# Solving the Hubble tension at intermediate redshifts with dynamical dark
energy

Isaac Tutusaus<EMAIL_ADDRESS>Institut de Recherche en
Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES,
14 Av. Edouard Belin, 31400 Toulouse, France
Martin Kunz<EMAIL_ADDRESS>Département de Physique Théorique and Center
for Astroparticle Physics, Université de Genève, Quai E. Ansermet 24, CH-1211
Genève 4, Switzerland  Léo Favre  Département de Physique Théorique and Center
for Astroparticle Physics, Université de Genève, Quai E. Ansermet 24, CH-1211
Genève 4, Switzerland

###### Abstract

The current expansion rate of the Universe, the Hubble constant $H_{0}$, is an
important cosmological quantity. However, two different ways to measure its
value do not agree – building a low-redshift distance ladder leads to a higher
value of $H_{0}$ than inferring it from high-redshift observations in a
$\Lambda$CDM cosmology. Most approaches to solve this tension either act at
very low redshift by modifying the local distance ladder, or at high redshift
by introducing new physics that changes the normalization of the inverse
distance ladder. Here we discuss a way to address the Hubble tension at
intermediate redshifts instead. By keeping the low- and high-redshift
normalizations unchanged, we find a violation of the distance duality in the
redshift range where luminosity and angular diameter distances overlap. We
‘solve’ this problem by introducing a redshift-dependent systematic effect
that brings the luminosity distance into agreement with the angular diameter
distance. The resulting expansion history is no longer compatible with
$\Lambda$CDM, but this can be fixed with a dynamical dark energy component. In
this way, we are able to solve the Hubble tension at intermediate redshifts.

##  I Introduction

Since the discovery of the accelerated expansion 25 years ago [1, 2], the
cosmological standard model has been the Lambda-Cold-Dark-Matter
($\Lambda$CDM) model. $\Lambda$CDM has weathered many precision tests like the
measurements of the anisotropies in the cosmic microwave background (CMB) by
the Planck satellite [3] or measurements of the distance-redshift relation
using type-Ia supernovae (SN-Ia, e.g. [4]) and baryonic acoustic oscillations
(BAO, e.g. [5]).

With advancements in experimental sensitivity, it is reasonable to expect
tensions between different data sets to emerge, either due to unknown
systematic effects in the data, or because we have reached a level of
precision where we are able to uncover new physics beyond the standard model.
Because of the latter, it is important to investigate and understand the
source of such disagreements. The currently most significant tension between
data sets comes from the determination of the Hubble constant $H_{0}$ that
describes the late-time expansion rate of the Universe, see e.g. [6]: Local
measurements with Cepheids and SN-Ia find $H_{0}=73.04\pm 1.04$ km/s/Mpc [7]
while the Planck data in combination with the $\Lambda$CDM model predicts
$H_{0}=67.7\pm 0.42$ km/s/Mpc [8], a number that is in agreement with the
value obtained from BAO data combined with constraints on the baryon density
$\omega_{b}$ from big-bang nucleosynthesis, without CMB anisotropies [5]. The
difference of more than five standard deviations between these measurements
has led to a great interest in the community, as this might be the first sign
of a breakdown of the $\Lambda$CDM model. Unfortunately, so far no systematic
effects have been identified that can explain the tension fully, but also no
model has been proposed that provides a convincing explanation for the
observed discrepancy.

In this paper we focus particularly on intermediate redshifts where we have
both luminosity distances from SN-Ia and angular diameter distances from BAO.
As we will show below, these measurements are not compatible at equal
redshifts. This is a puzzle, since they should simply be related by the
distance duality relation (DDR) in any metric theory of gravity if photons are
conserved [9, 10]. Since SN-Ia are anchored at low redshifts and BAO at high
redshifts, most proposals aim to change one of these anchors, either through
systematic effects affecting the SN-Ia normalisation or through extensions of
the standard model, like early dark energy, that affect the sound horizon at
high $z$. Here we will instead assume that a systematic effect at intermediate
redshifts changes the luminosity distance so that it agrees with the angular
diameter distance. This is motivated by earlier studies that have considered a
possible redshift dependence of the intrinsic SN-Ia luminosity. One example is
the luminosity dependence on the local star formation rate. Although some
analyses do not detect a significant dependence of the luminosity on this
quantity, see e.g. [11], others do find a significant local environmental
dependence of the intrinsic luminosity, see e.g. [12] and references therein.
Given the redshift dependence of the star formation rate, this effect would
lead to a redshift-dependent intrinisic SN-Ia luminosity. Another example is
the dependence on the metallicity of the host galaxy. Metallicity evolves as a
function of redshift, therefore, any luminosity dependence on the metallicity
would introduce a redshift dependence, see e.g. [13, 14]. Observationally, a
detection of a redshift-dependence of the intrinsic SN-Ia luminosity was
recently claimed in [15]. Allowing for a redshift-dependent intrinsic SN-Ia
luminosity can have a large impact on the cosmological conclusions derived
from the observations, see e.g. [16, 17]. As we will see, the “deformation” of
the luminosity distance required to match the BAO distances then requires
additionally late-time dark energy. The combination of a redshift-dependent
systematic effect affecting the SN-Ia and a time-evolving dark energy was
previously studied in [18], but the authors found that the tension could not
be fully removed. Very recently [19] also considered scenarios including both
systematic effects and an evolving dark energy component and found different
possibilities to alleviate the tension. In this publication we argue that
bringing the BAO and SN-Ia distances into agreement is not optional, and we
design a scenario that uses a redshift-dependent intrinsic SN-Ia luminosity to
enforce the DDR, together with a late-time dark energy to explain the
resulting distance-redshift relation. In this way, our scenario solves the
tension by construction.

##  II Cosmology and Data sets

In this article, we will use distance data to constrain the expansion rate of
the Universe. We will only consider flat models and choose the radiation
density parameter $\Omega_{r}$ in agreement with the CMB temperature. The
expansion rate as a function of redshift $z$ is then

$H(z)^{2}=H_{0}^{2}\left(\Omega_{r}(1+z)^{4}+\Omega_{m}(1+z)^{3}+\Omega_{\rm
DE}(z)\right)\,.$ (1)

For the $\Lambda$CDM standard model, the dark energy density parameter is
given by

$\Omega_{\rm DE}(z)=(1-\Omega_{r}-\Omega_{m})\,,$ (2)

while for a general dark energy component with time-evolving equation of state
parameter $w(z)=\bar{p}(z)/\bar{\rho}(z)$ it becomes

$\Omega_{\rm
DE}(z)=(1-\Omega_{r}-\Omega_{m})\exp\left\\{3\int_{0}^{z}\frac{1+w(s)}{1+s}ds\right\\}\,.$
(3)

###  II.1 Type Ia supernovae

Type Ia supernovae are standardizable candles, i.e. they measure the
luminosity distance

$D_{L}(z)=(1+z)\int_{0}^{z}\frac{ds}{H(s)}\,,$ (4)

which is usually expressed in terms of the distance modulus

$\mu(z)=5\log_{10}\left(\frac{D_{L}(z)}{10\mathrm{pc}}\right)\,.$ (5)

Here we use the Pantheon+ data [4], for which the observed distance modulus
can be obtained as

$\mu_{\rm obs}=m_{B}-M\,,$ (6)

where $m_{B}$ is the observed magnitude from a light-curve fit and $M$ stands
for the absolute luminosity of SN-Ia. The authors in [4] provide the observed
magnitudes after accounting for several systematic effects and marginalizing
over different nuisance parameters, like the stretch of the light-curve, its
colour, or the host galaxy mass.

We further include the SH0ES Cepheid host distance anchors [7] to break the
degeneracy between $M$ and $H_{0}$. The distance modulus residuals that enter
the SN-Ia likelihood are then given by

$\displaystyle\mu_{\rm obs}^{i}-\mu_{\rm Cepheid}^{i}\,,$ $\displaystyle\text{
if }i\in\text{Cepheid host,}$ (7) $\displaystyle\mu_{\rm
obs}^{i}-\mu(z^{i})\,,$ otherwise, (8)

where $\mu_{\rm Cepheid}^{i}$ corresponds to the Cepheid calibrated host-
galaxy distance.

When systematic effects introducing a redshift-dependent absolute magnitude of
SN-Ia are included, we add an additional term into the observed distance
modulus, leading to the residuals

$\displaystyle\mu_{\rm obs}^{i}+\Delta\mu(z^{i})-\mu_{\rm Cepheid}^{i}\,,$
$\displaystyle\text{ if }i\in\text{Cepheid host,}$ (9) $\displaystyle\mu_{\rm
obs}^{i}+\Delta\mu(z^{i})-\mu(z^{i})\,,$ otherwise. (10)

###  II.2 Baryon Acoustic Oscillations

Transverse baryon acoustic oscillations measure the comoving angular diameter
distance,

$D_{M}(z)=D_{L}(z)/(1+z)\,.$ (11)

This relation is also called the distance duality and holds in any metric
theory of gravity (see e.g. [10]).

Like the SN-Ia distances, the BAO distances are also relative to a
normalization, in this case the sound horizon scale at the drag epoch. This
normalization can be affected by various mechanisms that act at high redshift,
like early dark energy [20]. Here we assume that the physics at high redshift
is not modified, so that we can use the standard values for that
normalization.

In practice, we consider the legacy results of eBOSS [5], which contain the
transverse BAO measurements from BOSS galaxies [21], eBOSS LRGs [22, 23],
eBOSS ELGs [24, 25], eBOSS quasars [26, 27], and the BOSS+eBOSS Lyman-$\alpha$
auto-correlation and cross-correlation [28]. In addition to the transverse
measurements, we also include the radial BAO measurements from these data
sets, as well as the angle-averaged measurement from the main galaxy sample of
SDSS [29, 30]. We follow [5] and use the covariance matrix for the BOSS
galaxies measurements, as well as the covariance for the eBOSS LRGs and eBOSS
quasars measurements. For the eBOSS ELGs and the Lyman-$\alpha$ forest
measurements we employ instead the full likelihood provided by the
collaboration, given its non-Gaussianity. The different data sets are
considered to be independent from one another.

###  II.3 Cosmic Microwave Background

The temperature fluctuations observed with CMB anisotropies are not a simple
distance measurement, and predicting them requires the use of a Boltzmann
code. However, most of the information contained in the CMB can be compressed
into a few numbers [31]:

  * •

the CMB shift parameter $R=\sqrt{\Omega_{m}H_{0}^{2}}D_{M}(z_{*})$ where
$z_{*}$ is the redshift for which the optical depth becomes unity;

  * •

the angular scale of the sound horizon at last scattering (equivalent to
$\theta_{*}$), $\ell_{a}=\pi D_{M}(z_{*})/r_{s}(z_{*})$ where $r_{s}$ is the
coming size of the sound horizon;

  * •

as well as the baryon density $\omega_{b}$ and the scalar spectral index
$n_{s}$. In this work, we keep $\omega_{b}$ but marginalize over $n_{s}$ since
our other probes do not depend on it.

These numbers are effectively observables for the CMB. Their distribution is
Gaussian to a good approximation, and is also valid for smooth dark energy
components (or non-zero curvature) even when derived in a $\Lambda$CDM model.
For the 2015 Planck data release, the mean values and covariance matrix was
computed in [32]. Here, we derive an updated likelihood based on the Planck
2018 $\Lambda$CDM chains [8] – as shown in [31], the compressed likelihood
does not depend on the model used.

More specifically, we consider the 2018 plik TTTEEE+lowl+lowE Planck
likelihood and assume a flat $\Lambda$CDM model. We further compute the
radiation density parameter considering the approximation presented in [33]
with one massive neutrino of mass 0.06 eV and a number of effective
relativistic species in the early universe of $N_{\rm eff}=3.046$. We
additionally consider the fitting function presented in [34] to compute the
ratio of baryon and photon densities that appears in the sound speed, assuming
a CMB temperature of 2.728 K. Finally, we set the redshift of the last
scattering epoch at $z_{*}=1089$. With these approximations, we obtain the
following mean values for the reduced parameters:111During the final stages of
this work, an alternative compression of the CMB likelihood appeared in
Appendix F of [35], based on flat $w$CDM chains from Planck.

$(\ell_{a},R,\omega_{b})=(301.205,1.7478,0.02236)\,,$ (12)

together with the covariance

$\text{cov}=10^{-5}\times\left(\begin{array}[]{ccc}712.9489&-1.7156&0.2656\\\
-1.7156&2.1569&-0.0459\\\ 0.2656&-0.0459&0.0023\end{array}\right)\,.$ (13)

###  II.4 Compatibility of the distances

Figure 1: Residuals of the distance moduli for SN-Ia and transverse BAO with
respect to the BAO best-fit $\Lambda$CDM model. The discrepancy between the
two data sets illustrates the Hubble tension.

In Fig. 1 we show the redshift-binned luminosity distance moduli of SN-Ia,
together with the BAO distance moduli. The latter have been obtained from the
(comoving) BAO distances multiplied by the factor $(1+z)$ that should
transform them to luminosity distances according to the distance duality. It
is obvious that the distances do not agree in the shared redshift range. We
note that such an inconsistency has already been identified in previous
analyses [36, 37, 38], where the authors reconstructed an energy component to
better fit both data sets at the same time. However, while dark energy can
improve the overall fit, it cannot solve the discrepancy between the distance
measurements, as it affects all distances. Similar results were found in [39,
40], where, even with a very flexible parameterization for the dark energy
equation of state, no solution to the Hubble tension could be found with new
physics at low-redshift. For this reason, we instead restore the distance
duality by invoking the presence of systematic uncertainties to reconcile the
two data sets without modifying their anchors. We also note that the use of
the cosmic distance duality has been considered recently to constrain a
possible redshift evolution of the intrinsic SN-Ia luminosity [41]. The
authors of this work combined SN-Ia and strong lensing measurements, but the
distance measurements of these two data sets are compatible and led to no
detection of a redshift evolution of the intrinsic SN-Ia luminosity.

##  III Schematic analysis

Here we concoct a simple, schematic example of how to address the $H_{0}$
discrepancy at intermediate redshifts. Although the two distances do not
agree, each distance agrees separately with a $\Lambda$CDM model, for
different values of $H_{0}$. We assume that there is a systematic effect that
affects the observed luminosity distances so that they transition from
distances corresponding to one value of $H_{0}$ to another. More precisely, we
use a transition function $s(z)$ that changes from $s=0$ to $s=1$,

$s(z)=\frac{1}{1+e^{-(z-d)c}}\,,$ (14)

where $c,d$ are parameters that describe the transition: $d$ the location in
$z$ and $c$ the rapidity. We then assume that there are two $\Lambda$CDM
expansion rates given at late times by

$H^{[i]}(z)=H^{[i]}_{0}\left(\Omega_{m}^{[i]}(1+z)^{3}+(1-\Omega_{m}^{[i]})\right)^{1/2}$
(15)

for two different values of $H^{[i]}_{0},\Omega_{m}^{[i]},\,i=1,2$. The
effective expansion rate for the luminosity distance is then

$H_{\rm SN-Ia}(z)=(1-s(z))H^{[1]}(z)+s(z)H^{[2]}(z)\,.$ (16)

We compute the luminosity distance and the distance modulus from this
expansion rate as well as from $H^{[1]}(z)$ – which corresponds to SN-Ia
without systematic effects, and consider the difference of the distance moduli
$\Delta\mu(z)$ as the necessary systematic effect that can reconcile the data
sets. We show an example for $\Delta\mu(z)$ that is able to fit the data well
in Fig. 2.

Figure 2: The change of SN-Ia distance moduli, $\Delta\mu(z)$, necessary to
reconcile BAO and SN-Ia distances.

The resulting luminosity distance is able to reconcile SN-Ia and BAO
distances, see Fig. 3. However, it does not correspond to the expansion rate
of a $\Lambda$CDM model, because of the transition. But for any expansion rate
$H(z)$ and a given $\Omega_{m}$, there is a dark energy equation of state
$w(z)$ that reproduces $H(z)$ [42]. Since we have an explicit form for $H(z)$
from Eq. (16), we can easily derive the corresponding equation of state
parameter. By adjusting $\Omega_{m}$, we find that we can bring it into a form
where $w(z)$ is constant at small and large redshifts, and has a bump around
the transition redshift $z=d$. This suggests that we should be able to model
such a shape with two asymptotic values, $w_{1}$ and $w_{2}$, and a Gaussian
peak,

$w(z)=w_{1}(1-s(z))+w_{2}s(z)+ue^{-\frac{1}{2}(z-d)^{2}/v^{2}}\,.$ (17)

Here $u$ is the amplitude of the bump and $v$ controls the width, while the
location is given by $d$.

Figure 3: A comparison of distance moduli for SN-Ia and BAO distances (blue
points), relative to the best fit $w$CDM model for the (SN-Ia+syst)+BAO+CMB
points (black line). The orange points show the SN-Ia distance moduli from the
Pantheon+ data without systematic effect. The green points show the result of
applying the correction shown in Fig. 2. Clearly the green points are in much
better agreement with the blue BAO points than the orange points. This figure
uses a dark energy component with the $w(z)$ shown in Fig. 4.

Summarizing, we consider a cosmological model that accounts for some
systematic effects through $\Delta\mu(z)$ and dark energy characterized by an
equation of state $w(z)$. The former can be extracted from the difference of
distance moduli from Eqs. (15,16). Therefore, the systematic effects are
parametrized by the nuisance parameters $c,d$, as well as the Hubble constant
from the SN-Ia best-fit, $H_{0}^{[1]}$, the Hubble constant from the BAO+CMB
best-fit, $H_{0}^{[2]}\equiv H_{0}^{[1]}+\Delta H_{0}$, the matter density
from the SN-Ia best-fit, $\Omega_{m}^{[1]}$, and the matter density from the
BAO+CMB best-fit, $\Omega_{m}^{[2]}$. Concerning $w(z)$, we add the two
amplitudes, $w_{1},w_{2}$, and the nuisance parameters $u,v$. Finally, we also
need to account for the Hubble constant of our model, $H_{0}$, as well as the
matter density, $\Omega_{m}$, the baryon density, $\Omega_{b}$, and the
absolute magnitude of SN-Ia, $M$. This leads to a total number of 14 free
parameters. However, in order to simplify the analysis and avoid large
degeneracies, we fix $H_{0}^{[1]}=73.4$ km/s/Mpc, $\Omega_{m}^{[1]}=0.331$,
$\Delta H_{0}=5.8$ km/s/Mpc, and $\Omega_{m}^{[2]}=0.312$, which correspond to
the best-fit values obtained from the fit of a $\Lambda$CDM model to the
individual data sets. Maximizing the goodness of fit (leading to
$\Omega_{m}\approx 0.27$), we obtain the shape shown in Fig. 4, where $w$
remains close to $-1$ down to $z\approx 0.1$ and then exhibits a ‘phantom
bump’ around $z\approx 0.05$ (in agreement with the general considerations of
[43]).

Figure 4: The equation of state parameter $w(z)$ required to fit the full
Hubble diagram that corresponds to the green and blue points in Fig. 3.

Overall our scenario improves the goodness of fit of the data sets by
$\Delta\chi^{2}\approx 40$ and brings SN-Ia, BAO and CMB data into agreement,
at the price of introducing two new ingredients at low redshifts, a systematic
effect affecting the SN-Ia distances and a late-time dark energy. While
requiring two ingredients may not appear compelling, each one of them is not
very exotic per se, and a key point of this analysis is to show that the
combination of the two provides an alternative scenario that provides a good
fit to the data.

##  IV Conclusion

In this paper we show that it is possible in principle to fit simultaneously
the normalized supernova data, the BAO data and the main features of the CMB
with late-time dark energy, if there is a systematic effect affecting the
supernova luminosities, even if that effect does not change the late-time
$H_{0}$. If we neither want to change the normalisation of the SN-Ia, nor
introduce a deviation from $\Lambda$CDM at high redshifts, then the presence
of such a systematic effect is effectively unavoidable as the distance duality
requires luminosity and angular diameter distances to agree.

This systematic effect needs to reach about 0.2 magnitudes at high redshift,
and needs to reach about 80% of this change by redshift $z\approx 0.3$ in
order to bring the SN-Ia distances into agreement with the BAO distances. One
future way to test for such a systematic effect could be a comparison between
the gravitational wave (GW) inferred luminosity distance and the SN-Ia
luminosity distance as soon as we are able to observe a GW event in a galaxy
that also hosted a SN-Ia [44]. This would be a crucial test whether the
$H_{0}$ tension can be explained by a mechanism that affects neither the
distance anchors nor the primordial universe.

The systematic effect changes the true distance-redshift relation at low $z$
away from $\Lambda$CDM, requiring now the existence of a late-time evolving
dark energy component. The best-fit $w(z)$ from our schematic analysis shows a
relatively large excursion to $w\approx-4$ below $z\approx 0.1$. This might be
visible in other data sets, but it is challenging to observe as it happens at
very low redshifts.

Another potential tension concerns the amplitude of fluctuations on scales of
$8h^{-1}$ Mpc, $\sigma_{8}$. The value of $\sigma_{8}$ inferred from the CMB
in a $\Lambda$CDM model is higher than the value found by weak lensing
measurements (see e.g. [45, 46, 47]), and some mechanisms that aim to solve
the Hubble tension aggravate the $\sigma_{8}$ tension [48]. We computed the
value of $\sigma_{8}$ in our scenario where $w$ varies, and it turns out to be
about 5% higher than in the corresponding $\Lambda$CDM case
($\sigma_{8}\approx 0.834$ compared to $\sigma_{8}\approx 0.793$). However,
our preferred matter density is also somewhat lower than in the standard
model, so that our value of $S_{8}\approx 0.79$ is compatible with the weak
lensing observations of [49] at 1$\sigma$.

###### Acknowledgements.

MK acknowledges financial support from the Swiss National Science Foundation.

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# The $I^{G}J^{PC}=0^{+}1^{-+}$ Tetraquark State

Hua-Xing Chen1,2,3<EMAIL_ADDRESS>Atsushi Hosaka2
<EMAIL_ADDRESS>Shi-Lin Zhu1<EMAIL_ADDRESS>1Department of
Physics, Peking University, Beijing 100871, China
2Research Center for Nuclear Physics, Osaka University, Ibaraki 567–0047,
Japan
3State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871, China

###### Abstract

We study the tetraquark state with $I^{G}J^{PC}=0^{+}1^{-+}$ in the QCD sum
rule. We exhaust all possible flavor structures by using a diquark-antidiquark
construction and find that the flavor structure
$(\mathbf{\bar{3}}\otimes\mathbf{\bar{6}})\oplus(\mathbf{6}\otimes\mathbf{3})$
is preferred. There are altogether four independent currents which have the
quark contents $qs\bar{q}\bar{s}$. By using both the Shifman-Vainshtein-
Zakharov (SVZ) sum rule and the finite energy sum rule, these currents lead to
mass estimates around $1.8-2.1$ GeV, where the uncertainty is due to the
mixing of two single currents. Its possible decay modes are $S$-wave
$b_{1}(1235)\eta$ and $b_{1}(1235)\eta^{\prime}$, and $P$-wave $KK$,
$\eta\eta$, $\eta\eta^{\prime}$ and $\eta^{\prime}\eta^{\prime}$, etc. The
decay width is around 150 MeV through a rough estimation.

exotic mesons, tetraquark, QCD sum rule

###### pacs:

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

Manifestly exotic hadron states which are not reached by three quarks for
baryons and a quark-antiquark pair for mesons provide one of the most
important subjects in hadron physics. The confirmation of their existence (or
nonexistence) and the study of their structure are of great importance for the
understanding of strong interaction dynamics at low energy exotic .

Quantum numbers can tell whether a hadron is exotic or not. For instance
baryons with strangeness $S=+1$ and mesons with $J^{PC}=1^{-+}$ are such
states. For the baryon sector, the pentaquark $\Theta^{+}$ has been studied
intensively since 2003 Nakano:2003qx . But the existence is still
controversial. For the meson sector, the $\pi_{1}$ mesons of
$I^{G}J^{PC}=1^{-}1^{-+}$ are listed as manifestly exotic states in the PDG
for some time Yao:2006px ; experiments ; Lu:2004yn , and a lot of theoretical
considerations have been made theory ; isoscalar . So far, many of them are
for the isovector $I=1$ states. In principle, an isoscalar state is also
possible, though not observed experimentally isoscalar . We have performed the
QCD sum rule analyses of the light scalar mesons ($\sigma$, $\kappa$, $f_{0}$
and $a_{0}$), $Y(2175)$ and $\pi_{1}$ mesons Chen ; Chen:2008qw . All our
results are consistent with the experimental observations. Encouraged by this,
we would like to extend the QCD sum rule analysis using tetraquark currents
for these $I^{G}J^{PC}=0^{+}1^{-+}$ states.

Figure 1: Weight diagrams for
$\mathbf{6_{f}}\otimes\mathbf{\bar{6}_{f}}(\mathbf{S})$ (top panel),
$\mathbf{\bar{3}_{f}}\otimes\mathbf{3_{f}}(\mathbf{A})$ (middle panel), and
$\mathbf{\bar{3}_{f}}\otimes\mathbf{\bar{6}_{f}}(\mathbf{M})$ (bottom panel).
The weight diagram for $\mathbf{6_{f}}\otimes\mathbf{3_{f}}(\mathbf{M})$ is
the charge-conjugation transformation of the bottom one.

The QCD sum rule requires a computation of a two-point correlation function in
the form of operator product expansion (OPE), which is then fitted by a
phenomenological function to extract physical hadron properties sumrule . To
calculate the OPE, we need employ an interpolating field (current) which
couples to the physical state we consider. For tetraquarks, there are several
independent currents and it is important to establish how one or some of them
should be chosen. We have systematically performed the classification of
currents by using the diquark-antidiquark ($(qq)(\bar{q}\bar{q})$)
construction Chen ; Chen:2008qw . The currents constructed from the quark-
antiquark pairs ($(\bar{q}q)(\bar{q}q)$) can be written as a combination of
these ($(qq)(\bar{q}\bar{q})$) currents. We note here that the mixing can
happen between hybrid states, tetraquark states, and even six-quark states,
while the currents can also couple to all these states. However, it always
makes sense to clarify what a single channel problem tells us before entering
more sophisticated coupled channel problems. Therefore, here we concentrate
exclusively on the tetraquark properties with some details.

The tetraquark currents with the quantum numbers $J^{PC}=1^{-+}$ have been
constructed in our previous paper Chen:2008qw . Now we need construct the
isoscalar ones. The flavor structures are shown in Fig. 1 in terms of $SU(3)$
weight diagrams. The ideal mixing scheme is used since it is expected to work
well for hadrons except for the pseudoscalar mesons. In order to have a
definite charge-conjugation parity, the diquark and antidiquark inside can
have the same flavor symmetry, which is either symmetric
$\mathbf{6_{f}}\otimes\mathbf{\bar{6}_{f}}$ ($\mathbf{S}$) or antisymmetric
$\mathbf{\bar{3}_{f}}\otimes\mathbf{3_{f}}$ ($\mathbf{A}$). Another option is
the combination of $\mathbf{\bar{3}_{f}}\otimes\mathbf{\bar{6}_{f}}$ and
$\mathbf{6_{f}}\otimes\mathbf{3_{f}}$ ($\mathbf{M}$), which can also have a
definite charge-conjugation parity.

From Fig. 1, we find that there are althgether six isospin singlets:

$\displaystyle
qq\bar{q}\bar{q}(\mathbf{S})\,,qs\bar{q}\bar{s}(\mathbf{S})\,,ss\bar{s}\bar{s}(\mathbf{S})\sim\mathbf{6_{f}}\otimes\mathbf{\bar{6}_{f}}~{}~{}~{}(\mathbf{S})\,,$
$\displaystyle
qq\bar{q}\bar{q}(\mathbf{A})\,,qs\bar{q}\bar{s}(\mathbf{A})\sim\mathbf{\bar{3}_{f}}\otimes\mathbf{3_{f}}~{}~{}~{}(\mathbf{A})\,,$
(1) $\displaystyle
qs\bar{q}\bar{s}(\mathbf{M})\sim(\mathbf{\bar{3}_{f}}\otimes\mathbf{\bar{6}_{f}})\oplus(\mathbf{6_{f}}\otimes\mathbf{3_{f}})~{}(\mathbf{M})\,,$

where $q$ represents an $up$ or $down$ quark, and $s$ represents a $strange$
quark. For each state, there are several independent currents. We list them in
the following.

  1. 1.

For the three isospin singlets of $\mathbf{6}_{f}\otimes\mathbf{\bar{6}}_{f}$
($\mathbf{S}$):

$\displaystyle\begin{array}[]{ll}\eta^{S}_{1\mu}&\sim
u_{a}^{T}C\gamma_{5}d_{b}(\bar{u}_{a}\gamma_{\mu}\gamma_{5}C\bar{d}_{b}^{T}+\bar{u}_{b}\gamma_{\mu}\gamma_{5}C\bar{d}_{a}^{T})\\\
&+u_{a}^{T}C\gamma_{\mu}\gamma_{5}d_{b}(\bar{u}_{a}\gamma_{5}C\bar{d}_{b}^{T}+\bar{u}_{b}\gamma_{5}C\bar{d}_{a}^{T})\,,\\\
\eta^{S}_{2\mu}&\sim
u_{a}^{T}C\gamma^{\nu}d_{b}(\bar{u}_{a}\sigma_{\mu\nu}C\bar{d}_{b}^{T}-\bar{u}_{b}\sigma_{\mu\nu}C\bar{d}_{a}^{T})\\\
&+u_{a}^{T}C\sigma_{\mu\nu}d_{b}(\bar{u}_{a}\gamma^{\nu}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma^{\nu}C\bar{d}_{a}^{T})\,,\end{array}$
(6) $\displaystyle\begin{array}[]{ll}\eta^{S}_{3\mu}&\sim
u_{a}^{T}C\gamma_{5}s_{b}(\bar{u}_{a}\gamma_{\mu}\gamma_{5}C\bar{s}_{b}^{T}+\bar{u}_{b}\gamma_{\mu}\gamma_{5}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\gamma_{\mu}\gamma_{5}s_{b}(\bar{u}_{a}\gamma_{5}C\bar{s}_{b}^{T}+\bar{u}_{b}\gamma_{5}C\bar{s}_{a}^{T})\,,\\\
\eta^{S}_{4\mu}&\sim
u_{a}^{T}C\gamma^{\nu}s_{b}(\bar{u}_{a}\sigma_{\mu\nu}C\bar{s}_{b}^{T}-\bar{u}_{b}\sigma_{\mu\nu}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\sigma_{\mu\nu}s_{b}(\bar{u}_{a}\gamma^{\nu}C\bar{s}_{b}^{T}-\bar{u}_{b}\gamma^{\nu}C\bar{s}_{a}^{T})\,.\end{array}$
(11) $\displaystyle\begin{array}[]{ll}\eta^{S}_{5\mu}&\sim
s_{a}^{T}C\gamma_{5}s_{b}(\bar{s}_{a}\gamma_{\mu}\gamma_{5}C\bar{s}_{b}^{T}+\bar{s}_{b}\gamma_{\mu}\gamma_{5}C\bar{s}_{a}^{T})\\\
&+s_{a}^{T}C\gamma_{\mu}\gamma_{5}s_{b}(\bar{s}_{a}\gamma_{5}C\bar{s}_{b}^{T}+\bar{s}_{b}\gamma_{5}C\bar{s}_{a}^{T})\,,\\\
\eta^{S}_{6\mu}&\sim
s_{a}^{T}C\gamma^{\nu}s_{b}(\bar{s}_{a}\sigma_{\mu\nu}C\bar{s}_{b}^{T}-\bar{s}_{b}\sigma_{\mu\nu}C\bar{s}_{a}^{T})\\\
&+s_{a}^{T}C\sigma_{\mu\nu}s_{b}(\bar{s}_{a}\gamma^{\nu}C\bar{s}_{b}^{T}-\bar{s}_{b}\gamma^{\nu}C\bar{s}_{a}^{T})\,.\end{array}$
(16)

where $\eta^{S}_{1\mu}$ and $\eta^{S}_{2\mu}$ are the two independent currents
containing only light flavors; $\eta^{S}_{3\mu}$ and $\eta^{S}_{4\mu}$ are the
two independent ones containing one $s\bar{s}$ pair; $\eta^{S}_{5\mu}$ and
$\eta^{S}_{6\mu}$ are the two independent ones containing two $s\bar{s}$
pairs.

  2. 2.

For the two isospin singlets of $\mathbf{\bar{3}}_{f}\otimes\mathbf{3}_{f}$
($\mathbf{A}$):

$\displaystyle\begin{array}[]{ll}\eta^{A}_{1\mu}&\sim
u_{a}^{T}C\gamma_{5}d_{b}(\bar{u}_{a}\gamma_{\mu}\gamma_{5}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma_{\mu}\gamma_{5}C\bar{d}_{a}^{T})\\\
&+u_{a}^{T}C\gamma_{\mu}\gamma_{5}d_{b}(\bar{u}_{a}\gamma_{5}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma_{5}C\bar{d}_{a}^{T})\,,\\\
\eta^{A}_{2\mu}&\sim
u_{a}^{T}C\gamma^{\nu}d_{b}(\bar{u}_{a}\sigma_{\mu\nu}C\bar{d}_{b}^{T}+\bar{u}_{b}\sigma_{\mu\nu}C\bar{d}_{a}^{T})\\\
&+u_{a}^{T}C\sigma_{\mu\nu}d_{b}(\bar{u}_{a}\gamma^{\nu}C\bar{d}_{b}^{T}+\bar{u}_{b}\gamma^{\nu}C\bar{d}_{a}^{T})\,,\end{array}$
(21) $\displaystyle\begin{array}[]{ll}\eta^{A}_{3\mu}&\sim
u_{a}^{T}C\gamma_{5}s_{b}(\bar{u}_{a}\gamma_{\mu}\gamma_{5}C\bar{s}_{b}^{T}-\bar{u}_{b}\gamma_{\mu}\gamma_{5}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\gamma_{\mu}\gamma_{5}s_{b}(\bar{u}_{a}\gamma_{5}C\bar{s}_{b}^{T}-\bar{u}_{b}\gamma_{5}C\bar{s}_{a}^{T})\,,\\\
\eta^{A}_{4\mu}&\sim
u_{a}^{T}C\gamma^{\nu}s_{b}(\bar{u}_{a}\sigma_{\mu\nu}C\bar{s}_{b}^{T}+\bar{u}_{b}\sigma_{\mu\nu}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\sigma_{\mu\nu}s_{b}(\bar{u}_{a}\gamma^{\nu}C\bar{s}_{b}^{T}+\bar{u}_{b}\gamma^{\nu}C\bar{s}_{a}^{T})\,,\end{array}$
(26)

where $\eta^{A}_{1\mu}$ and $\eta^{A}_{2\mu}$ are the two independent currents
containing only light flavors; $\eta^{A}_{3\mu}$ and $\eta^{A}_{4\mu}$ are the
two independent ones containing one $s\bar{s}$ pair.

  3. 3.

For the isospin singlet of
$(\mathbf{\bar{3}}_{f}\otimes\mathbf{\bar{6}}_{f})\oplus(\mathbf{6}_{f}\otimes\mathbf{3}_{f})$
($\mathbf{M}$),

$\displaystyle\begin{array}[]{ll}\eta^{M}_{1\mu}&\sim
u_{a}^{T}C\gamma_{\mu}s_{b}(\bar{u}_{a}C\bar{s}_{b}^{T}+\bar{u}_{b}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}Cs_{b}(\bar{u}_{a}\gamma_{\mu}C\bar{s}_{b}^{T}+\bar{u}_{b}\gamma_{\mu}C\bar{s}_{a}^{T})\,,\\\
\eta^{M}_{2\mu}&\sim
u_{a}^{T}C\sigma_{\mu\nu}\gamma_{5}s_{b}(\bar{u}_{a}\gamma^{\nu}\gamma_{5}C\bar{s}_{b}^{T}+\bar{u}_{b}\gamma^{\nu}\gamma_{5}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\gamma^{\nu}\gamma_{5}s_{b}(\bar{u}_{a}\sigma_{\mu\nu}\gamma_{5}C\bar{s}_{b}^{T}+\bar{u}_{b}\sigma_{\mu\nu}\gamma_{5}C\bar{s}_{a}^{T})\,,\\\
\eta^{M}_{3\mu}&\sim
u_{a}^{T}Cs_{b}(\bar{u}_{a}\gamma_{\mu}C\bar{s}_{b}^{T}-\bar{u}_{b}\gamma_{\mu}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\gamma_{\mu}s_{b}(\bar{u}_{a}C\bar{s}_{b}^{T}-\bar{u}_{b}C\bar{s}_{a}^{T})\,,\\\
\eta^{M}_{4\mu}&\sim
u_{a}^{T}C\gamma^{\nu}\gamma_{5}s_{b}(\bar{u}_{a}\sigma_{\mu\nu}\gamma_{5}C\bar{s}_{b}^{T}-\bar{u}_{b}\sigma_{\mu\nu}\gamma_{5}C\bar{s}_{a}^{T})\\\
&+u_{a}^{T}C\sigma_{\mu\nu}\gamma_{5}s_{b}(\bar{u}_{a}\gamma^{\nu}\gamma_{5}C\bar{s}_{b}^{T}-\bar{u}_{b}\gamma^{\nu}\gamma_{5}C\bar{s}_{a}^{T})\,,\end{array}$
(35)

where $\eta^{M}_{i\mu}$ are the four independent ones containing one
$s\bar{s}$ pair. The above structure has some implications on their decay
patterns.

The expressions of Eqs. (6)-(35) are not exactly correct, since they do not
have a definite isospin. For instance, the current $\eta^{A}_{3\mu}$ should
contain $(us\bar{u}\bar{s}+ds\bar{d}\bar{s})$ in order to have $I=0$. However,
in the following QCD sum rule analysis, we find that there is no difference
between these two cases in the limit that the masses and condensates of the
$up$ and $down$ quarks are the same. Actually we also ignore a small quark
mass effect ($m_{u}\sim m_{d}\lesssim 10$ MeV).

By using these tetraquark currents, we have performed the OPE calculation up
to dimension 12. Values for various condensates and $m_{s}$ follow the
references Yao:2006px ; values . There are altogether 14 currents. It turns
out that some of them lead to the same results of OPEs as the previous ones in
Ref. Chen:2008qw : $\eta^{S}_{1,2,3,4\mu}\sim\eta^{S}_{1,2,3,4\mu}$
Chen:2008qw , $\eta^{A}_{3,4\mu}\sim\eta^{A}_{1,2\mu}$ Chen:2008qw , and
$\eta^{M}_{1,2,3,4\mu}\sim\eta^{M}_{5,6,7,8\mu}$ Chen:2008qw . Therefore, we
just need calculate the OPEs of $\eta^{S}_{5,6\mu}$ and $\eta^{A}_{1,2\mu}$.
The full OPE expressions are too lengthy and are omitted here.

In our previous paper Chen:2008qw  we have found that the OPEs of the currents
$\eta^{S}_{i\mu}$’s and $\eta^{A}_{i\mu}$’s lead to unphysical results where
the spectral densities $\rho(s)$ become negative in the region of $2$ GeV2
$\lesssim s\lesssim 4$ GeV2. We find this to be the case also for the
isoscalar currents. Therefore, our QCD sum rule analysis does not support a
tetraquark state which has a flavor structure either
$\mathbf{6_{f}}\otimes\mathbf{\bar{6}_{f}}$ or
$\mathbf{\bar{3}_{f}}\otimes\mathbf{3_{f}}$ and a mass less than 2 GeV.

We shall discuss only the currents of the mixed flavor symmetry. We find there
is only one set of four independent currents as given in Eqs. (35), unlike the
isovector case which have two sets. The spectral densities calculated by the
mixed currents are positive for a wide range of $s$, and the convergence of
OPE is very good in the region of $2$ GeV${}^{2}<M_{B}^{2}<$ 5GeV2 as in our
previous study Chen:2008qw . In general, the pole contribution should be large
enough in the SVZ sum rule. However, the pole contributions of multiquark
states are rather small due to the large continuum contribution. Therefore a
careful choice of the threshold parameter is important in order to subtract
the continuum contribution.

When using the SVZ sum rule, the mass is obtained as functions of Borel mass
$M_{B}$ and threshold value $s_{0}$. As an example, we show the mass
calculated from currents $\eta^{M}_{2\mu}$ in Fig. 2. The Borel mass
dependence is weak, as shown in the upper figure; the $s_{0}$ dependence has a
minimum where the stability is the best, as shown in the bottom figure. The
minimum is around 2.0 GeV, which we choose to be our prediction. The other
three independent currents $\eta^{M}_{1\mu}$, $\eta^{M}_{3\mu}$ and
$\eta^{M}_{4\mu}$ lead to similar results, which are around 2.1 GeV, 1.9 GeV
and 2.0 GeV respectively.

Figure 2: The mass of the state $qs\bar{q}\bar{s}$ calculated by using the
current $\eta^{M}_{2\mu}$, as functions of $M_{B}^{2}$ (left) and $s_{0}$
(right) in units of GeV.

When using the finite energy sum rule, the mass is obtained as a function of
the threshold value $s_{0}$, which is shown in Fig. 3. There is also a mass
minimum around 2.1 GeV, 1.9 GeV, 1.9 GeV and 2.0 GeV for currents
$\eta^{M}_{1\mu}$, $\eta^{M}_{2\mu}$, $\eta^{M}_{3\mu}$ and $\eta^{M}_{4\mu}$
respectively. In a short summary, we have performed a QCD sum rule analysis
for $qs\bar{q}\bar{s}$. The mass obtained is around 2.0 GeV. We label this
state $\sigma_{1}(2000)$.

Figure 3: The mass calculated using the finite energy sum rule. The labels
besides the lines indicate the suffix i of the current $\eta^{M}_{i\mu}$
($i=1,\cdots,4$).

We can also study the mixing of these four currents. The currents
$\eta^{M}_{1\mu}$ and $\eta^{M}_{3\mu}$ have the largest mass difference, so
we study their mixing as an example:

$\displaystyle\eta^{M}_{mix}={\rm cos}\theta\eta^{M}_{1\mu}+{\rm
sin}\theta\eta^{M}_{3\mu}\,,$ (36)

where $\theta$ is the mixed angle. We calculate its OPE, and find that the
resulting spectral density is just:

$\displaystyle\rho^{M}_{mix}={\rm cos}^{2}\theta\rho^{M}_{1\mu}+{\rm
sin}^{2}\theta\rho^{M}_{3\mu}\,,$ (37)

The obtained mass is shown in Fig. 4 as functions of $\theta$. When we take
$s_{0}=3.5$ GeV2 (solid line), the mass maximum is 2.05 GeV, and the minimum
is 1.85 GeV. Therefore, we arrive at the similar result which produces the
mass around 2 GeV. We can also consider the mixing of other currents, which
would not change the results significantly due to the similarity of single
currents. The mass estimates are around $1.8-2.1$ GeV, where the uncertainty
is due to the mixing of two single currents.

Figure 4: The mass calculated using the finite energy sum rule, and for the
mixed current $\eta^{M}_{mix}$. The curves are obtained by setting $s_{0}=3.5$
GeV2 (solid line), 4 GeV2 (short-dashed line) and 5 GeV2 (long-dashed line).

Now let us discuss its decay properties as expected from a naive fall-apart
process. As shown in Eqs. (35) the currents contain one $s\bar{s}$ pair.
Therefore, we expect that the final states should also contain one $s\bar{s}$
pair. In order to spell out the possible spin of decaying particles and their
orbital angular momentum, we need perform a Fierz rearrangement to change
$(qq)(\bar{q}\bar{q})$ currents to $(\bar{q}q)(\bar{q}q)$ ones. For
illustration, we use one of the four independent $(\bar{q}q)(\bar{q}q)$
currents Chen:2008qw :

$\displaystyle\xi^{M}_{2\mu}$ $\displaystyle=$
$\displaystyle(\bar{s}_{a}\gamma^{\mu}\gamma_{5}s_{a})(\bar{u}_{b}\gamma_{5}u_{b})-(\bar{s}_{a}\gamma_{5}s_{a})(\bar{u}_{b}\gamma^{\mu}\gamma_{5}u_{b})$
(38) $\displaystyle+\cdots\,.$

All terms of this current have the structure
$(\bar{q}_{a}\gamma^{\mu}\gamma_{5}q_{a})(\bar{q}_{b}\gamma_{5}q_{b})$.
Therefore, the expected decay patterns are: (1) $1^{+}$ and $0^{-}$ particles
with relative angular momentum $L=0$, and (2) $0^{-}$ and $0^{-}$ particles
with $L=1$.

For the $S$-wave decay, we expect the following two-body decay patterns

$\displaystyle\sigma_{1}(I^{G}J^{PC}=0^{+}1^{-+})$ $\displaystyle\rightarrow$
$\displaystyle a_{1}(1260)\eta,a_{1}\eta^{\prime},\cdots\,,$ (39)
$\displaystyle b_{1}(1235)\eta,b_{1}\eta^{\prime}\cdots\,.$

If we consider, however, the $G$ parity conservation, the fist line is
forbidden and the second line is the only one allowed. These modes can be
observed in the final states $\omega\pi\eta$ and $\omega\pi\eta^{\prime}$.

For the $P$-wave decay, we expect (with the $G$ parity conservation):

$\displaystyle\sigma_{1}(I^{G}J^{PC}=0^{+}1^{-+})$ $\displaystyle\rightarrow$
$\displaystyle
KK,\eta\eta,\eta\eta^{\prime},\eta^{\prime}\eta^{\prime}\cdots\,.$ (40)

We can also estimate the (partial) decay width through the comparison with the
observed $\pi_{1}(2015)$ Lu:2004yn , which has $\Gamma_{\rm tot}\sim 230$ MeV.
Assuming that the decay of $\pi_{1}(2015)$ solely goes through $S$-wave
$b_{1}\pi$ and that of $\sigma_{1}(2000)$ through $b_{1}\eta$, we expect
$\Gamma_{\sigma_{1}\rightarrow b_{1}\eta}\sim 160$ MeV, as they are
proportional to the $S$-wave phase space. For the $P$-wave decay there is an
information $\pi_{1}(2015)\rightarrow\eta^{\prime}\pi$, which corresponds to
$\sigma_{1}(2000)\rightarrow\eta^{\prime}\eta$ (Because both $\pi_{1}(1600)$
and $\pi_{1}(2015)$ have been observed in the final states $\pi\eta^{\prime}$
other than $\pi\eta$, we choose $\eta\eta^{\prime}$ to be the final states of
$\sigma_{1}(2000)$ other than $KK$ and $\eta\eta$). Assuming once again that
this is the unique decay mode, we expect that the decay width is approximately
130 MeV. If the decay occurs $50\%$ through $b_{1}\pi$ ($b_{1}\eta$) and
$50\%$ through $\eta^{\prime}\pi$ ($\eta^{\prime}\eta$), we expect that
$\Gamma_{\sigma_{1}}\sim 150$ MeV.

In summary, we have performed the QCD sum rule analysis of the exotic
tetraquark states with $I^{G}J^{PC}=0^{+}1^{-+}$. We test all possible flavor
structures in the diquark-antidiquark $(qq)(\bar{q}\bar{q})$ construction,
$\mathbf{6}\otimes\mathbf{\bar{6}}$, $\mathbf{\bar{3}}\otimes\mathbf{3}$ and
$(\mathbf{\bar{3}}\otimes\mathbf{\bar{6}})\oplus(\mathbf{6}\otimes\mathbf{3})$.
We find that only the mixed currents of the flavor structure
$(\mathbf{\bar{3}}\otimes\mathbf{\bar{6}})\oplus(\mathbf{6}\otimes\mathbf{3})$
allow a positive and convergent OPE, and there is only one choice with the
quark content $qs\bar{q}\bar{s}$, which have four independent currents. We
have then performed both the SVZ sum rule and the finite energy sum rule. The
mass estimates are around $1.8-2.1$ GeV, where the uncertainty is due to the
mixing of two single currents. The possible decay modes are $S$-wave
$b_{1}(1235)\eta$ and $b_{1}(1235)\eta^{\prime}$, and $P$-wave $KK$,
$\eta\eta$, $\eta\eta^{\prime}$ and $\eta^{\prime}\eta^{\prime}$, etc. The
decay width is around 150 MeV through a rough estimation. Here we want to note
that we do not know how to determine the mixing angle, which is an interesting
problem.

## Acknowledgments

H.X.C. is grateful for Monkasho support for his stay at the Research Center
for Nuclear Physics where this work was done. This project was supported by
the National Natural Science Foundation of China under Grants No. 10625521,
10721063, the Ministry of Education of China, and the Grant for Scientific
Research ((C) No. 19540297) from the Ministry of Education, Culture, Science
and Technology, Japan.

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