}~{}6_{F}\otimes\bar{6}_{F}~{}~{}~{}(S)\,,$
$\displaystyle qs\bar{q}\bar{s}(A)~{}~{}$ $\displaystyle\in$
$\displaystyle~{}~{}\bar{3}_{F}\otimes 3_{F}~{}~{}~{}(A)\,,$ (17)
$\displaystyle qq\bar{q}\bar{q}(M),qs\bar{q}\bar{s}(M)~{}~{}$
$\displaystyle\in$
$\displaystyle~{}~{}(\bar{3}_{F}\otimes\bar{6}_{F})\oplus(6_{F}\otimes
3_{F})~{}~{}~{}(M)\,.$

We do not differentiate $up$ and $down$ quarks and denote them by $q$. We are
only interested in those neutral components. The other states do not carry
definite C-parity. It turns out that the neutral isovector and isoscalar
states have the same QCD sum rules. Our following discussions are valid for
both of them. Taking the charge-conjugation transformation, we get

$\mathbb{C}S_{6}\mathbb{C}^{-1}=V_{6}\,,\mathbb{C}A_{6}\mathbb{C}^{-1}=P_{6}\,,\mathbb{C}A_{3}\mathbb{C}^{-1}=P_{3}\,,\mathbb{C}S_{3}\mathbb{C}^{-1}=V_{3}\,,\mathbb{C}T_{6}\mathbb{C}^{-1}=T_{6}\,,\mathbb{C}T_{3}\mathbb{C}^{-1}=T_{3}\,.$
(18)

$T_{6}$ and $T_{3}$ have even charge-conjugation parity. We conclude that the
currents with $J^{PC}=0^{--}$ are:

$\displaystyle\eta^{(S)}$ $\displaystyle=$ $\displaystyle
S_{6}-V_{6}=q_{1a}^{T}Cq_{2b}(\bar{q}_{3a}\gamma_{5}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\gamma_{5}C\bar{q}^{T}_{4a})-q_{1a}^{T}C\gamma_{5}q_{2b}(\bar{q}_{3a}C\bar{q}^{T}_{4b}+\bar{q}_{3b}C\bar{q}^{T}_{4a})\,,$
$\displaystyle\eta^{(M)}_{1}$ $\displaystyle=$ $\displaystyle
A_{6}-P_{6}=q_{1a}^{T}C\gamma_{\mu}q_{2b}(\bar{q}_{3a}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4a})-q_{1a}^{T}C\gamma_{\mu}\gamma_{5}q_{2b}(\bar{q}_{3a}\gamma^{\mu}C\bar{q}^{T}_{4b}+\bar{q}_{3b}\gamma^{\mu}C\bar{q}^{T}_{4a})\,,$
$\displaystyle\eta^{(M)}_{2}$ $\displaystyle=$ $\displaystyle
A_{3}-P_{3}=q_{1a}^{T}C\gamma_{\mu}q_{2b}(\bar{q}_{3a}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\gamma^{\mu}\gamma_{5}C\bar{q}^{T}_{4a})-q_{1a}^{T}C\gamma_{\mu}\gamma_{5}q_{2b}(\bar{q}_{3a}\gamma^{\mu}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\gamma^{\mu}C\bar{q}^{T}_{4a})\,,$
(19) $\displaystyle\eta^{(A)}$ $\displaystyle=$ $\displaystyle
S_{3}-V_{3}=q_{1a}^{T}Cq_{2b}(\bar{q}_{3a}\gamma_{5}C\bar{q}^{T}_{4b}-\bar{q}_{3b}\gamma_{5}C\bar{q}^{T}_{4a})-q_{1a}^{T}C\gamma_{5}q_{2b}(\bar{q}_{3a}C\bar{q}^{T}_{4b}-\bar{q}_{3b}C\bar{q}^{T}_{4a})\,.$

Adding different quauk contents as shown in Eq. (II.1), there are altogether
seven independent currents as shown :

  1. 1.

For $6_{F}\otimes\bar{6}_{F}~{}(S)$:

$\displaystyle\eta_{1}$ $\displaystyle=$ $\displaystyle
S_{6}(qq\bar{q}\bar{q})-V_{6}(qq\bar{q}\bar{q})=u^{T}_{a}Cd_{b}(\bar{u}_{a}\gamma_{5}C\bar{d}^{T}_{b}+\bar{u}_{b}\gamma_{5}C\bar{d}^{T}_{a})-u^{T}_{a}C\gamma_{5}d_{b}(\bar{u}_{a}C\bar{d}^{T}_{b}+\bar{u}_{b}C\bar{d}^{T}_{a})\,,$
$\displaystyle\eta_{2}$ $\displaystyle=$ $\displaystyle
S_{6}(qs\bar{q}\bar{s})-V_{6}(qs\bar{q}\bar{s})=u^{T}_{a}Cs_{b}(\bar{u}_{a}\gamma_{5}C\bar{s}^{T}_{b}+\bar{u}_{b}\gamma_{5}C\bar{s}^{T}_{a})-u^{T}_{a}C\gamma_{5}s_{b}(\bar{u}_{a}C\bar{s}^{T}_{b}+\bar{u}_{b}C\bar{s}^{T}_{a})\,,$
(20)

where $\eta_{1}$ belongs to the $\mathbf{27_{F}}$ representation and contains
up and down quarks only while $\eta_{2}$ belongs to the $\mathbf{8_{F}}$
representation and contains one $s\bar{s}$ quark pair.

  2. 2.

For $(\bar{3}_{F}\otimes\bar{6}_{F})\oplus(6_{F}\otimes 3_{F})~{}(M)$:

$\displaystyle\eta_{3}$ $\displaystyle=$ $\displaystyle
A_{6}(qq\bar{q}\bar{q})-P_{6}(qq\bar{q}\bar{q})=u^{T}_{a}C\gamma_{\mu}d_{b}(\bar{u}_{a}\gamma^{\mu}\gamma_{5}C\bar{d}^{T}_{b}+\bar{u}_{b}\gamma^{\mu}\gamma_{5}C\bar{d}^{T}_{a})-u^{T}_{a}C\gamma_{\mu}\gamma_{5}d_{b}(\bar{u}_{a}\gamma^{\mu}C\bar{d}^{T}_{b}+\bar{u}_{b}\gamma^{\mu}C\bar{d}^{T}_{a})\,,$
$\displaystyle\eta_{4}$ $\displaystyle=$ $\displaystyle
A_{6}(qs\bar{q}\bar{s})-P_{6}(qs\bar{q}\bar{s})=u^{T}_{a}C\gamma_{\mu}s_{b}(\bar{u}_{a}\gamma^{\mu}\gamma_{5}C\bar{s}^{T}_{b}+\bar{u}_{b}\gamma^{\mu}\gamma_{5}C\bar{s}^{T}_{a})-u^{T}_{a}C\gamma_{\mu}\gamma_{5}s_{b}(\bar{u}_{a}\gamma^{\mu}C\bar{s}^{T}_{b}+\bar{u}_{b}\gamma^{\mu}C\bar{s}^{T}_{a})\,,$
$\displaystyle\eta_{5}$ $\displaystyle=$ $\displaystyle
A_{3}(qq\bar{q}\bar{q})-P_{3}(qq\bar{q}\bar{q})=u^{T}_{a}C\gamma_{\mu}d_{b}(\bar{u}_{a}\gamma^{\mu}C\bar{d}^{T}_{b}-\bar{u}_{b}\gamma^{\mu}C\bar{d}^{T}_{a})-u^{T}_{a}C\gamma_{\mu}\gamma_{5}d_{b}(\bar{u}_{a}\gamma^{\mu}C\bar{d}^{T}_{b}-\bar{u}_{b}\gamma^{\mu}C\bar{d}^{T}_{a})\,,$
(21) $\displaystyle\eta_{6}$ $\displaystyle=$ $\displaystyle
A_{3}(qs\bar{q}\bar{s})-P_{3}(qs\bar{q}\bar{s})=u^{T}_{a}C\gamma_{\mu}s_{b}(\bar{u}_{a}\gamma^{\mu}C\bar{s}^{T}_{b}-\bar{u}_{b}\gamma^{\mu}C\bar{s}^{T}_{a})-u^{T}_{a}C\gamma_{\mu}\gamma_{5}s_{b}(\bar{u}_{a}\gamma^{\mu}C\bar{s}^{T}_{b}-\bar{u}_{b}\gamma^{\mu}C\bar{s}^{T}_{a})\,,$

where $\eta_{3}$ and $\eta_{5}$ belong to the $\mathbf{\bar{10}_{F}}$
representation and contain only u, d quarks while $\eta_{4}$ and $\eta_{6}$
belong to the $\mathbf{8_{F}}$ representation and contain one $s\bar{s}$ quark
pair.

  3. 3.

For $\bar{3}_{F}\otimes 3_{F}~{}(A)$:

$\displaystyle\eta_{7}=S_{3}(qs\bar{q}\bar{s})-V_{3}(qs\bar{q}\bar{s})=u^{T}_{a}Cs_{b}(\bar{u}_{a}\gamma_{5}C\bar{s}^{T}_{b}-\bar{u}_{b}\gamma_{5}C\bar{s}^{T}_{a})-u^{T}_{a}C\gamma_{5}s_{b}(\bar{u}_{a}C\bar{s}^{T}_{b}-\bar{u}_{b}C\bar{s}^{T}_{a})\,.$
(22)

where $\eta_{7}$ belongs to the $\mathbf{8_{F}}$ and contains one $s\bar{s}$
quark pair.

It is understood that there exists the other part $\pm[u\leftrightarrow d]$ in
the expressions of $\eta_{2,4,6,7}$.

###  II.2 The $J^{PC}=0^{+-}$ Tetraquark Currents

Now we move on to the $J^{PC}=0^{+-}$ case. There are also ten independent
scalar tetraquark currents without derivative:

$\displaystyle
S^{\prime}_{6}=q_{1a}^{T}Cq_{2b}(\bar{q}_{3a}C\bar{q}^{T}_{4b}+\bar{q}_{3b}C\bar{q}^{T}_{4a})\,,$
$\displaystyle
V^{\prime}_{6}=q_{1a}^{T}\gamma_{\mu}Cq_{2b}(\bar{q}_{3a}C\gamma^{\mu}\bar{q}^{T}_{4b}+\bar{q}_{3b}C\gamma^{\mu}\bar{q}^{T}_{4a})\,,$
$\displaystyle
T^{\prime}_{6}=q_{1a}^{T}\sigma_{\mu\nu}Cq_{2b}(\bar{q}_{3a}C\sigma^{\mu\nu}\bar{q}^{T}_{4b}+\bar{q}_{3b}C\sigma^{\mu\nu}\bar{q}^{T}_{4a})\,,$
$\displaystyle
A^{\prime}_{6}=q_{1a}^{T}\gamma_{\mu}\gamma_{5}Cq_{2b}(\bar{q}_{3a}C\gamma^{\mu}\gamma_{5}\bar{q}^{T}_{4b}+\bar{q}_{3b}C\gamma^{\mu}\gamma_{5}\bar{q}^{T}_{4a})\,,$
$\displaystyle
P^{\prime}_{6}=q_{1a}^{T}\gamma_{5}Cq_{2b}(\bar{q}_{3a}C\gamma_{5}\bar{q}^{T}_{4b}+\bar{q}_{3b}C\gamma_{5}\bar{q}^{T}_{4a})\,,$
$\displaystyle
S^{\prime}_{3}=q_{1a}^{T}Cq_{2b}(\bar{q}_{3a}C\bar{q}^{T}_{4b}-\bar{q}_{3b}C\bar{q}^{T}_{4a})\,,$
(23) $\displaystyle
V^{\prime}_{3}=q_{1a}^{T}\gamma_{\mu}Cq_{2b}(\bar{q}_{3a}C\gamma^{\mu}\bar{q}^{T}_{4b}-\bar{q}_{3b}C\gamma^{\mu}\bar{q}^{T}_{4a})\,,$
$\displaystyle
T^{\prime}_{3}=q_{1a}^{T}\sigma_{\mu\nu}Cq_{2b}(\bar{q}_{3a}C\sigma^{\mu\nu}\bar{q}^{T}_{4b}-\bar{q}_{3b}C\sigma^{\mu\nu}\bar{q}^{T}_{4a})\,,$
$\displaystyle
A^{\prime}_{3}=q_{1a}^{T}\gamma_{\mu}\gamma_{5}Cq_{2b}(\bar{q}_{3a}C\gamma^{\mu}\gamma_{5}\bar{q}^{T}_{4b}-\bar{q}_{3b}C\gamma^{\mu}\gamma_{5}\bar{q}^{T}_{4a})\,,$
$\displaystyle
P^{\prime}_{3}=q_{1a}^{T}\gamma_{5}Cq_{2b}(\bar{q}_{3a}C\gamma_{5}\bar{q}^{T}_{4b}-\bar{q}_{3b}C\gamma_{5}\bar{q}^{T}_{4a})\,.$

The flavor structure is again fixed due to the Pauli principle. To have a
charge-conjugation parity, we fix the quark contents to be: $q_{1}=q_{3}$ and
$q_{2}=q_{4}$ (or $q_{1}=q_{4}$ and $q_{2}=q_{3}$). After performing the
charge-conjugation transformation, we find that they all have an even charge-
conjugation parity, for example:

$\mathbb{C}S^{\prime}_{6}\mathbb{C}^{-1}=+S^{\prime}_{6}\,.$ (24)

Therefore, the $J^{PC}=0^{+-}$ tetraquark interpolating currents without
derivatives do not exist.

##  III The Spectral Density

We consider the two-point correlation function in the framework of QCD sum
rule Shifman:1978bx ; Reinders:1984sr :

$\Pi(q^{2})\equiv\int d^{4}xe^{iqx}\langle
0|T\eta(x)\eta^{\dagger}(0)|0\rangle,$ (25)

where $\eta$ is an interpolating current. We can calculate $\Pi(q^{2})$ at the
quark gluon level using the propagator:

$\displaystyle iS^{ab}_{q}$ $\displaystyle\equiv$ $\displaystyle\langle
0|T[q^{a}(x)\bar{q}^{b}(0)]|0\rangle$ (26) $\displaystyle=$
$\displaystyle\frac{i\delta^{ab}}{2\pi^{2}x^{4}}\hat{x}+\frac{i}{32\pi^{2}}\frac{\lambda^{n}_{ab}}{2}gG_{\mu\nu}^{n}\frac{1}{x^{2}}(\sigma^{\mu\nu}\hat{x}+\hat{x}\sigma^{\mu\nu})-\frac{\delta^{ab}}{12}\langle\bar{q}q\rangle$
$\displaystyle+\frac{\delta^{ab}x^{2}}{192}\langle g_{s}\bar{q}\sigma
Gq\rangle-\frac{m_{q}\delta^{ab}}{4\pi^{2}x^{2}}+\frac{i\delta^{ab}m_{q}\langle\bar{q}q\rangle}{48}\hat{x}+\frac{i\delta^{ab}m_{q}^{2}\langle\bar{q}q\rangle}{8\pi^{2}x^{2}}\hat{x},$

where $\hat{x}\equiv\gamma_{\mu}x^{\mu}$. With the dispersion relation
$\Pi(q^{2})$ is related to the observable at the hadron level

$\Pi(p^{2})=\int_{0}^{\infty}\frac{\rho(s)}{s-p^{2}-i\varepsilon}ds,$ (27)

where

$\rho(s)\equiv\sum_{n}\delta(s-M^{2}_{n})\langle 0|\eta|n\rangle\langle
n|\eta^{{\dagger}}|0\rangle=f^{2}_{X}\delta(s-M^{2}_{X})+\mbox{continuum}\;.$
(28)

Here, the usual pole plus continuum parametrization of the hadronic spectral
density is adopted. Up to dimension 12, the spectral density $\rho_{i}(s)$ at
the quark and gluon level reads:

$\displaystyle\rho_{1}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{15360\pi^{6}}-\frac{m_{q}^{2}}{192\pi^{6}}s^{3}-(\frac{\langle
g_{s}^{2}GG\rangle}{3072\pi^{6}}-\frac{m_{q}\langle\bar{q}q\rangle}{24\pi^{4}})s^{2}+[\frac{\langle
g_{s}^{2}GG\rangle m_{q}^{2}}{256\pi^{6}}+\frac{\langle
g_{s}^{3}fGGG\rangle}{768\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)]s$ (29)
$\displaystyle-(\frac{3m_{q}^{2}\langle\bar{q}q\rangle^{2}}{2\pi^{2}}+\frac{\langle
g_{s}^{2}GG\rangle
m_{q}\langle\bar{q}q\rangle}{192\pi^{4}})+(\frac{16}{9}m_{q}\langle\bar{q}q\rangle^{3}-\frac{1}{\pi^{2}}m_{q}^{2}\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma Gq\rangle)\delta(s)\,,$ $\displaystyle\rho_{2}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{15360\pi^{6}}-\frac{m_{s}^{2}}{384\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{64\pi^{6}}+\frac{m_{s}\langle\bar{s}s\rangle}{48\pi^{4}}-\frac{\langle
g_{s}^{2}GG\rangle}{3072\pi^{6}})s^{2}$ (30) $\displaystyle+[\frac{\langle
g_{s}^{3}fGGG\rangle}{768\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)-(\frac{m_{s}^{3}\langle\bar{s}s\rangle}{8\pi^{4}}-\frac{m_{s}^{2}\langle
g_{s}^{2}GG\rangle}{512\pi^{6}})]s$
$\displaystyle+(\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{12\pi^{2}}-\frac{m_{s}^{2}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}-\frac{m_{s}\langle\bar{s}s\rangle\langle
g_{s}^{2}GG\rangle}{384\pi^{4}})-(\frac{m_{s}^{2}\langle\bar{u}u\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}{6\pi^{2}}-\frac{8}{9}m_{s}\langle\bar{s}s\rangle\langle\bar{q}q\rangle^{2})\delta(s)\,,$
$\displaystyle\rho_{3}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{3840\pi^{6}}-\frac{m_{q}^{2}}{48\pi^{6}}s^{3}+(\frac{5\langle
g_{s}^{2}GG\rangle}{1536\pi^{6}}+\frac{m_{q}\langle\bar{q}q\rangle}{6\pi^{4}})s^{2}+[\frac{\langle
g_{s}^{3}fGGG\rangle}{192\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)-\frac{5\langle
g_{s}^{2}GG\rangle m_{q}^{2}}{128\pi^{6}}]s$ (31)
$\displaystyle-(\frac{6m_{q}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}-\frac{5\langle
g_{s}^{2}GG\rangle
m_{q}\langle\bar{q}q\rangle}{96\pi^{4}})+(\frac{64}{9}m_{q}\langle\bar{q}q\rangle^{3}-\frac{4}{\pi^{2}}m_{q}^{2}\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma Gq\rangle)\delta(s)\,,$ $\displaystyle\rho_{4}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{3840\pi^{6}}-\frac{m_{s}^{2}}{96\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{16\pi^{6}}+\frac{m_{s}\langle\bar{s}s\rangle}{12\pi^{4}}+\frac{5\langle
g_{s}^{2}GG\rangle}{1536\pi^{6}})s^{2}$ (32) $\displaystyle+[\frac{\langle
g_{s}^{3}fGGG\rangle}{192\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)-(\frac{m_{s}^{3}\langle\bar{s}s\rangle}{2\pi^{4}}+\frac{5m_{s}^{2}\langle
g_{s}^{2}GG\rangle}{256\pi^{6}})]s$
$\displaystyle+(\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{3\pi^{2}}-\frac{4m_{s}^{2}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}+\frac{5m_{s}\langle\bar{s}s\rangle\langle
g_{s}^{2}GG\rangle}{192\pi^{4}})-(\frac{2m_{s}^{2}\langle\bar{u}u\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}{3\pi^{2}}-\frac{32}{9}m_{s}\langle\bar{s}s\rangle\langle\bar{q}q\rangle^{2})\delta(s)\,,$
$\displaystyle\rho_{5}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}-\frac{m_{q}^{2}}{96\pi^{6}}s^{3}+(\frac{\langle
g_{s}^{2}GG\rangle}{1536\pi^{6}}+\frac{m_{q}\langle\bar{q}q\rangle}{12\pi^{4}})s^{2}+[\frac{\langle
g_{s}^{3}fGGG\rangle}{384\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)-\frac{\langle
g_{s}^{2}GG\rangle m_{q}^{2}}{128\pi^{6}}]s$ (33)
$\displaystyle-(\frac{3m_{q}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}-\frac{\langle
g_{s}^{2}GG\rangle
m_{q}\langle\bar{q}q\rangle}{96\pi^{4}})+(\frac{32}{9}m_{q}\langle\bar{q}q\rangle^{3}-\frac{2}{\pi^{2}}m_{q}^{2}\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma Gq\rangle)\delta(s)\,,$ $\displaystyle\rho_{6}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}-\frac{m_{s}^{2}}{192\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{32\pi^{6}}+\frac{m_{s}\langle\bar{s}s\rangle}{24\pi^{4}}+\frac{\langle
g_{s}^{2}GG\rangle}{1536\pi^{6}})s^{2}$ (34) $\displaystyle+[\frac{\langle
g_{s}^{3}fGGG\rangle}{384\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)-(\frac{m_{s}^{3}\langle\bar{s}s\rangle}{4\pi^{4}}+\frac{m_{s}^{2}\langle
g_{s}^{2}GG\rangle}{256\pi^{6}})]s$
$\displaystyle+(\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{6\pi^{2}}-\frac{2m_{s}^{2}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}+\frac{m_{s}\langle\bar{s}s\rangle\langle
g_{s}^{2}GG\rangle}{192\pi^{4}})-(\frac{m_{s}^{2}\langle\bar{u}u\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}{3\pi^{2}}-\frac{16}{9}m_{s}\langle\bar{s}s\rangle\langle\bar{q}q\rangle^{2})\delta(s)\,,$
$\displaystyle\rho_{7}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{30720\pi^{6}}-\frac{m_{s}^{2}}{768\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{128\pi^{6}}+\frac{m_{s}\langle\bar{s}s\rangle}{96\pi^{4}}+\frac{\langle
g_{s}^{2}GG\rangle}{3072\pi^{6}})s^{2}$ (35) $\displaystyle+[\frac{\langle
g_{s}^{3}fGGG\rangle}{1536\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)-(\frac{m_{s}^{3}\langle\bar{s}s\rangle}{16\pi^{4}}+\frac{m_{s}^{2}\langle
g_{s}^{2}GG\rangle}{512\pi^{6}})]s$
$\displaystyle+(\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{24\pi^{2}}-\frac{m_{s}^{2}\langle\bar{q}q\rangle^{2}}{6\pi^{2}}+\frac{m_{s}\langle\bar{s}s\rangle\langle
g_{s}^{2}GG\rangle}{384\pi^{4}})-(\frac{m_{s}^{2}\langle\bar{u}u\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}{12\pi^{2}}-\frac{4}{9}m_{s}\langle\bar{s}s\rangle\langle\bar{q}q\rangle^{2})\delta(s)\,.$

It is interesting to note several important features of the above spectral
densities:

  * •

First the special Lorentz structure of the $J^{PC}=0^{--}$ interpolating
currents forbids the appearance of the four-quark type of condensates
$\langle\bar{q}q\rangle^{2}$, $\langle\bar{q}q\rangle$ $\langle
g_{s}\bar{q}\sigma Gq\rangle$ and $\langle g_{s}\bar{q}\sigma Gq\rangle^{2}$.
Usually these terms play an important role in the multiquark sum rules. The
Feynman diagrams for the dimension 10 condensate $\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}$ are shown in Fig. 1.

|  |
---|---|---
Figure 1: Feynman diagrams for the quark gluon mixed condensate.

  * •

The dominant non-perturbative correction arises from the gluon condensate,
which is destructive for $\rho_{1-2}(s)$ and constructive for $\rho_{3-7}(s)$.
Moreover there are corrections from the tri-gluon condensate $\langle
g_{s}^{3}f^{abc}G^{a}G^{b}G^{c}\rangle$ as shown in Fig.2. In the above
expressions we use the short-hand notation $\langle g_{s}^{3}fGGG\rangle$ to
denote the tri-gluon condensate. There are three types of Feynman diagrams.
The first class of Feynman diagrams vanishes because of the product of the
color matrices. The second class is proportional to $m_{q}$ and could be
omitted in the chiral limit. Only the third class leads to non-vanishing tri-
gluon correction. In fact the gluon condensates become the only power
corrections in the chiral limit.

|  |
---|---|---
Figure 2: Feynman diagrams for the tri-gluon condensate.

  * •

The second term in each $\rho_{i}(s)$ is destructive, which renders the
spectral density negative when $s$ is small. This $-m_{q}^{2}s^{3}$ piece is
an artefact of the expansion of the quark propagator ${i\over{\hat{p}}-m_{q}}$
in terms of the quark mass $m_{q}$ perturbatively. Without making such an
expansion, the perturbative contribution to the spectral density is always
positive-definite. Such a destructive term will sometimes produce an
artificial plateau and stability window in the sum rule analysis, which must
be removed.

  * •

Although the tree-level four-quark condensate vanishes, one may wonder whether
the four-quark condensate $g_{s}^{2}\langle\bar{q}q\rangle^{2}$ plays a role
since the latter is very important in the $q\bar{q}$ meson sum rules
Shifman:1978bx ; Reinders:1984sr . Two types of Feynman diagrams could produce
such a correction. The first class of Feynman diagrams is very similar to that
in the $q\bar{q}$ meson case where a gluon propagator is attached between two-
quark condensates, as Fig.3 shown. It’s easy to check that they vanish due to
the special Lorentz structure of the correlation function.

|
---|---
Figure 3: One set of Feynman diagrams for the four-quark condensate.

One of the second class of diagrams is shown in Fig.4. In this case, we use
the mesonic type interpolating currents in the appendixA to simplify the
derivation. After making Wick-contraction to the correlation function,

$\displaystyle\bar{\psi}_{3}(x)\Gamma^{\prime}_{1}\psi_{4}(x)\bar{\psi}_{1}(x)\Gamma_{1}\psi_{2}(x)\bar{\psi}_{1}(z_{1})gt^{a}\gamma^{\mu}\psi_{1}(z_{1})A_{\mu}^{a}(z_{1})\bar{\psi}_{2}(z_{2})gt^{b}\gamma^{\nu}\psi_{2}(z_{2})A_{\nu}^{b}(z_{2})\bar{\psi}_{2}(y)\Gamma_{2}\psi_{1}(y)\bar{\psi}_{4}(y)\Gamma^{\prime}_{2}\psi_{3}(y)$

we get

$\displaystyle
Tr[-\Gamma^{\prime}_{1}S_{Q}(x-y)\Gamma^{\prime}_{2}S_{Q}(y-x)]Tr[-S_{Q}(x-z_{2})\gamma^{\nu}S_{Q}(z_{2}-y)\Gamma_{2}S_{Q}(y-z_{1})\gamma^{\mu}S_{Q}(z_{1}-x)\Gamma_{1}\times
g_{\mu\nu}\times S_{G}(z_{2}-z_{1})].$

where $S_{Q}$ is the quark propagator and $S_{G}$ is the gluon propagator.
{$\Gamma_{1}$,$\Gamma_{2}$} could be either {$I$, $\gamma_{5}$} or
{$\gamma_{\alpha}$, $\gamma_{5}\gamma_{\alpha}$}.
$S_{Q}(y-z_{1})\propto\langle\bar{q}q\rangle$. In fact, there would be three
$\gamma$-matrices or three $\gamma$-matrices plus $\gamma_{5}$ left in the
latter trace. Therefore this piece also vanishes.

---
Figure 4: Feynman diagrams for the four-quark condensate.

##  IV Numerical Analysis

In the chiral limit ($m_{s}=m_{q}=0$) the spectral density reads

$\displaystyle\rho_{1-2}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{15360\pi^{6}}-\frac{\langle
g_{s}^{2}GG\rangle}{3072\pi^{6}}s^{2}+\frac{\langle
g_{s}^{3}fGGG\rangle}{768\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)s,$
$\displaystyle\rho_{3-4}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{3840\pi^{6}}-\frac{5\langle
g_{s}^{2}GG\rangle}{1536\pi^{6}}s^{2}+\frac{\langle
g_{s}^{3}fGGG\rangle}{192\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)s,$
$\displaystyle\rho_{5-6}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}-\frac{\langle
g_{s}^{2}GG\rangle}{1536\pi^{6}}s^{2}+\frac{\langle
g_{s}^{3}fGGG\rangle}{384\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)s,$
$\displaystyle\rho_{7}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{30720\pi^{6}}-\frac{\langle
g_{s}^{2}GG\rangle}{3072\pi^{6}}s^{2}+\frac{\langle
g_{s}^{3}fGGG\rangle}{1536\pi^{6}}(3\ln(\frac{s}{\tilde{\mu}^{2}})-5)s$ (36)

where $\tilde{\mu}=1$ GeV. Requiring the pole contribution is larger than
$40\%$, one gets the upper bound $M^{2}_{\mbox{max}}$ of the Borel parameter
$M_{B}^{2}$. The convergence of the operator expansion product leads to the
lower bound $M^{2}_{\mbox{min}}$ of the Borel parameter. In the present case,
we require that the two gluon condensate correction be less than one third of
the perturbative term and the tri-gluon condensate correction less than one
third of the gluon condensate correction. The working region of $M_{B}^{2}$ in
the sum rule analysis is [$M^{2}_{\mbox{min}}$, $M^{2}_{\mbox{max}}$], which
is dependent on the threshold $s_{0}$.

In order to study the sensitivity of the sum rule to the condensate values, we
adopt two sets of the gluon condensate values in our numerical analysis. One
set is from Ioffe’s recent review gluon1 : $\langle
g_{s}^{2}GG\rangle=(0.20\pm 0.16)~{}\mbox{GeV}^{4}$, $\langle
g_{s}^{3}fGGG\rangle=0.12~{}\mbox{GeV}^{6}$. We also use the original SVZ
values Shifman:1978bx : $\langle g_{s}^{2}GG\rangle=(0.48\pm
0.14)~{}\mbox{GeV}^{4}$, $\langle
g_{s}^{3}fGGG\rangle=0.045~{}\mbox{GeV}^{6}$. The working regions of the sum
rules with the above two sets of gluon condensates and $s_{0}=7$ GeV2 are
listed in Table 1.

$\diagdown$ | [$M^{2}_{\mbox{min}}$, $M^{2}_{\mbox{max}}$](SVZ) | [$M^{2}_{\mbox{min}}$, $M^{2}_{\mbox{max}}$] (Ioffe)
---|---|---
$\rho_{1-2}$ | $0.77\sim 1.50$ | $0.90\sim 1.68$
$\rho_{3-4}$ | $1.22\sim 1.90$ | $1.40\sim 1.65$
$\rho_{5-6}$ | $1.05\sim 1.77$ | $1.55\sim 1.74$
$\rho_{7}$ | $1.10\sim 1.85$ | $1.50\sim 1.75$
Table 1: The working region of $M_{B}^{2}$ with Ioffe’s and SVZ’s gluon
condensates and $s_{0}=7$ GeV2.

The working region of the sum rule is very narrow even with $s_{0}=7$ GeV2.
The variation of $M_{X}$ with $M_{B}^{2}$ and $s_{0}$ is shown in Figs. 5-8
for the interpolating currents $\eta_{1-2}$, $\eta_{3-4}$, $\eta_{5-6}$,
$\eta_{7}$ respectively using Ioffe’s gluon condensate values. The variation
of $M_{X}$ with $M_{B}^{2}$ and $s_{0}$ and SVZ’s gluon condensate values is
presented in Figs.9-12.

For a genuine hadron state, one expects that the extracted mass from the sum
rule analysis is stable with the reasonable variation of the Borel parameter
and the continuum threshold. In other words, there should exists dual
stability in $M_{B}^{2}$ and $s_{0}$ in the working region of $M_{B}^{2}$.
From all these figures we notice none of the mass curves satisfy the stability
requirement. These interpolating currents do not support a low-lying resonant
signal.

|
---|---
Figure 5: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{1-2}$ using Ioffe’s gluon condensate values. |
---|---
Figure 6: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{3-4}$ using Ioffe’s gluon condensate values. |
---|---
Figure 7: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{5-6}$ using Ioffe’s gluon condensate values. |
---|---
Figure 8: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{7}$ using Ioffe’s gluon condensate values. |
---|---
Figure 9: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{1-2}$ using SVZ’s gluon condensate values. |
---|---
Figure 10: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{3-4}$ using SVZ’s gluon condensate values. |
---|---
Figure 11: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$ (Right) for the current $\eta_{5-6}$ using SVZ’s gluon condensate values. |
---|---
Figure 12: The variation of $M_{X}$ with $M^{2}_{B}$ (Left) and $s_{0}$
(Right) for the current $\eta_{7}$ using SVZ’s gluon condensate values.

##  V Conclusion

The exotic state with $J^{PC}=0^{--}$ can not be composed of a pair of gluons
nor $q\bar{q}$. In order to explore the possible existence of these
interesting states, we first construct the tetraquark type interpolating
operators systematically. As a byproduct, we notice that the $J^{PC}=0^{+-}$
tetraquark operators without derivatives do not exist. Then we make the
operator product expansion and extract the spectral density. The gluon
condensate becomes the dominant power correction. Usually the four-quark type
of condensates $\langle\bar{q}q\rangle^{2}$, $\langle\bar{q}q\rangle$ $\langle
g_{s}\bar{q}\sigma Gq\rangle$ and $\langle g_{s}\bar{q}\sigma Gq\rangle^{2}$
are the dominant nonperturbative corrections in the multiquark sum rules.
However these terms vanish because of the special Lorentz structure imposed by
the exotic $0^{--}$ quantum numbers.

Within the framework of the SVZ sum rule, we note that the absence of the
$\langle\bar{q}q\rangle^{2}$, $\langle\bar{q}q\rangle$ $\langle
g_{s}\bar{q}\sigma Gq\rangle$ and $\langle g_{s}\bar{q}\sigma Gq\rangle^{2}$
terms destabilize the sum rule. There does not exist stability in either
$M_{B}^{2}$ or $s_{0}$ in the working region of $M_{B}^{2}$. Therefore we
conclude that none of these independent interpolating currents support a
resonant signal below 2 GeV, which is consistent with the current experimental
measurement Amsler:2008zzb .

## Acknowledgments

The authors are grateful to Professor Wei-Zhen Deng for useful discussions.
This project was supported by the National Natural Science Foundation of China
under Grants 10625521, 10721063 and Ministry of Science and Technology of
China (2009CB825200).

## References

  * (1) C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008).
  * (2) E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007).
  * (3) G. S. Adams et al. [E862 Collaboration], Phys. Lett. B 657, 27 (2007).
  * (4) A. Abele et al. [Crystal Barrel Collaboration], Phys. Lett. B 446, 349 (1999); A. Abele et al. [Crystal Barrel Collaboration], Phys. Lett. B 423, 175 (1998).
  * (5) D. R. Thompson et al. [E852 Collaboration], Phys. Rev. Lett. 79, 1630 (1997).
  * (6) H. X. Chen, A. Hosaka and S. L. Zhu, Phys. Rev. D 78, 054017 (2008); D78, 117502 (2008).
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##  Appendix A Interpolating Currents in $(\bar{q}q)(\bar{q}q)$ Basis

For $6_{F}\otimes\bar{6}_{F}~{}(S)$:

$\displaystyle\eta^{(S)(1)}_{m}$ $\displaystyle=$
$\displaystyle(\bar{q}_{1a}\gamma_{\mu}q_{1a})(\bar{q}_{2b}\gamma^{\mu}\gamma_{5}q_{2b})+(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{1a})(\bar{q}_{2b}\gamma^{\mu}q_{2b})+(\bar{q}_{1a}\gamma_{\mu}q_{2a})(\bar{q}_{2b}\gamma^{\mu}\gamma_{5}q_{1b})+(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{2a})(\bar{q}_{2b}\gamma^{\mu}q_{1b}),$
$\displaystyle\eta^{(S)(8)}_{m}$ $\displaystyle=$
$\displaystyle\lambda_{ab}\lambda_{cd}\\{(\bar{q}_{1a}\gamma_{\mu}q_{1b})(\bar{q}_{2c}\gamma^{\mu}\gamma_{5}q_{2d})+(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{1b})(\bar{q}_{2d}\gamma^{\mu}q_{2d})+(\bar{q}_{1a}\gamma_{\mu}q_{2b})(\bar{q}_{2c}\gamma^{\mu}\gamma_{5}q_{1d})+(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{2b})(\bar{q}_{2c}\gamma^{\mu}q_{1d})\\},$

For $(\bar{3}_{F}\otimes\bar{6}_{F})\oplus(6_{F}\otimes 3_{F})~{}(M)$:

$\displaystyle\ \eta^{(M)(1)}_{1m}$ $\displaystyle=$
$\displaystyle(\bar{q}_{1a}q_{1a})(\bar{q}_{2b}\gamma_{5}q_{2b})-(\bar{q}_{1a}\gamma_{5}q_{1a})(\bar{q}_{2b}q_{2b}),$
$\displaystyle\ \eta^{(M)(8)}_{1m}$ $\displaystyle=$
$\displaystyle\lambda_{ab}\lambda_{cd}\\{(\bar{q}_{1a}q_{1b})(\bar{q}_{2c}\gamma_{5}q_{2d})-(\bar{q}_{1a}\gamma_{5}q_{1b})(\bar{q}_{2c}q_{2d})\\},$
$\displaystyle\ \eta^{(M)(1)}_{2m}$ $\displaystyle=$
$\displaystyle(\bar{q}_{1a}\gamma_{\mu}q_{1a})(\bar{q}_{2b}\gamma^{\mu}\gamma_{5}q_{2b})-(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{1a})(\bar{q}_{2b}\gamma^{\mu}q_{2b}),$
$\displaystyle\ \eta^{(M)(8)}_{2m}$ $\displaystyle=$
$\displaystyle\lambda_{ab}\lambda_{cd}\\{(\bar{q}_{1a}\gamma_{\mu}q_{1b})(\bar{q}_{2c}\gamma^{\mu}\gamma_{5}q_{2d})-(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{1b})(\bar{q}_{2c}\gamma^{\mu}q_{2d})\\},$

For $\bar{3}_{F}\otimes 3_{F}~{}(A)$:

$\displaystyle\eta^{(A)(1)}_{m}$ $\displaystyle=$
$\displaystyle(\bar{q}_{1a}\gamma_{\mu}q_{1a})(\bar{q}_{2b}\gamma^{\mu}\gamma_{5}q_{2b})+(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{1a})(\bar{q}_{2b}\gamma^{\mu}q_{2b})-(\bar{q}_{1a}\gamma_{\mu}q_{2a})(\bar{q}_{2b}\gamma^{\mu}\gamma_{5}q_{1b})-(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{2a})(\bar{q}_{2b}\gamma^{\mu}q_{1b}),$
$\displaystyle\eta^{(A)(8)}_{m}$ $\displaystyle=$
$\displaystyle\lambda_{ab}\lambda_{cd}\\{(\bar{q}_{1a}\gamma_{\mu}q_{1b})(\bar{q}_{2c}\gamma^{\mu}\gamma_{5}q_{2d})+(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{1b})(\bar{q}_{2d}\gamma^{\mu}q_{2d})-(\bar{q}_{1a}\gamma_{\mu}q_{2b})(\bar{q}_{2c}\gamma^{\mu}\gamma_{5}q_{1d})-(\bar{q}_{1a}\gamma_{\mu}\gamma_{5}q_{2b})(\bar{q}_{2c}\gamma^{\mu}q_{1d})\\},$

where the indices $(1)$, $(8)$ represent the color singlet and octet. Now we
get eight mesonic currents. Then we introduce the formula of the interchange
of the color indices:

$\displaystyle(q_{1a}q_{2b}\bar{q}_{3a}\bar{q}_{4b})$ $\displaystyle=$
$\displaystyle\frac{1}{3}(q_{1a}q_{2b}\bar{q}_{3b}\bar{q}_{4a})+\frac{1}{2}\lambda_{ab}\lambda_{cd}(q_{1a}q_{2c}\bar{q}_{3d}\bar{q}_{4b}),$
$\displaystyle\lambda_{ab}\lambda_{cd}(q_{1a}q_{2c}\bar{q}_{3b}\bar{q}_{4d})$
$\displaystyle=$
$\displaystyle\frac{16}{9}(q_{1a}q_{2b}\bar{q}_{3b}\bar{q}_{4a})-\frac{1}{3}\lambda_{ab}\lambda_{cd}(q_{1a}q_{2c}\bar{q}_{3d}\bar{q}_{4b}),$
(37)

Next, we perform the Fierz rearrangement in the Lorrentz indices with the
formula

$\displaystyle(\bar{a}b)(\bar{b}a)=\frac{1}{4}(\bar{a}a)(\bar{b}b)+\frac{1}{4}(\bar{a}\gamma_{5}a)(\bar{b}\gamma_{5}b)+\frac{1}{4}(\bar{a}\gamma_{\mu}a)(\bar{b}\gamma^{\mu}b)-\frac{1}{4}(\bar{a}\gamma_{5}\gamma_{\mu}a)(\bar{b}\gamma_{5}\gamma^{\mu}b)+\frac{1}{8}(\bar{a}\sigma_{\mu\nu}a)(\bar{b}\sigma^{\mu\nu}b),$
(38)

For example, we have

$\displaystyle(q_{1a}^{T}Cq_{2b})(\bar{q}_{3a}\gamma_{5}C\bar{q}^{T}_{4b})$
$\displaystyle=$
$\displaystyle-\frac{1}{4}(q_{1a}^{T}C\gamma_{5}C\bar{q}^{T}_{4b})(\bar{q}_{3a}q_{2b})-\frac{1}{4}(q_{1a}^{T}C\gamma_{\mu}\gamma_{5}C\bar{q}^{T}_{4b})(\bar{q}_{3a}\gamma^{\mu}q_{2b})$
$\displaystyle-\frac{1}{8}(q_{1a}^{T}C\sigma_{\mu\nu}\gamma_{5}C\bar{q}^{T}_{4b})(\bar{q}_{3a}\sigma^{\mu\nu}q_{2b})+\frac{1}{4}(q_{1a}^{T}C\gamma_{\mu}\gamma_{5}\gamma_{5}C\bar{q}^{T}_{4b})(\bar{q}_{3a}\gamma^{\mu}\gamma_{5}q_{2b})$
$\displaystyle-\frac{1}{4}(q_{1a}^{T}C\gamma_{5}\gamma_{5}C\bar{q}^{T}_{4b})(\bar{q}_{3a}\gamma_{5}q_{2b})$
$\displaystyle=$
$\displaystyle-\frac{1}{4}(\bar{q}_{4b}\gamma_{5}q_{1a})(\bar{q}_{3a}q_{2b})-\frac{1}{4}(\bar{q}_{4b}\gamma_{\mu}\gamma_{5}q_{1a})(\bar{q}_{3a}\gamma^{\mu}q_{2b})$
$\displaystyle+\frac{1}{8}(\bar{q}_{4b}\sigma_{\mu\nu}\gamma_{5}q_{1a})(\bar{q}_{3a}\sigma^{\mu\nu}q_{2b})-\frac{1}{4}(\bar{q}_{4b}\gamma_{\mu}q_{1a})(\bar{q}_{3a}\gamma^{\mu}\gamma_{5}q_{2b})$
$\displaystyle-\frac{1}{4}(\bar{q}_{4b}q_{1a})(\bar{q}_{3a}\gamma_{5}q_{2b}),$

There are only four independent currents among those eight mesonic currents.
Any four currents are independent and can be expressed by the other four.

$\displaystyle\eta^{(S)(8)}_{m}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\eta^{(S)(1)}_{m},$ $\displaystyle\eta^{(M)(8)}_{1m}$
$\displaystyle=$
$\displaystyle-\frac{2}{3}\eta^{(M)(1)}_{1m}-\eta^{(M)(1)}_{2m},$
$\displaystyle\eta^{(M)(8)}_{2m}$ $\displaystyle=$
$\displaystyle-4\eta^{(M)(1)}_{1m}-\frac{2}{3}\eta^{(M)(1)}_{2m},$
$\displaystyle\eta^{(A)(8)}_{m}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}\eta^{(A)(1)}_{m},$

We establish the relations between the diquark currents and the mesonic
currents using the Fierz transformation. For instance, we can verify the
relations

$\displaystyle\eta^{(S)(1)}_{m}$ $\displaystyle=$
$\displaystyle-2\eta^{S}_{d},$ $\displaystyle\eta^{(M)(1)}_{1m}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\eta^{M}_{1d}+\frac{1}{4}\eta^{M}_{2d},$
$\displaystyle\eta^{(M)(1)}_{2m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\eta^{M}_{1d}+\frac{1}{2}\eta^{M}_{2d},$
$\displaystyle\eta^{(A)(1)}_{m}$ $\displaystyle=$
$\displaystyle-2\eta^{A}_{d}.$

##  Appendix B Finite energy sum rule

Sometimes the finite energy sum rule is also employed in the numerical
analysis. One first defines the $n$th moment using the spectral density

$\displaystyle W(n,s_{0})=\int^{s_{0}}_{0}\rho(s)s^{n}ds.$ (40)

With the quark-hadron duality, we have

$\displaystyle W(n,s_{0})|_{Hadron}=W(n,s_{0})|_{OPE}.$ (41)

The mass of the ground state can be obtained as

$\displaystyle M_{X}^{2}(n,s_{0})=\frac{W(n+1,s_{0})}{W(n,s_{0})}.$ (42)

We have plotted the variation of $M_{X}$ with $s_{0}$ for all the seven
interpolating currents in Fig. 13. The left and right diagrams correspond to
Ioffe’s and SVZ’s gluon condensate values respectively. It seems that there
exists a minimum of $M_{X}$ for each current. However, a reasonable sum rule
requires that the operator product expansion should converge well. In other
words, we require that the two-gluon power correction be less than one third
of the perturbative term and the tri-gluon power correction less than one
third of two-gluon power correction in $W(0,s_{0})$, which leads to the
working window of this finite energy sum rule as:

$\begin{array}[]{c|c|c}\diagdown&s_{0}(\mbox{SVZ})&s_{0}(\mbox{Ioffe})\\\
\hline\cr\rho_{1-2}&4.0&7.0\\\ \hline\cr\rho_{3-4}&4.2&5.7\\\
\hline\cr\rho_{5-6}&4.0&7.0\\\ \hline\cr\rho_{7}&4.9&6.0\end{array}$

Clearly for each current the minimum of the mass curve lies outside of the
working region in both of the figures and is not a real resonant signal.
Starting from 4.0 GeV2, each mass curve grows monotonically with $s_{0}$.
Thus, there does not exist a resonant signal for every interpolating current.

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Figure 13: The variation of $M_{X}$ with $s_{0}$ and $n=0$ from the finite
energy sum rule. The left and right diagrams correspond to Ioffe’s and SVZ’s
gluon condensate values respectively.