# Strange stars with a mirror-dark-matter core confronting with the
observations of compact stars

Shu-Hua Yanga<EMAIL_ADDRESS>Chun-Mei Pib Xiao-Ping Zhenga,c a Institute
of Astrophysics, Central China Normal University, Wuhan 430079, China  b
School of Physics and Mechanical & Electrical Engineering, Hubei University of
Education, Wuhan 430205, China  c Department of Astronomy, School of physics,
Huazhong University of Science and Technology, Wuhan 430074, China

###### Abstract

We investigate the structure and the tidal deformability of strange stars
(SSs) with a mirror-dark-matter (MDM) core for the standard MIT bag model. We
find that to explain the observations of PSR J0740+6620, PSR J0030+0451 and
GW170817 simultaneously, SSs in GW170817 should have a MDM core although it is
unnecessary for PSR J0740+6620 and PSR J0030+0451 to contain a MDM core. Our
study leads to the result that for the standard MIT bag model, the
observations of compact stars mentioned above confirm the existence of a dark-
matter core inside SSs.

###### keywords:

Strange quark matter, Mirror dark matter, Neutron star

††journal: Physics Letters B††journal: Journal of LaTeX Templates

##  1 Introduction

As hypothesized by Itoh [1], Bodmer [2], Witten [3], and Terazawa [4], strange
quark matter (SQM) consisting of up ($u$), down ($d$) and strange ($s$) quarks
and electrons may be the true ground state of baryonic matter. According to
this hypothesis, compact stars made entirely of SQM, referred to as SSs, ought
to exist in the universe [5, 6, 7, 8, 9, 10].

Yang et al. [11, 12] found that both for the standard MIT bag model and for
the density dependent quark mass model, the existence of SSs is ruled out by
the mass of PSR J0740+6620 ($2.14_{-0.09}^{+0.10}\,M_{\odot}$ for a 68.3%
credibility interval; $2.14_{-0.18}^{+0.20}\,M_{\odot}$ for a 95.4%
credibility interval) [13] and the dimensionless tidal deformability of a
$1.4\,M_{\odot}$ star of GW170817 ($\Lambda(1.4)=190_{-120}^{+390}$) [14, 15].
However, Yang et al. [11, 12] demonstrated that if non-Newtonian gravity
effects are considered, SSs can exist for certain ranges of the values of the
non-Newtonian gravity parameter. In this Letter, instead of employing the non-
Newtonian gravity effects, we propose an alternative explanation to the above
mentioned astrophysical observations which supposes that SSs have a mirror-
dark-matter (MDM) core.

Compact stars might contain a dark-matter core made of self-interacting dark
matter [16, 17, 18]. Neutron stars (NSs) and SSs with a dark-matter core have
been studied extensively [e.g., 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35]. Especially, Neutron stars with a MDM core have been
studied [36, 37]. Ciarcelluti and Sandin [37] found that the discrepancy
between the mass and radius data of EXO 0748-676 [38] and the group 4U
1608-52, 4U 1820-30 and EXO 1745-248 can be interpreted as the signature of a
dark-matter core inside them. One key point of their work is that a MDM core
inside NSs leads to an apparent softening of the EOS, and the relative amount
of MDM in NSs strongly effect the mass-radius relationship of the star.
Another key point is that the equilibrium sequence of NSs is non-unique and
history dependent, because the relative amount of MDM trapped in each NS could
be different, which depends on the individual history, starting from the
formation of the progenitor star and continuing through the evolutionary
phases until present age.

We will study the structure and the tidal deformability of SSs with a MDM core
and explain the observations of PSR J0740+6620, GW170817 and the mass and
radius of PSRJ0030+0451 observed by NICER [39, 40] simultaneously. This Letter
is organized as follows: In Sec. 2, we briefly review the EOS of SQM and MDM.
In Sec. 3, we present the theoretical framework the structure and the tidal
deformability of SSs with a MDM core. In Sec. 4, numerical results and
discussions are presented. Finally, the summary and conclusions are given in
Sec. 5.

##  2 EOS of SQM and MDM

Following Yang et al. [11], we briefly review the phenomenological model for
the EOS employed in this paper, namely, the standard bag model [5, 6, 7, 10].
In that model, $u$ and $d$ quarks are treated as massless particles but $s$
quarks have a finite mass, $m_{s}$. First-order perturbative corrections in
the strong interaction coupling constant $\alpha_{S}$ are taken into account.
The thermodynamic potential for the $u$, $d$ and $s$ quarks, and for the
electrons ($\Omega_{i}$ with $i=u,d,s,e$) are the same as Refs.[6, 11] with
the renomormalization constant $\sigma=300$ MeV, and the current mass of $s$
quark is taken as $m_{s}=93$ MeV [41]. The number density of each species is
given by

$n_{i}=-\frac{\partial\Omega_{i}}{\partial\mu_{i}},$ (1)

where $\mu_{i}$ ($i=u,d,s,e$) are the chemical potentials. For SQM, chemical
equilibrium is maintained by the weak-interaction, which leads for the
chemical potentials to the following conditions,

$\displaystyle\mu_{d}$ $\displaystyle=$ $\displaystyle\mu_{s},$ (2)
$\displaystyle\mu_{s}$ $\displaystyle=$ $\displaystyle\mu_{u}+\mu_{e}.$ (3)

The electric charge neutrality condition is given by

$\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{e}=0.$ (4)

The energy density is given by

$\epsilon_{Q}=\sum_{i=u,d,s,e}(\Omega_{i}+\mu_{i}n_{i})+B,$ (5)

and the corresponding pressure is obtained from

$p_{Q}=-\sum_{i=u,d,s,e}\Omega_{i}-B,$ (6)

where $B$ denotes the bag constant.

MDM is a stable and self-interacting DM candidate that emerges from the parity
symmetric extension of the Standard Model of particles [42]. The idea of the
possible existence of MDM can be traced back to 1956, when Lee and Yang
proposed that the weak interactions is not parity symmetric [43], they also
pointed that even if the interactions of the known particle were to violate
parity, the symmetry could be restored if a set of mirror particles existed.
For details about MDM, see Refs. [44, 45, 46, 47, 48, 49, 50, 51, 52].

In the minimal parity-symmetric extension of the standard model [42, 53], the
group structure is $G\otimes G$, where G is the gauge group of the standard
model. In this model the two sectors are described by the same lagrangians,
but where ordinary particles have left-handed interactions, mirror particles
have right-handed interactions. Thus, the microphysics of MDM is the same as
that of ordinary matter, and we will use the same EOS for SQM and MDM.

Following [36, 37], we neglect the direct interaction between MDM and the
ordinary matter, although these two sectors interact through the gravitational
interaction. Except for gravity, MDM could interact with ordinary matter via
so-called kinetic mixing of gauge bosons, or via unknown fields that carry
both ordinary and mirror charges. However, if such interactions exist, they
must be weak and can be well ignored in our study [36].

##  3 The structure and tidal deformability of SSs with a MDM core

In the following, we use geometrized units $G=c=1$, and use the subscript $Q$
for SQM and $D$ for MDM.

To study the properties of SSs with a MDM core, we employ a two-fluid
formalism where SQM and MDM sectors do not interact directly. However, these
two sectors interact through the gravitational interaction in this formalism.

In the two-fluid formalism, the Tolman-Oppenheimer-Volkoff (TOV) equations are
[e.g., 32, 33, 36, 37]

$\frac{dm(r)}{dr}=4\pi\epsilon(r)r^{2},$ (7)

$\frac{dp_{Q}(r)}{dr}=-\frac{[m(r)+4\pi
r^{3}p(r)][\epsilon_{Q}(r)+p_{Q}(r)]}{r[r-2m(r)]},$ (8)

$\frac{dp_{D}(r)}{dr}=-\frac{[m(r)+4\pi
r^{3}p(r)][\epsilon_{D}(r)+p_{D}(r)]}{r[r-2m(r)]},$ (9)

where,

$\displaystyle\epsilon(r)$ $\displaystyle=$
$\displaystyle\epsilon_{Q}(r)+\epsilon_{D}(r),$ (10) $\displaystyle p(r)$
$\displaystyle=$ $\displaystyle p_{Q}(r)+p_{D}(r),$ (11)

The dimensionless tidal deformability is defined as
$\Lambda\equiv\lambda/M^{5}$, where $\lambda$ denotes the tidal deformability
parameter, which can be expressed in terms of the dimensionless tidal Love
number $k_{2}$ as $\lambda=\frac{2}{3}k_{2}R^{5}$ [54, 55, 56, 57]. Thus, one
has

$\Lambda=\frac{2}{3}k_{2}\beta^{-5},$ (12)

where $\beta$ is compactness of the star, and it is defined as $\beta\equiv
M/R$.

In the two-fluid formalism, the tidal Love number $k_{2}$ can be calculated
using the expression [58]

$k_{2}=\frac{8}{5}\frac{\beta^{5}z}{F},$ (13)

with

$\displaystyle z\equiv(1-2\beta)^{2}[2-y_{R}+2\beta(y_{R}-1)]$ (14)

and

$\displaystyle F$ $\displaystyle\equiv$ $\displaystyle
6\beta(2-y_{R})+6\beta^{2}(5y_{R}-8)+4\beta^{3}(13-11y_{R})$ (15)
$\displaystyle+4\beta^{4}(3y_{R}-2)+8\beta^{5}(1+y_{R})+3z\textrm{ln}(1-2\beta).~{}~{}~{}~{}$

In Eqs. (14) and (15), $y_{R}\equiv y(R)-4\pi R^{3}\epsilon_{s}/M$, where
$y(R)$ is the value of $y(r)$ at the surface of the star, and the second term
of right hand side exists because there is a nonzero energy density
$\epsilon_{s}$ just inside the surface of SSs [59]. The quantity $y(r)$
satisfies the differential equation

$\frac{dy(r)}{dr}=-\frac{y(r)^{2}}{r}-\frac{y(r)-6}{r-2m(r)}-rQ(r),$ (16)

with [e.g., 32, 33]

$\displaystyle Q(r)$ $\displaystyle\equiv$ $\displaystyle\frac{4\pi
r}{r-2m(r)}\bigg{[}[5-y(r)]\epsilon(r)+[9+y(r)]p(r)$ (17)
$\displaystyle+\frac{\epsilon_{Q}(r)+p_{Q}(r)}{\partial
p_{Q}(r)/\partial\epsilon_{Q}(r)}+\frac{\epsilon_{D}(r)+p_{D}(r)}{\partial
p_{D}(r)/\partial\epsilon_{D}(r)}\bigg{]}$
$\displaystyle-4\bigg{[}\frac{m(r)+4\pi r^{3}p(r)}{r[r-2m(r)]}\bigg{]}^{2},$

For a given EOS, Eq. (16) can be calculated together with the TOV equations
with the boundary conditions $y(0)=2$, $p(R)=0$, $m(0)=0$ for a given pressure
at the center of the star $p(0)$. Note that there is another energy density
jump $\epsilon_{sD}$ at the surface of the MDM core. Therefore, a correction
of $-4\pi R_{D}^{3}\epsilon_{sD}/M(R_{D})$ is added to $y(R_{D})$, where
$R_{D}$ is the radius of the MDM core.

##  4 results and discussions

We investigate the allowed parameter space of the standard MIT bag model
according to the following five constraints [e.g., 11, 12, 60, 61, 62, 63]:

First, the existence of SSs is based on the idea that the presence of strange
quarks lowers the energy per baryon of a mixture of $u$, $d$ and $s$ quarks in
beta equilibrium below the energy of the most stable atomic nucleus, 56Fe
($E/A\sim 930$ MeV) [3]. This constraint results in the 3-flavor lines shown
in Fig. 1.

The second constraint is given by assuming that non-strange quark matter
(i.e., two-flavor quark matter made of only $u$ and $d$ quarks) in bulk has an
energy per baryon higher than the one of 56Fe, plus a 4 MeV correction coming
from surface effects [5, 9, 63]. By imposing $E/A\geq 934$ MeV on non-strange
quark matter, one ensures that atomic nuclei do not dissolve into their
constituent quarks. This leads to the 2-flavor lines in Fig. 1. The areas
between the 3-flavor lines and the 2-flavor lines in Fig. 1 show the allowed
$B^{1/4}$–$\alpha_{S}$ parameter regions where the first and the second
constraints described just above are fulfilled.

The above two constraints are from nuclear structure. As we discuss the
constraints from astrophysics in the following, these two constraints must be
fulfilled, which means that we are only interested in the region between the
3-flavor line and the 2-flavor line.

The third constraint is that the maximum mass of SSs must be greater than the
mass of PSR J0740+6620, $M_{\rm max}\geq 2.14\,M_{\odot}$. By employing this
constraint, the allowed parameter space is limited to the region below the
green line in Fig. 1.

The fourth constraint is form the observational data of NICER. The NICER
observations of the isolated pulsar PSR J0030+0451 produced two independent
measurements of the pulsar’s mass and equatorial radius:
$M=1.34_{-0.16}^{+0.15}\,M_{\odot}$ and $R_{\rm eq}=12.71_{-1.19}^{+1.14}$ km
[39], and $M=1.44_{-0.14}^{+0.15}\,M_{\odot}$ and $R_{\rm
eq}=13.02_{-1.06}^{+1.24}$ km [40]. In Fig. 1, these data on the $M$–$R$ plane
is translated into the $B^{1/4}$–$\alpha_{S}$ space, which lead to the region
between the blue solid and blue dashed lines.

The cyan-shadowed area Fig. 1 marks the parameter space which satisfy all the
four constraints mentioned above simultaneously.

The last constraint follows from the tidal deformability observation of
GW170817, $\Lambda(1.4)=190_{-120}^{+390}$, where $\Lambda(1.4)$ is the
dimensionless tidal deformability of a $1.4\,M_{\odot}$ star. The parameter
space satisfies this constraint corresponds to the magenta-shadowed area in
Fig. 1.

Here we want to stress that Fig. 1 is plotted for SSs without a MDM core. As
can be seen from Fig. 1, the cyan-shadowed area does not overlap with the
magenta-shadowed area, which means that SSs without a MDM core can not agree
with the observations of PSR J0740+6620, PSR J0030+0451 and GW170817
simultaneously.

Fig. 2 shows the mass-radius relation of SSs for $m_{s}=93$ MeV and
$\alpha_{S}=0.7$ with various values of $f_{D}$. The mass fraction of MDM
($f_{D}$) is defined by $f_{D}=M_{D}/M$, where $M$ is the total mass of the
star and $M_{D}$ is the mass of the MDM core. As can be seen in Fig. 2, both
the maximum mass and the radius of the $1.4\,M_{\odot}$ star decrease with the
increasing of $f_{D}$.

Fig. 3 shows the relation between the dimensionless tidal deformability of a
$1.4\,M_{\odot}$ star ($\Lambda(1.4)$) and the mass fraction of MDM ($f_{D}$)
for $m_{s}=93$ MeV and $\alpha_{S}=0.7$. One can easily find that
$\Lambda(1.4)$ decrease with the increasing of $f_{D}$, and it is bigger for
smaller value of $B^{1/4}$.

Fig. 4 shows the constraints to the parameters of the EOS of SQM for
$m_{s}=93$ MeV for SSs with various values of $f_{D}$. Note that the cyan-
shadowed areas are the same as these in Fig. 1, which are for the case of SSs
without a MDM core (in other words, for $f_{D}=0$%) and satisfy both $M_{\rm
max}\geq 2.14\,M_{\odot}$ and the observational data of NICER. We can see that
the parameter space region which satisfy $\Lambda(1.4)<580$ (the region above
the $\Lambda(1.4)=580$ line) shifts downward as $f_{D}$ increases and it
begins to overlap with the cyan-shadowed area for the value of $f_{D}=2.5$%
for the analysis of NICER data by Riley et al. [39] (Fig.4a), and $f_{D}=3.1$%
for Miller et al. [40] (Fig.4b). Thus, assuming PSR J0740+6620 and PSR
J0030+0451 are SSs without a MDM core, there exists allowed parameter space
for which SSs agree with the observations of PSR J0740+6620, PSR J0030+0451
and GW170817 simultaneously in the case that SSs in GW170817 have a MDM core
with $f_{D}>2.5$% (for the case of Riley et al., and $f_{D}>3.1$% for the case
of Miller et al.).

##  5 Summary and conclusions

In this Letter, we study the structure and the tidal deformability of SSs with
a MDM core and explain the observations of PSR J0740+6620, PSRJ0030+0451 and
GW170817 simultaneously. As realized by [36, 37], The mass fraction of MDM
($f_{D}$) of each SS could be different, which depends on the individual
history. We show that all the above observations could be explained
simultaneously if one assumes that PSR J0740+6620 and PSR J0030+0451 are SSs
without a MDM core, and SSs in GW170817 have a MDM core with $f_{D}>2.5$% (for
the case of Riley et al., and $f_{D}>3.1$% for the case of Miller et al.). In
fact, it is easy to deduce that in order to fulfill all these observations, it
is not necessary to assume that PSR J0740+6620 and PSR J0030+0451 are SSs
without a MDM core. PSR J0740+6620 and PSR J0030+0451 could be SSs with a MDM
core, but SSs in GW170817 should have a larger MDM core than them in this
case. As a conclusion, to explain of all the observations, SSs in GW170817
should have a MDM core.

Finally, as pointed out by Ciarcelluti and Sandin [37], although our results
are calculated for MDM, they are qualitatively valid for other kinds of dark
matter that could form stable cores inside SSs. Therefore, our study leads to
the result that for the standard MIT bag model, the observations of PSR
J0740+6620, PSR J0030+0451 and GW170817 confirm the existence of a dark-matter
core inside SSs.

## Acknowledgments

This work is supported by the Scientific Research Program of the National
Natural Science Foundation of China (NSFC, grant Nos. 12033001, 11773011, and
11447012), and National SKA Program of China No. 2020SKA0120300.

## References

## References

  * [1] N. Itoh, Hydrostatic equilibrium of hypothetical quark stars, Prog. Theor. Phys. 44 (1970) 291–292.  doi:10.1143/PTP.44.291.
  * [2] A. R. Bodmer, Collapsed nuclei, Phys. Rev. D 4 (1971) 1601–1606.  doi:10.1103/PhysRevD.4.1601.
  * [3] E. Witten, Cosmic separation of phases, Phys. Rev. D 30 (1984) 272–285.  doi:10.1103/PhysRevD.30.272.
  * [4] H. Terazawa, Super-hypernuclei in the quark-shell model, J. Phys. Soc. Japan 58 (1989) 3555.  doi:10.1143/JPSJ.58.3555.
  * [5] E. Farhi, R. L. Jaffe, Strange matter, Phys. Rev. D 30 (1984) 2379–2390.  doi:10.1103/PhysRevD.30.2379.
  * [6] C. Alcock, E. Farhi, A. Olinto, Strange stars, Astrophys. J. 310 (1986) 261.  doi:10.1086/164679.
  * [7] P. Haensel, J. L. Zdunik, R. Schaefer, Strange quark stars, Astron. Astrophys. 160 (1986) 121–128.
  * [8] C. Alcock, A. Olinto, Exotic phases of hadronic matter and their astrophysical application, Ann. Rev. Nucl. Part. Sci. 38 (1988) 161–184.  doi:10.1146/annurev.ns.38.120188.001113.
  * [9] J. Madsen, Physics and astrophysics of strange quark matter, Lect.Notes Phys. 516 (1999) 162–203.  doi:10.1007/BFb0107314.
  * [10] F. Weber, Strange quark matter and compact stars, Prog. Part. Nucl. Phys. 54 (2005) 193–288.  doi:10.1016/j.ppnp.2004.07.001.
  * [11] S.-H. Yang, C.-M. Pi, X.-P. Zheng, F. Weber, Non-newtonian gravity in strange quark stars and constraints from the observations of PSR J0740-6620 and GW170817, Astrophys. J. 902 (2020) 32.  doi:10.3847/1538-4357/abb365.
  * [12] S.-H. Yang, C.-M. Pi, X.-P. Zheng, F. Weber, Constraints from compact star observations on non-Newtonian gravity in strange stars based on a density dependent quark mass model, Phys. Rev. D 103 (2021) 043012.  doi:10.1103/PhysRevD.103.043012.
  * [13] H. Cromartie, E. Fonseca, S. M. Ransom, P. B. Demorest, Z. Arzoumanian, et al., Relativistic shapiro delay measurements of an extremely massive millisecond pulsar, Nat. Astron. 4 (2020) 72–76.  doi:10.1038/s41550-019-0880-2.
  * [14] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, et al., GW170817: observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119 (2017) 161101.  doi:10.1103/PhysRevLett.119.161101.
  * [15] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, et al., GW170817: measurements of neutron star radii and equation of state, Phys. Rev. Lett. 121 (2018) 161101.  doi:10.1103/physrevlett.121.161101.
  * [16] D. N. Spergel, P. J. Steinhardt, Observational evidence for self-interacting cold dark matter, Phys. Rev. Lett. 84 (2000) 3760–3763.  doi:10.1103/PhysRevLett.84.3760.
  * [17] S. Tulin, H.-B. Yu, Dark matter self-interactions and small scale structure, Phys. Rep. 730 (2018) 1–57.  doi:10.1016/j.physrep.2017.11.004.
  * [18] G. Bertone, T. M. P. Tait, A new era in the search for dark matter, Nature 562 (2018) 51–56.  doi:10.1038/s41586-018-0542-z.
  * [19] S.-C. Leung, M.-C. Chu, L.-M. Lin, Dark-matter admixed neutron stars, Phys. Rev. D 84 (2011) 107301.  doi:10.1103/PhysRevD.84.107301.
  * [20] X. Y. Li, F. Y. Wang, K. S. Cheng, Gravitational effects of condensate dark matter on compact stellar objects, J. Cosmol. Astropart. Phys. 1210 (2012) 31\.  doi:10.1088/1475-7516/2012/10/031.
  * [21] A. Li, F. Huang, R.-X. Xu, Too massive neutron stars: The role of dark matter?, Astropart. Phys. 37 (2012) 70–74.  doi:10.1016/j.astropartphys.2012.07.006.
  * [22] Q.-F. Xiang, W.-Z. Jiang, D.-R. Zhang, R.-Y. Yang, Effects of fermionic dark matter on properties of neutron stars, Phys. Rev. C 89 (2014) 025803.  doi:10.1103/PhysRevC.89.025803.
  * [23] S. Mukhopadhyay, D. Atta, K. Imam, D. N. Basu, C. Samanta, Compact bifluid hybrid stars: hadronic matter mixed with self-interacting fermionic asymmetric dark matter, Eur. Phys. J. C 77 (2017) 440.  doi:10.1140/epjc/s10052-017-5006-3.
  * [24] J. Ellis, G. Hutsi, K. Kannike, L. Marzola, M. Raidal, V. Vaskonen, Dark matter effects on neutron star properties, Phys. Rev. D 97 (2018) 123007.  doi:10.1103/PhysRevD.97.123007.
  * [25] J. Ellis, A. Hektor, G. Hutsi, K. Kannike, L. Marzola, M. Raidal, V. Vaskonen, Search for dark matter effects on gravitational signals from neutron star mergers, Phys. Lett. B 781 (2018) 607–610.  doi:10.1016/j.physletb.2018.04.048.
  * [26] Z. Rezaei, Neutron stars with spin polarized self-interacting dark matter, Astropart. Phys. 101 (2018) 1–7.  doi:10.1016/j.astropartphys.2018.03.002.
  * [27] X. D. Wang, B. Qi, G. L. Yang, N. B. Zhang, S. Y. Wang, Possible maximum mass of dark matter existing in compact stars based on the self-interacting fermionic model, Int. J. Mod. Phys. D 28 (2019) 1950148.  doi:10.1142/S0218271819501487.
  * [28] M. Deliyergiyev, A. D. Popolo, L. Tolos, M. L. Delliou, X. Lee, F. Burgio, Dark compact objects: An extensive overview, Phys. Rev. D 99 (2019) 063015.  doi:10.1103/PhysRevD.99.063015.
  * [29] M. Bezares, D. Vigano, C. Palenzuela, Gravitational wave signatures of dark matter cores in binary neutron star mergers by using numerical simulations, Phys. Rev. D 100 (2019) 044049.  doi:10.1103/PhysRevD.100.044049.
  * [30] O. Ivanytskyi, V. Sagun, I. Lopes, Neutron stars: New constraints on asymmetric dark matter, Phys. Rev. D 102 (2020) 063028.  doi:10.1103/PhysRevD.102.063028.
  * [31] A. Bauswein, G. Guo, J.-H. Lien, Y.-H. Lin, M.-R. Wu, Compact dark objects in neutron star mergers.arXiv:2012.11908.
  * [32] A. Das, T. Malik, A. C. Nayak, Dark matter admixed neutron star properties in the light of gravitational wave observations: a two fluid approach.arXiv:2011.01318.
  * [33] R. Ciancarella, F. Pannarale, A. Addazi, A. Marciano, Constraining mirror dark matter inside neutron stars.arXiv:2010.12904.
  * [34] P. Mukhopadhyay, J. Schaffner-Bielich, Quark stars admixed with dark matter, Phys. Rev. D 93 (2016) 083009.  doi:10.1103/PhysRevD.93.083009.
  * [35] G. Panotopoulos, I. Lopes, Gravitational effects of condensed dark matter on strange stars, Phys. Rev. D 96 (2017) 023002.  doi:10.1103/PhysRevD.96.023002.
  * [36] F. Sandin, P. Ciarcelluti, Effects of mirror dark matter on neutron stars, Astropart. Phys. 32 (2009) 278–284.  doi:10.1016/j.astropartphys.2009.09.005.
  * [37] P. Ciarcelluti, F. Sandin, Have neutron stars a dark matter core?, Phys. Lett. B 695 (2011) 19–21.  doi:10.1016/j.physletb.2010.11.021.
  * [38] F. Ozel, Soft equations of state for neutron-star matter ruled out by EXO 0748 - 676, Nature 441 (2006) 1115.  doi:10.1038/nature04858.
  * [39] T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, et al., A NICER view of PSR J0030+0451: millisecond pulsar parameter estimation, Astrophys. J. Lett. 887 (2019) L21.  doi:10.3847/2041-8213/ab481c.
  * [40] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, et al., PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter, Astrophys. J. Lett. 887 (2019) L24.  doi:10.3847/2041-8213/ab50c5.
  * [41] P. A. Zyla, R. M. Barnett, J. Beringer, O. Dahl, D. A. Dwyer, et al., Review of particle physics, Prog. Theor. Exp. Phys. 2020 (2020) 083C01.  doi:10.1093/ptep/ptaa104.
  * [42] R. Foot, H. Lew, R. R. Volkas, A model with fundamental improper spacetime symmetries, Phys. Lett. B 272 (1991) 67–70.  doi:10.1016/0370-2693(91)91013-L.
  * [43] T. D. Lee, C. N. Yang, Question of parity conservation in weak interactions, Phys. Rev. 104 (1956) 254–258.  doi:10.1103/PhysRev.104.254.
  * [44] S. I. Blinnikov, M. Y. Khlopov, On the possible signatures of mirror particles, Sov. J. Nucl. Phys. 36 (1982) 472–474.
  * [45] S. I. Blinnikov, M. Y. Khlopov, Possible astronomical effects of mirror particles, Sov. Astron. 27 (1983) 371–375.
  * [46] M. Y. Khlopov, G. M. Beskin, N. G. Bochkarev, L. A. Pustilnik, S. A. Pustilnik, Observational physics of the mirror world,, Sov. Astron. 35 (1991) 21–30.
  * [47] R. Foot, Mirror matter-type dark matter, Int. J. Mod. Phys. D 13 (2004) 2161–2192.  doi:10.1142/S0218271804006449.
  * [48] Z. Berezhiani, Mirror world and its cosmological consequences, Int. J. Mod. Phys. A 19 (2004) 3775–3806.  doi:10.1142/S0217751X04020075.
  * [49] Z. Berezhiani, Through the looking-glass: Alice’s adventures in mirror world, in: From Fields to Strings: Circumnavigating Theoretical Physics,Vol. 3, eds. M. Shifman et al. (World Scientific) (2005) 2147–2195doi:10.1142/9789812775344_0055.
  * [50] L. B. Okun, Mirror particles and mirror matter: 50 years of speculation and searching, Phys. Usp. 50 (2007) 380–389.  doi:10.1070/pu2007v050n04abeh006227.
  * [51] P. Ciarcelluti, Cosmology with mirror dark matter, Int. J. Mod. Phys. D 19 (2010) 2151–2230.  doi:10.1142/S0218271810018438.
  * [52] R. Foot, Mirror dark matter: Cosmology, galaxy structure and direct detection, Int. J. Mod. Phys. A 29 (2014) 1430013.  doi:10.1142/S0217751X14300130.
  * [53] M. Pavsic, External inversion, internal inversion, and reflection invariance, Int. J. Theor. Phys. 9 (1974) 229–244.  doi:10.1007/BF01810695.
  * [54] T. Hinderer, Tidal love numbers of neutron stars, Astrophys. J. 677 (2008) 1216–1220.  doi:10.1086/533487.
  * [55] E. E. Flanagan, T. Hinderer, Constraining neutron-star tidal love numbers with gravitational-wave detectors, Phys. Rev. D 77 (2008) 021502.  doi:10.1103/PhysRevD.77.021502.
  * [56] T. Damour, A. Nagar, Relativistic tidal properties of neutron stars, Phys. Rev. D 80 (2009) 084035.  doi:10.1103/PhysRevD.80.084035.
  * [57] T. Hinderer, B. D. Lackey, R. N. Lang, J. S. Read, Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral, Phys. Rev. D 81 (2010) 123016.  doi:10.1103/PhysRevD.81.123016.
  * [58] J. M. Lattimer, M. Prakash, The equation of state of hot, dense matter and neutron stars, Phys. Rep. 621 (2016) 127–164.  doi:10.1016/j.physrep.2015.12.005.
  * [59] S. Postnikov, M. Prakash, J. M. Lattimer, Tidal love numbers of neutron and self-bound quark stars, Phys. Rev. D 82 (2010) 024016.  doi:10.1103/PhysRevD.82.024016.
  * [60] C. Schaab, B. Hermann, F. Weber, M. K. Weigel, Are strange stars distinguishable from neutron stars by their cooling behaviour?, J. Phys. G: Nucl. Part. Phys. 23 (1997) 2029–2037.  doi:10.1088/0954-3899/23/12/027.
  * [61] S. Weissenborn, I. Sagert, G. Pagliara, M. Hempel, J. Schaffner-Bielich, Quark matter in massive compact stars, Astrophys. J. Lett. 740 (2011) L14.  doi:10.1088/2041-8205/740/1/L14.
  * [62] C.-M. Pi, S.-H. Yang, X.-P. Zheng, R-mode instability of strange stars and observations of neutron stars in LMXBs, Res. Astron. Astrophys. 15 (2015) 871\.  doi:10.1088/1674-4527/15/6/009.
  * [63] E.-P. Zhou, X. Zhou, A. Li, Constraints on interquark interaction parameters with GW170817 in a binary strange star scenario, Phys. Rev. D 97 (2018) 083015\.  doi:10.1103/PhysRevD.97.083015.

Figure 1: (Color online) The constraints to the parameters of the EOS of SQM,
namely, $B^{1/4}$ and $\alpha_{S}$ for $m_{s}=93$ MeV for the case of SSs
without a MDM core. The magenta-shadowed area corresponds to the allowed
parameter space according to the dimensionless tidal deformability of a
$1.4\,M_{\odot}$ star of GW170817 ($\Lambda(1.4)=190_{-120}^{+390}$). The
cyan-shadowed area marks the parameter space which satisfy both $M_{\rm
max}\geq 2.14\,M_{\odot}$ and the observational data of NICER (Fig.1a for the
analysis by Riley et al. [39] and Fig.1b for Miller et al. [40]). The blue
regions merely satisfy $M_{\rm max}\geq 2.14\,M_{\odot}$, and the green
regions merely satisfy the observational data of NICER. The red dots in Fig.1b
mark the two parameter sets (125.1,0.7) and (137.3, 0.7).

Figure 2: (Color online) The mass-radius relation of SSs for $m_{s}=93$ MeV
and $\alpha_{S}=0.7$ with various values of the mass fraction of MDM
($f_{D}$). The black lines are for $B^{1/4}=125.1$ MeV, and the red lines are
for $B^{1/4}=137.3$ MeV. The solid, dashed, dotted, dash-dotted lines are for
$f_{D}=0$%, 10%, 20%, and 30%, respectively. The blue and green regions show
the mass and radius estimates of PSR J0030+0451 derived from NICER data by
Riley et al. [39] ($R=12.71_{-1.19}^{+1.14}$ km,
$M=1.34_{-0.16}^{+0.15}\,M_{\odot}$) and Miller et al. [40]
($R=13.02_{-1.06}^{+1.24}$ km, $M=1.44_{-0.14}^{+0.15}\,M_{\odot}$).

Figure 3: (Color online) Relation between the dimensionless tidal
deformability of a $1.4\,M_{\odot}$ star ($\Lambda(1.4)$) and the mass
fraction of MDM ($f_{D}$) for $m_{s}=93$ MeV and $\alpha_{S}=0.7$. The shaded
region corresponds to $70<\Lambda(1.4)<580$, which is given by the observation
of GW170817.

Figure 4: (Color online) The constraints to the parameters of the EOS of SQM,
namely, $B^{1/4}$ and $\alpha_{S}$ for $m_{s}=93$ MeV for SSs with various
values of the mass fraction of MDM ($f_{D}$). Similar to Fig.1, the cyan-
shadowed area marks the parameter space which satisfy both $M_{\rm max}\geq
2.14\,M_{\odot}$ and the observational data of NICER (Fig.4a for the analysis
by Riley et al. [39] and Fig.4b for Miller et al. [40]) for SSs without a MDM
core (in other words, for $f_{D}=0$%). The red lines are for
$\Lambda(1.4)=580$, with solid, dashed, dotted, dash-dotted lines for
$f_{D}=0$%, 5%, 10%, and 20%, respectively. The blue line is for $f_{D}=2.5$%
in Fig.4a, and is for $f_{D}=3.1$% in Fig.4b.


# Light Scalar Meson $\sigma(600)$ in QCD Sum Rule with Continuum

Hua-Xing Chen1,2<EMAIL_ADDRESS>Atsushi Hosaka2
<EMAIL_ADDRESS>Hiroshi Toki2<EMAIL_ADDRESS>Shi-Lin Zhu1
<EMAIL_ADDRESS>1Department of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China
2Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka
567–0047, Japan

###### Abstract

The light scalar meson $\sigma(600)$ is known to appear at low excitation
energy with very large width on top of continuum states. We investigate it in
the QCD sum rule as an example of resonance structures appearing above the
corresponding thresholds. We use all the possible local tetraquark currents by
taking linear combinations of five independent local ones. We ought to
consider the $\pi$-$\pi$ continuum contribution in the phenomenological side
of the QCD sum rule in order to obtain a good sum rule signal. We study the
stability of the extracted mass against the Borel mass and the threshold value
and find the $\sigma(600)$ mass at 530 MeV $\pm$ 40 MeV. In addition we find
the extracted mass has an increasing tendency with the Borel mass, which is
interpreted as caused by the width of the resonance.

scalar meson, tetraquark, QCD sum rule

###### pacs:

12.39.Mk, 12.38.Lg, 12.40.Yx

##  I Introduction

The light scalar mesons, $\sigma(600)$, $\kappa(800)$, $f_{0}(980)$ and
$a_{0}(980)$, have been intensively discussed for many years Amsler:2004ps ;
Bugg:2004xu ; Klempt:2007cp . However, their nature is still not fully
understood Caprini:2005zr ; Hatsuda:1994pi ; Oller:1997ti ; Sugiyama:2007sg ;
Prelovsek:2010gm . They have the same quantum numbers $J^{PC}=0^{++}$ as the
vacuum, and hence the structure of these states is a very important subject in
order to understand non-perturbative properties of the QCD vacuum such as
spontaneous chiral symmetry breaking. They compose of the flavor $SU(3)$ nonet
with the mass below 1 GeV, and have a mass ordering which is difficult to be
explained by using a $q\bar{q}$ configuration in the conventional quark model
Amsler:2008zzb ; Aitala:2000xu ; Ablikim:2004qna ; Aston:1987ir ;
Akhmetshin:1999di . Therefore, several different pictures have been proposed,
such as tetraquark states and meson-meson bound states, etc. Here we note that
hadrons with complex structures such as tetraquarks may exist in the continuum
above the threshold energy of two hadrons with simple quark structure.

The tetraquark structure of the scalar mesons was proposed long time ago by
Jaffe with an assumption of strong diquark correlations Jaffe:1976ig ;
Jaffe:1976ih . It can naturally explain their mass ordering and decay
properties Alford:2000mm ; Maiani:2004uc ; Weinstein:1990gu . Yet the basic
assumption of diquark correlation is not fully established. In this letter, we
study $\sigma(600)$ as a tetraquark state in the QCD sum rule approach as an
example of resonances in the continuum states above the $\pi$-$\pi$ threshold.
In the QCD sum rule, we calculate matrix elements from the QCD (OPE) and
relate them to observables by using dispersion relations. Under suitable
assumptions, the QCD sum rule has proven to be a very powerful and successful
non-perturbative method in the past decades Shifman:1978bx ; Reinders:1984sr .
Recently, this method has been applied to the study of tetraquarks by many
authors Bracco:2005kt ; Narison:2005wc ; Lee:2006vk ; Chen:2007xr .

In our previous paper Chen:2007xr , we have found that the QCD sum rule
analysis with tetraquark currents implies the masses of scalar mesons in the
region of 600 – 1000 MeV with the ordering
$m_{\sigma}<m_{\kappa}<m_{f_{0},a_{0}}$, while the conventional $\bar{q}q$
current is considerably heavier (larger than 1 GeV). To get this result, first
we find there are five independent local tetraquark currents, and then we use
one of these currents or linear combinations of two currents to perform the
QCD sum rule analysis. But these interpolating currents do not describe the
full space of tetraquark currents. In order to complete our previous study, we
use more general currents by taking linear combinations of all these currents.
It describes the full space of local tetraquark currents which can couple to
$\sigma(600)$. Since $\sigma(600)$ meson is closely related to the $\pi$-$\pi$
continuum and it has a wide decay width, we also consider the contribution of
the $\pi$-$\pi$ continuum as well as the effect of the finite decay width.

This paper is organized as follows. In Sec. II, we establish five independent
local tetraquark currents, and perform a QCD sum rule analysis by using linear
combinations of five single currents. In Sec. III, we perform a numerical
analysis, and we also study the contribution of $\pi$-$\pi$ continuum. In Sec.
IV, we consider the effect of the finite decay width. Sec. V is devoted to
summary.

##  II QCD Sum Rule

The local tetraquark currents for $\sigma(600)$ have been worked out in Ref
Chen:2007xr . There are two types of currents: diquark-antidiquark currents
$(qq)(\bar{q}\bar{q})$ and meson-meson currents $(\bar{q}q)(\bar{q}q)$. These
two constructions can be proved to be equivalent, and they can both describe
the full space of local tetraquark currents Chen:2007xr . Therefore we shall
just use the first ones. Since we use their linear combinations to perform the
QCD sum rule analysis, we can not distinguish whether it is a diquark-
antidiquark state or a meson-meson bound state. However, we find that
tetraquark currents with a single term do not lead to a reliable QCD sum rule
result which means that $\sigma(600)$ probably has a complicated structure.
The five independent local currents are given by:

$\displaystyle S^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\gamma_{5}d_{b})(\bar{u}_{a}\gamma_{5}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma_{5}C\bar{d}_{a}^{T})\,,$
$\displaystyle V^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\gamma_{\mu}\gamma_{5}d_{b})(\bar{u}_{a}\gamma^{\mu}\gamma_{5}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma^{\mu}\gamma_{5}C\bar{d}_{a}^{T})\,,$
$\displaystyle T^{\sigma}_{6}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\sigma_{\mu\nu}d_{b})(\bar{u}_{a}\sigma^{\mu\nu}C\bar{d}_{b}^{T}+\bar{u}_{b}\sigma^{\mu\nu}C\bar{d}_{a}^{T})\,,$
(1) $\displaystyle A^{\sigma}_{6}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\gamma_{\mu}d_{b})(\bar{u}_{a}\gamma^{\mu}C\bar{d}_{b}^{T}+\bar{u}_{b}\gamma^{\mu}C\bar{d}_{a}^{T})\,,$
$\displaystyle P^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}Cd_{b})(\bar{u}_{a}C\bar{d}_{b}^{T}-\bar{u}_{b}C\bar{d}_{a}^{T})\,.$

The summation is taken over repeated indices ($\mu$, $\nu,\cdots$ for Dirac,
and $a,b,\cdots$ for color indices). The currents $S$, $V$, $T$, $A$ and $P$
are constructed by scalar, vector, tensor, axial-vector, pseudoscalar diquark
and antidiquark fields, respectively. The subscripts $3$ and $6$ show that the
diquarks (antidiquarks) are combined into the color representations,
$\mathbf{\bar{3}_{c}}$ and $\mathbf{6_{c}}$ ($\mathbf{3_{c}}$ and
$\mathbf{\bar{6}_{c}}$), respectively.

These five diquark-antidiquark currents $(qq)(\bar{q}\bar{q})$ are
independent. In this work we use general currents by taking linear
combinations of these five currents:

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
t_{1}e^{i\theta_{1}}S^{\sigma}_{3}+t_{2}e^{i\theta_{2}}V^{\sigma}_{3}+t_{3}e^{i\theta_{3}}T^{\sigma}_{6}+t_{4}e^{i\theta_{4}}A^{\sigma}_{6}+t_{5}e^{i\theta_{5}}P^{\sigma}_{3}\,,$
(2)

where $t_{i}$ and $\theta_{i}$ are ten mixing parameters, whose linear
combination describes the full space of local currents which can couple to
$\sigma(600)$. We can not determine them in advance and therefore we choose
them randomly for the study of the QCD sum rule.

By using the current in Eq. (2), we calculate the OPE up to dimension eight.
To simplify our calculation, we neglect several condensates, such as $\langle
g^{3}G^{3}\rangle$, etc., and we do not consider the $\alpha_{s}$ correction,
such as $g^{2}\langle\bar{q}q\rangle^{2}$, etc. The obtained OPE are shown in
the following. We find that most of the crossing terms are not important such
as $\rho_{13}$, and even more some of them disappear: $\rho_{15}=0$, etc. For
the most cases, we find that the OPE terms of Dim=6 and Dim=8 give major
contributions in the OPE series in our region of interest. This is because the
condensates $\langle\bar{q}q\rangle^{2}$ (D=6) and
$\langle\bar{q}q\rangle\langle g\bar{q}\sigma Gq\rangle$ (D=8) are much larger
than others.

Since the OPE series should be convergent to give a reliable QCD sum rule, we
also calculate the OPE of Dim=10 and Dim=12. However, we find that these terms
are not important. Using the parameter set (2) and the the values of the
condensates of the next section as an example, we show the convergence of the
two-point correlation function
$\Pi(M_{B},s_{0})\equiv\int_{0}^{s_{0}}\rho(s)e^{-s/M_{B}^{2}}ds$ in Fig. 1 as
functions of $M_{B}^{2}$. The threshold value is taken to be $s_{0}=1$ GeV2,
and we show its behavior up to certain dimensions. We find that the OPE up to
Dim=0 and Dim=2 are very small; the OPE of Dim=4 gives a minor contribution;
the OPE of Dim=6 and Dim=8 are both important; the OPE of Dim=10 and Dim=12
are both small, and so we shall neglect them in the following analysis.

Figure 1: The convergence of the two-point correlation function
$\Pi(M_{B},s_{0})$. The threshold value is taken to be $s_{0}=1$ GeV2, and we
show its behavior up to certain dimensions, as functions of $M_{B}^{2}$. The
solid line is for $\Pi(M_{B},s_{0})$ up to Dim=8. The short-dashed line around
it is for $\Pi(M_{B},s_{0})$ up to Dim=10, and the long-dashed line around it
is for $\Pi(M_{B},s_{0})$ up to Dim=12.

$\displaystyle\rho(s)$ $\displaystyle=$ $\displaystyle
t_{1}^{2}\rho_{11}(s)+t_{2}^{2}\rho_{22}(s)+t_{3}^{2}\rho_{33}(s)+t_{4}^{2}\rho_{44}(s)+t_{5}^{2}\rho_{55}(s)$
$\displaystyle+2t_{1}t_{2}\cos{(\theta_{1}-\theta_{2})}\rho_{12}(s)+2t_{1}t_{3}\cos{(\theta_{1}-\theta_{3})}\rho_{13}(s)+2t_{1}t_{4}\cos{(\theta_{1}-\theta_{4})}\rho_{14}(s)$
$\displaystyle+2t_{2}t_{3}\cos{(\theta_{2}-\theta_{3})}\rho_{23}(s)+2t_{2}t_{4}\cos{(\theta_{2}-\theta_{4})}\rho_{24}(s)+2t_{2}t_{5}\cos{(\theta_{2}-\theta_{5})}\rho_{25}(s)$
$\displaystyle+2t_{3}t_{4}\cos{(\theta_{3}-\theta_{4})}\rho_{34}(s)+2t_{3}t_{5}\cos{(\theta_{3}-\theta_{5})}\rho_{35}(s)\,,$

where

$\displaystyle\rho_{11}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}+(-\frac{{m_{u}}^{2}}{1536\pi^{6}}+\frac{{m_{u}}m_{d}}{1536\pi^{6}}-\frac{{m_{d}}^{2}}{1536\pi^{6}})s^{3}+(\frac{\langle
g^{2}GG\rangle}{6144\pi^{6}}-\frac{{m_{u}}\langle\bar{q}q\rangle}{192\pi^{4}}-\frac{{m_{d}}\langle\bar{q}q\rangle}{192\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}+\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}-\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{12\pi^{2}})s$
$\displaystyle-\frac{7m_{u}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}+\frac{m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{4\pi^{2}}-\frac{7m_{d}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{768\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{768\pi^{4}}+\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{12\pi^{2}}\,,$ $\displaystyle\rho_{22}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{15360\pi^{6}}+(-\frac{{m_{u}}^{2}}{384\pi^{6}}-\frac{{m_{u}}m_{d}}{768\pi^{6}}-\frac{{m_{d}}^{2}}{384\pi^{6}})s^{3}+(\frac{\langle
g^{2}GG\rangle}{3072\pi^{6}}+\frac{{m_{u}}\langle\bar{q}q\rangle}{24\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{24\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}+\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}-\frac{\langle\bar{q}q\rangle^{2}}{6\pi^{2}})s$
$\displaystyle+\frac{11m_{u}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}+\frac{2m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{11m_{d}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}-\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{6\pi^{2}}\,,$ $\displaystyle\rho_{33}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{1280\pi^{6}}+(-\frac{{m_{u}}^{2}}{32\pi^{6}}-\frac{{m_{d}}^{2}}{32\pi^{6}})s^{3}+(\frac{11\langle
g^{2}GG\rangle}{768\pi^{6}}+\frac{{m_{u}}\langle\bar{q}q\rangle}{4\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{4\pi^{4}})s^{2}$
$\displaystyle+(-\frac{11m_{u}^{2}\langle
g^{2}GG\rangle}{128\pi^{6}}-\frac{11m_{d}^{2}\langle
g^{2}GG\rangle}{128\pi^{6}})s$
$\displaystyle+\frac{5m_{u}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{20m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{5m_{d}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{11m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{96\pi^{4}}+\frac{11m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{96\pi^{4}}\,,$
$\displaystyle\rho_{44}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}+(-\frac{{m_{u}}^{2}}{192\pi^{6}}+\frac{{m_{u}}m_{d}}{384\pi^{6}}-\frac{{m_{d}}^{2}}{192\pi^{6}})s^{3}+\frac{5\langle
g^{2}GG\rangle}{3072\pi^{6}}s^{2}$ $\displaystyle+(-\frac{5m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{5m_{u}m_{d}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{5m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{16\pi^{4}}-\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{16\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{3\pi^{2}})s$
$\displaystyle-\frac{m_{u}^{2}\langle\bar{q}q\rangle^{2}}{6\pi^{2}}+\frac{8m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}-\frac{m_{d}^{2}\langle\bar{q}q\rangle^{2}}{6\pi^{2}}+\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}+\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}+\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{3\pi^{2}}\,,$ $\displaystyle\rho_{55}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}+(-\frac{{m_{u}}^{2}}{1536\pi^{6}}-\frac{{m_{u}}m_{d}}{1536\pi^{6}}-\frac{{m_{d}}^{2}}{1536\pi^{6}})s^{3}+(\frac{\langle
g^{2}GG\rangle}{6144\pi^{6}}+\frac{{m_{u}}\langle\bar{q}q\rangle}{64\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{64\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}+\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}+\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}-\frac{\langle\bar{q}q\rangle^{2}}{12\pi^{2}})s$
$\displaystyle+\frac{17m_{u}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}+\frac{7m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}+\frac{17m_{d}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}+\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{256\pi^{4}}+\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{256\pi^{4}}-\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{12\pi^{2}}\,,$ $\displaystyle\rho_{12}(s)$
$\displaystyle=$
$\displaystyle(\frac{{m_{u}}^{2}}{3072\pi^{6}}+\frac{{m_{u}}m_{d}}{1536\pi^{6}}+\frac{{m_{d}}^{2}}{3072\pi^{6}})s^{3}+(-\frac{{m_{u}}\langle\bar{q}q\rangle}{48\pi^{4}}-\frac{{m_{d}}\langle\bar{q}q\rangle}{48\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}-\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{6\pi^{2}})s-\frac{5m_{u}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{2\pi^{2}}-\frac{5m_{d}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{6\pi^{2}}\,,$ $\displaystyle\rho_{13}(s)$
$\displaystyle=$ $\displaystyle-\frac{\langle
g^{2}GG\rangle}{1024\pi^{6}}s^{2}+(\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,,$ (13)
$\displaystyle\rho_{14}(s)$ $\displaystyle=$
$\displaystyle(\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}}+\frac{3m_{u}m_{d}\langle
g^{2}GG\rangle}{2048\pi^{6}}+\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,,$ (14)
$\displaystyle\rho_{23}(s)$ $\displaystyle=$
$\displaystyle(-\frac{9m_{u}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}}-\frac{9m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{9m_{d}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}})s+\frac{3m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}+\frac{3m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}\,,$ (15)
$\displaystyle\rho_{24}(s)$ $\displaystyle=$ $\displaystyle\frac{\langle
g^{2}GG\rangle}{1024\pi^{6}}s^{2}+(-\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}})s+\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}+\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,,$ (16)
$\displaystyle\rho_{25}(s)$ $\displaystyle=$
$\displaystyle(\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}}+\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{2048\pi^{6}}+\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}\,,$ (17)
$\displaystyle\rho_{34}(s)$ $\displaystyle=$
$\displaystyle(-\frac{{m_{u}}^{2}}{256\pi^{6}}-\frac{{m_{u}}m_{d}}{128\pi^{6}}-\frac{{m_{d}}^{2}}{256\pi^{6}})s^{3}+(\frac{{m_{u}}\langle\bar{q}q\rangle}{4\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{4\pi^{4}})s^{2}$
$\displaystyle+(-\frac{15m_{u}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}}-\frac{15m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{15m_{d}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}}+\frac{3m_{u}\langle g\bar{q}\sigma
Gq\rangle}{8\pi^{4}}+\frac{3m_{d}\langle g\bar{q}\sigma
Gq\rangle}{8\pi^{4}}-\frac{2\langle\bar{q}q\rangle^{2}}{\pi^{2}})s$
$\displaystyle+\frac{5m_{u}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{6m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{5m_{d}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{5m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}+\frac{5m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}-\frac{2\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{\pi^{2}}\,,$ $\displaystyle\rho_{35}(s)$
$\displaystyle=$ $\displaystyle-\frac{\langle
g^{2}GG\rangle}{1024\pi^{6}}s^{2}+(\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,.$ (19)

##  III Numerical Analysis

To perform the numerical analysis, we use the values for all the condensates
from Refs. Yang:1993bp ; Narison:2002pw ; Gimenez:2005nt ; Jamin:2002ev ;
Ioffe:2002be ; Ovchinnikov:1988gk :

$\displaystyle\langle\bar{q}q\rangle=-(0.240\mbox{ GeV})^{3}\,,$
$\displaystyle\langle\bar{s}s\rangle=-(0.8\pm 0.1)\times(0.240\mbox{
GeV})^{3}\,,$ $\displaystyle\langle g_{s}^{2}GG\rangle=(0.48\pm 0.14)\mbox{
GeV}^{4}\,,$ $\displaystyle m_{u}=5.3\mbox{ MeV}\,,m_{d}=9.4\mbox{ MeV}\,,$
$\displaystyle m_{s}(1\mbox{ GeV})=125\pm 20\mbox{ MeV}\,,$ (20)
$\displaystyle\langle g_{s}\bar{q}\sigma
Gq\rangle=-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$ $\displaystyle
M_{0}^{2}=(0.8\pm 0.2)\mbox{ GeV}^{2}\,.$

As usual we assume the vacuum saturation for higher dimensional operators such
as $\langle 0|\bar{q}q\bar{q}q|0\rangle\sim\langle 0|\bar{q}q|0\rangle\langle
0|\bar{q}q|0\rangle$. There is a minus sign in the definition of the mixed
condensate $\langle g_{s}\bar{q}\sigma Gq\rangle$, which is different with
some other QCD sum rule calculation. This is just because the definition of
coupling constant $g_{s}$ is different Yang:1993bp ; Hwang:1994vp .

Altogether we took randomly chosen 50 sets of $t_{i}$ and $\theta_{i}$. Some
of these sets of numbers lead to negative spectral densities in the low energy
region of interest, which should be, however, positive from their definition.
This is due to several reasons. One reason is that the convergence of OPE may
not be achieved yet for those currents for the tetraquark state. Another
reason is that some currents may not couple to the physical states properly.
Except them, there are fifteen sets which lead to positive spectral densities.
We show these fifteen sets of $t_{i}$ and $\theta_{i}$ in Table 1, and label
them as (01), (02), $\cdots$, (15). They are sorted by the fourth column “Pole
Contribution” (PC):

${\rm
Pole\,\,Contribution}\equiv\frac{\int_{0}^{s_{0}}e^{-s/M_{B}^{2}}\rho(s){\rm
d}s}{\int_{0}^{\infty}e^{-s/M_{B}^{2}}\rho(s){\rm d}s}\,.$ (21)

The pole contribution (PC) is an important quantity to check the validity of
the QCD sum rule analysis. Here, $\rho(s)$ denotes the spectral function. It
depends on the ten mixed parameters as well as $M_{B}$ and $s_{0}$. We note
that $\pi$-$\pi$ continuum which we shall study later is not included in the
pole contribution. By fixing $s_{0}=1$ GeV2, we show the PC values in Table 1
for the fifteen sets. “PC(0.5)”, “PC(0.8)” and “PC(1.2)” denote pole
contribution by setting $M_{B}^{2}=0.5$ GeV2, $0.8$ GeV2 and $1.2$ GeV2,
respectively. We find that the pole contribution decreases very rapidly as the
Borel Mass increases. Since we have discussed the convergence of OPE in the
previous section, and found that the Dim=10 and Dim=12 terms are much smaller
than the Dim=6 and Dim=8 terms, and so it is only the pole contribution which
gives a upper limitation on the Borel Mass. The Borel window is wider for the
former parameter sets (1), (2), $\cdots$, and narrower for the latter ones. It
almost disappears for the set (15), whose mass prediction is also much
different from others. The Borel window should be our working region. However,
since the Borel stability is always very good when $M_{B}^{2}>$ 0.5 GeV2, we
shall keep the idea of Borel window in mind and work in the region
$0.5<M_{B}^{2}<$ 2 GeV2. On the other side, we shall care more about the
threshold value $s_{0}$.

Table 1: Values for parameters $t_{i}$, $\theta_{i}$, the mass range $M_{\sigma}$, the pole contribution (PC) and the continuum amplitude $a(t_{i},\,\theta_{i})$. The meaning of these quantities are given in the text. There are altogether fifteen sets, which are sorted by the fourth column “PC”. “PC(0.5)”, “PC(0.8)” and “PC(1.2)” denote pole contribution by setting $M_{B}^{2}=0.5$ GeV2, $0.8$ GeV2 and $1.2$ GeV2, respectively. No | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $t_{5}$ | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{4}$ | $\theta_{5}$ |  $M_{\sigma}$(MeV) | PC(0.5) | PC(0.8) | PC(1.2) | a (GeV4)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | $0.03$ | $0.03$ | $0.73$ | $0.37$ | $0.24$ | $2.7$ | $3.4$ | $4.7$ | $5.5$ | $3.6$ | $510\sim 580$ | 92% | 52% | 13% | $1.2\times 10^{-7}$
(2) | $0.03$ | $0.92$ | $0.75$ | $0.70$ | $0.03$ | $5.6$ | $0.80$ | $4.1$ | $2.9$ | $2.5$ | $510\sim 590$ | 90% | 46% | 11% | $5.5\times 10^{-7}$
(3) | $0.25$ | $0.79$ | $0.16$ | $0.95$ | $0.22$ | $1.8$ | $1.2$ | $6.1$ | $0.44$ | $1.8$ | $510\sim 600$ | 87% | 44% | 11% | $3.6\times 10^{-7}$
(4) | $0.53$ | $0.26$ | $0.93$ | $0.24$ | $0.76$ | $2.9$ | $0.40$ | $2.0$ | $2.5$ | $3.3$ | $510\sim 610$ | 85% | 41% | 10% | $1.7\times 10^{-6}$
(5) | $0.74$ | $0.54$ | $0.74$ | $0.65$ | $0.67$ | $0.15$ | $3.1$ | $1.4$ | $2.7$ | $6.1$ | $520\sim 640$ | 81% | 36% | 8% | $1.9\times 10^{-6}$
(6) | $0.98$ | $0.50$ | $0.12$ | $0.33$ | $0.03$ | $2.0$ | $4.0$ | $6.3$ | $1.3$ | $1.6$ | $510\sim 590$ | 82% | 32% | 6% | $5.8\times 10^{-8}$
(7) | $0.98$ | $0.42$ | $0.84$ | $0.82$ | $0.72$ | $0.095$ | $1.5$ | $3.7$ | $2.4$ | $3.0$ | $540\sim 700$ | 70% | 26% | 6% | $4.2\times 10^{-6}$
(8) | $0.48$ | $0.68$ | $0.58$ | $0.96$ | $0.04$ | $1.8$ | $2.5$ | $3.0$ | $4.3$ | $3.7$ | $530\sim 690$ | 70% | 25% | 6% | $1.9\times 10^{-6}$
(9) | $0.53$ | $1.0$ | $0.99$ | $0.34$ | $0.86$ | $5.6$ | $4.8$ | $5.3$ | $4.1$ | $0.076$ | $540\sim 700$ | 68% | 24% | 5% | $4.5\times 10^{-6}$
(10) | $0.75$ | $0.96$ | $0.32$ | $0.12$ | $0.11$ | $4.3$ | $2.6$ | $0.93$ | $5.1$ | $2.9$ | $560\sim 760$ | 57% | 17% | 4% | $9.5\times 10^{-7}$
(11) | $0.31$ | $0.81$ | $0.71$ | $0$ | $0.10$ | $4.2$ | $1.8$ | $2.8$ | $5.4$ | $5.1$ | $570\sim 780$ | 55% | 17% | 4% | $3.2\times 10^{-6}$
(12) | $0.47$ | $0.40$ | $0$ | $0.46$ | $0.91$ | $0.18$ | $1.9$ | $1.9$ | $0.091$ | $0.94$ | $540\sim 730$ | 58% | 16% | 3% | $2.0\times 10^{-7}$
(13) | $0.60$ | $0.26$ | $0.44$ | $0.27$ | $0.24$ | $3.3$ | $3.6$ | $0.92$ | $5.9$ | $3.7$ | $620\sim 850$ | 43% | 13% | 3% | $1.7\times 10^{-6}$
(14) | $0.74$ | $0.73$ | $0.73$ | $0.32$ | $0.28$ | $1.3$ | $1.3$ | $4.6$ | $3.3$ | $5.6$ | $620\sim 850$ | 42% | 12% | 3% | $4.3\times 10^{-6}$
(15) | $0.65$ | $0.55$ | $0.92$ | $0.19$ | $0.96$ | $4.9$ | $5.2$ | $4.0$ | $5.5$ | $3.3$ | $730\sim 930$ | 25% | 7% | 2% | $5.4\times 10^{-6}$

By using these fifteen sets of numbers, we perform the QCD sum rule analysis.
There are two parameters, the Borel mass $M_{B}$ and the threshold value
$s_{0}$ in the QCD sum rule analysis. We find that the Borel mass stability is
usually good, but the threshold value stability is not always good. We show
the mass range of $\sigma(600)$, $M_{\sigma}$, in Table 1, where the working
region is taken to be $0.8$ GeV${}^{2}<s_{0}<1.2$ GeV2 and $0.8$
GeV${}^{2}<M_{B}^{2}<2$ GeV2. We find the mass range is small when the pole
contribution (PC) is large.

The parameter sets (01)-(06) lead to relatively good threshold value
stability. Taking the set (02) as an example, we show its spectral density
$\rho(s)$ in Fig 2 as function of $s$. It is positive definite, and has a
small value around $s\sim 1.2$ GeV2. Therefore, the threshold value dependence
is weak around this point, as shown in Fig. 3 for the extracted mass as
functions of both $M_{B}^{2}$ and $s_{0}$. We find all the curves are very
stable in the region $0.5$ GeV${}^{2}<M_{B}^{2}<2$ GeV2 and $0.6$
GeV${}^{2}<s_{0}<1.4$ GeV2. From the set (02) we can extract the mass of
$\sigma(600)$ around $550$ MeV. From other good cases, we find that the mass
of $\sigma(600)$ is around $550$ MeV as well.

Figure 2: The spectral density $\rho(s)$ calculated by the mixed current
$\eta$, as a function of $s$. We show the results of the parameter set (02) as
an example.

Figure 3: The extracted mass of $\sigma(600)$ as a tetraquark state calculated
by the mixed current $\eta$, as functions of the Borel mass $M_{B}$ and the
threshold value $s_{0}$. We show the results of the parameter set (02) as an
example. At the left panel, the solid, short-dashed and long-dashed curves are
obtained by setting $s_{0}=0.8,~{}1$ and $1.2$ GeV2, respectively. At the
right panel, the solid and dashed curves are obtained by setting
$M_{B}^{2}=0.5,~{}1$ and $2$ GeV2, respectively.

The parameter sets (07)-(15) lead to the threshold value stability, which is
not good. Taking the set (13) as an example, we show its spectral density in
Fig. 4 as a function of $s$ (left figure), and the extracted mass in Fig. 5 as
a function of $s_{0}$ (upper three curves). The mass increases with $s_{0}$
and we cannot extract the mass from this result.

Figure 4: The spectral density $\rho(s)$ calculated by the mixed current
$\eta$, as a function of $s$. We show the results of the parameter set (13) as
an example. The left figure shows the full spectral density as given on the
left hand side of Eq. (22), while the right figure is the one with
$\rho_{\pi\pi}(s)$ subtracted.

Figure 5: The extracted mass of $\sigma(600)$ as a tetraquark state calculated
by the mixed current $\eta$, as functions of the threshold value $s_{0}$. We
choose the parameter set (13) as an example. The solid, short-dashed and long-
dashed curves are obtained by setting $M_{B}^{2}=0.5,~{}1$ and $2$ GeV2,
respectively. The upper three curves are obtained without adding the
contribution of the $\pi$-$\pi$ continuum in the spectral density in the
phenomenological side, while the lower three curves are obtained after adding
the contribution of the $\pi$-$\pi$ continuum.

Many effects contribute to the mass dependence on the threshold value, but for
$\sigma(600)$ the $\pi$-$\pi$ continuum contribution is probably the dominant
one. Hence, we add a term $\rho_{\pi\pi}(s)$ in the spectral function in the
phenomenological side to describe the $\pi$-$\pi$ continuum:

$\displaystyle\rho(s)$ $\displaystyle=$ $\displaystyle
f^{2}_{Y}\delta(s-M^{2}_{Y})+\rho_{\pi\pi}(s)+\rho_{cont}\,.$ (22)

where $\rho_{cont}$ is the standard expression of the continuum contribution
except the $\pi$-$\pi$ continuum. To find an expression for
$\rho_{\pi\pi}(s)$, we introduce a coupling

$\displaystyle\lambda_{\pi\pi}$ $\displaystyle\equiv$ $\displaystyle\langle
0|\eta|\pi^{+}\pi^{-}\rangle\,.$ (23)

The correlation function of the $\pi$-$\pi$ continuum is

$\displaystyle\Pi_{\pi\pi}(p^{2})$ $\displaystyle=$ $\displaystyle
i\int{d^{4}q\over(2\pi)^{2}}{i\over(p+q)^{2}-m^{2}_{\pi}+i\epsilon}{i\over
q^{2}-m^{2}_{\pi}+i\epsilon}|\lambda_{\pi\pi}|^{2}\,,$ (24)

and the spectral density of the $\pi$-$\pi$ continuum is just its imaginary
part

$\displaystyle\rho_{\pi\pi}(s)={\rm Im}\Pi_{\pi\pi}(s)={1\over
16\pi^{2}}\sqrt{1-{4m_{\pi}^{2}\over s}}|\lambda_{\pi\pi}|^{2}\,.$ (25)

We may calculate $\lambda_{\pi\pi}$ by using the method of current algebra if
we know the property of the resonance state. However, this is not the topic of
this paper. Moreover, in this paper we use a general local tetraquark current
to test the full space of local tetraquark currents, so we again make some try
and error tests, and find that the following function leads to a reasonable
QCD sum rule result, $\lambda_{\pi\pi}\sim s$. Hence, we take the spectral
density of the $\pi$-$\pi$ continuum as

$\displaystyle\rho_{\pi\pi}(s)=a(t_{i},\theta_{i})s^{2}\sqrt{1-{4m_{\pi}^{2}\over
s}}\,.$ (26)

We add the continuum contribution $\rho_{\pi\pi}(s)$ in the phenomenological
side and perform the QCD sum rule analysis. The values of parameter
$a(t_{i},\theta_{i})$ are listed in Table 1. After adding the continuum
contribution, the threshold value stability becomes much better. Still taking
the set (13) as an example, we show its spectral density in Fig. 4 as a
function of $s$ (right figure), and the extracted mass in Fig. 5 as functions
of $s_{0}$ (lower three curves). We see that now the spectral density has a
small value around $s\sim 1.1$ GeV2, and the stability of the threshold value
is significantly improved.

Hence, we made the same analysis for all the other cases. We found all the
cases are good except one, which is the case (15), where we are not able to
get the desired stability as a function of $s_{0}$. The mass function has a
small stability region and increases rapidly with $s_{0}$. Hence, we consider
this case is between the good case and bad case, and remove it from the
further analysis in this paper. We show several results out of all the good
cases in Fig. 6, which are obtained by using the parameter sets (01), (03),
(06), (09), (12) and (14). We list the used $a(t_{i},\theta_{i})$ in Table 1
for all the cases. All the masses behave very nicely as functions of the Borel
mass and $s_{0}$ as shown in Fig. 6. In our working region $0.8$
GeV${}^{2}<s_{0}<1.2$ GeV2 and $0.8$ GeV${}^{2}<M_{B}^{2}<2$ GeV2, all the
cases lead to a mass within the region $495$ MeV$\sim 570$ MeV. From this mass
range, the mass of $\sigma(600)$ is extracted to be $530$ MeV $\pm$ 40 MeV.

Figure 6: The extracted mass of $\sigma(600)$ as a tetraquark state calculated
by the mixed currents $\eta$, as functions of the threshold value $s_{0}$. We
choose the parameter sets (01), (03), (06), (09), (12) and (14). The results
are shown in sequence. The solid, short-dashed and long-dashed curves are
obtained by setting $M_{B}^{2}=0.5,~{}1$ and $2$ GeV2, respectively.

##  IV The Effect of Finite Decay Width

After the $s_{0}$ stability has been improved, we notice now that the mass
increases systematically with the Borel mass as seen in Fig. 6 in all the
cases. We therefore try to consider a possible reason of this systematic
result. The $\sigma(600)$ meson has a large decay width. We parametrize it by
a Gaussian distribution instead of the $\delta$-function for the
$\sigma(600)$.

$\displaystyle\rho^{FDW}(s)={f^{2}_{X}\over\sqrt{2\pi}\sigma_{X}}\exp\big{(}-{(\sqrt{s}-M_{X})^{2}\over
2\sigma_{X}^{2}}\big{)}~{}.$ (27)

The Gaussian width $\sigma_{X}$ is related to the Breit-Wigner decay width
$\Gamma$ by $\sigma_{X}=\Gamma/2.4$. We set $\sigma_{X}=200$ MeV, and
$M_{X}=550$ MeV, and calculate the following “mass”:

$\displaystyle
M^{2}(M_{B},s_{0})={\int_{0}^{s_{0}}e^{-s/M_{B}^{2}}s\exp\big{(}-{(\sqrt{s}-M_{X})^{2}\over
2\sigma_{X}^{2}}\big{)}{ds\over
2\sqrt{s}}\over\int_{0}^{s_{0}}e^{-s/M_{B}^{2}}\exp\big{(}-{(\sqrt{s}-M_{X})^{2}\over
2\sigma_{X}^{2}}\big{)}{ds\over 2\sqrt{s}}}~{}.$ (28)

We find that the obtained mass $M$ is not just 550 MeV, but increases as
$M_{B}^{2}$ increases as shown in Fig. 7. Hence, the extracted mass in the QCD
sum rule analysis ought to depend on the Borel mass. The amount of the change
of the extracted mass in the QCD sum rule analysis is similar to the one found
in this model calculation. Moreover, we find that the finite decay width does
not change the final result significantly, which we have also noticed in our
previous paper Chen:2007xr .

Figure 7: The extracted “mass” considering a finite decay width. The solid,
short-dashed and long-dashed curves are obtained by setting
$M_{B}^{2}=0.5,~{}1$ and $2$ GeV2, respectively.

##  V Summary

In summary, we have studied the light scalar meson $\sigma(600)$ in the QCD
sum rule. We have used general local tetraquark currents which are linear
combinations of five independent local ones. This describes the full space of
local tetraquark currents which can couple to $\sigma(600)$ either strongly or
weakly. We find some cases where the stability of the Borel mass and threshold
value is both good, while in some cases the threshold value stability is not
so good. The resonance mass has an increasing trend with $s_{0}$, which
indicates a continuum contribution. Hence, we have introduced a contribution
from the $\pi$-$\pi$ continuum, and obtained a good threshold value stability.
The mass of $\sigma(600)$ is extracted to be $530$ MeV $\pm$ 40MeV. Very
interesting observation is that the mass increases slightly with the Borel
mass. We have made a model calculation by taking the Gaussian width of $200$
MeV centered at $550$ MeV and try to make a sum rule analysis. We see a
similar increase trend as seen in the QCD sum rule analysis.

The continuum contribution exists in the background of the $\sigma(600)$ meson
and it is very important to consider this fact in the QCD sum rule analysis
for exotic states. We have seen clear tendency of the mass increase with the
Borel mass after getting good signal of the threshold dependence. The decay
width of $\sigma(600)$ is related to this increase tendency. We are now trying
to calculate this by using the three-point correlation function within the QCD
sum rule approach. The present analysis is very encouraging to apply the QCD
sum rule including the continuum states for other scalar mesons. Moreover, the
continuum contribution should be important in many other resonances such as
$\Lambda(1405)$ etc, which lies in some continuum background. In the future,
we will use the QCD sum rule analysis with continuum to study various
resonances.

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grants No. 10625521 and No. 10721063, the Ministry of Science and
Technology of China (2009CB825200), the Ministry of Education research Grant:
Kakenhi (18540269), and the Grant for Scientific Research ((C) No. 19540297)
from the Ministry of Education, Culture, Science and Technology, Japan.

## References

  * (1) C. Amsler and N. A. Tornqvist, Phys. Rept. 389, 61 (2004).
  * (2) D. V. Bugg, Phys. Rept. 397, 257 (2004).
  * (3) E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007).
  * (4) I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. 96, 132001 (2006).
  * (5) T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994).
  * (6) J. A. Oller and E. Oset, Nucl. Phys. A 620, 438 (1997) [Erratum-ibid. A 652, 407 (1999)].
  * (7) J. Sugiyama, T. Nakamura, N. Ishii, T. Nishikawa and M. Oka, Phys. Rev. D 76, 114010 (2007).
  * (8) S. Prelovsek, T. Draper, C. B. Lang, M. Limmer, K. F. Liu, N. Mathur and D. Mohler, arXiv:1002.0193 [hep-ph].
  * (9) C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008).
  * (10) E. M. Aitala et al. [E791 Collaboration], Phys. Rev. Lett. 86, 770 (2001).
  * (11) M. Ablikim et al. [BES Collaboration], Phys. Lett. B 598, 149 (2004).
  * (12) D. Aston et al., Nucl. Phys. B 296, 493 (1988).
  * (13) R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B 462, 380 (1999).
  * (14) R. L. Jaffe, Phys. Rev. D 15, 267 (1977).
  * (15) R. L. Jaffe, Phys. Rev. D 15, 281 (1977).
  * (16) M. G. Alford and R. L. Jaffe, Nucl. Phys. B 578, 367 (2000).
  * (17) L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. Lett. 93, 212002 (2004).
  * (18) J. D. Weinstein and N. Isgur, Phys. Rev. D 41, 2236 (1990).
  * (19) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).
  * (20) L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127, 1 (1985).
  * (21) M. E. Bracco, A. Lozea, R. D. Matheus, F. S. Navarra and M. Nielsen, Phys. Lett. B 624, 217 (2005).
  * (22) S. Narison, Phys. Rev. D 73, 114024 (2006).
  * (23) H. J. Lee and N. I. Kochelev, Phys. Lett. B 642, 358 (2006).
  * (24) H. X. Chen, A. Hosaka and S. L. Zhu, Phys. Rev. D 76, 094025 (2007).
  * (25) K. C. Yang, W. Y. P. Hwang, E. M. Henley and L. S. Kisslinger, Phys. Rev. D 47, 3001 (1993).
  * (26) S. Narison, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 17, 1 (2002).
  * (27) V. Gimenez, V. Lubicz, F. Mescia, V. Porretti and J. Reyes, Eur. Phys. J. C 41, 535 (2005).
  * (28) M. Jamin, Phys. Lett. B 538, 71 (2002).
  * (29) B. L. Ioffe and K. N. Zyablyuk, Eur. Phys. J. C 27, 229 (2003).
  * (30) A. A. Ovchinnikov and A. A. Pivovarov, Sov. J. Nucl. Phys. 48, 721 (1988) [Yad. Fiz. 48, 1135 (1988)].
  * (31) W. Y. P. Hwang and K. C. Yang, Phys. Rev. D 49, 460 (1994).