#### \(B_s\pi \) – \(B\bar{K}\) interactions in finite volume and X(5568)

Eur. Phys. J. C
Jun-Xu Lu 1
Xiu-Lei Ren 0
Li-Sheng Geng 1
0 State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University , Beijing 100871 , China
1 School of Physics and Nuclear Energy Engineering, International Research Center for Nuclei and Particles in the Cosmos, Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University , Beijing 100191 , China
The recent observation of X (5568) by the D0 Collaboration has aroused a lot of interest both theoretically and experimentally. In the present work, we first point out that X (5568) and Ds∗0(2317) cannot simultaneously be of molecular nature, from the perspective of heavy-quark symmetry and chiral symmetry, based on a previous study of the lattice QCD scattering lengths of D K and its coupled channels. Then we compute the discrete energy levels of the Bs π and B K¯ system in finite volume using unitary chiral perturbation theory. The comparison with the latest lattice QCD simulation, which disfavors the existence of X (5568), supports our picture where the Bs π and B K¯ interactions are weak and X (5568) cannot be a Bs π and B K¯ molecular state. In addition, we show that the extended Weinberg compositeness condition also indicates that X (5568) cannot be a molecular state made from Bs π and B K¯ interactions.
1 Introduction
Recently, an apparently exotic mesonic state, the so-called
X (5568) state, was observed by the D0 Collaboration in the
Bs0π ± invariant mass spectrum [1]. The extracted mass and
width are M = 5567.8 ± 2.9−+10..99 MeV and = 21.9 ±
6.4−+25..05 MeV, respectively, and the preferred spin-parity is
J P = 0+. This state, being one of the exotic X Y Z states [2],
should contain at least four valence quark flavors u, d, s, and
b.
The experimental observation of X (5568) has inspired
much theoretical work. It has been proposed to be either
a tetraquark state [3–14], a triangular singularity [15], or a
molecular state [16]. Although these theoretical studies favor
the existence of a state that can be identified as the X (5568),
some other studies have yielded negative conclusions. For
instance, the difficulty to accommodate such a narrow
structure with a relatively low mass has been stressed by Burns
et al. [17] and Guo et al. [18]. A recent study in the chiral
quark model showed that neither diquark–antidiquark nor
meson–meson structures support the existence of X (5568)
[19]. In Ref. [20], the lowest-lying tetraquark s-wave state
was found to be 150 MeV higher than X (5568). On the
experimental side, neither the LHCb nor the CMS Collaboration
found a signal corresponding to X (5568) [21,22].
Approximate heavy-quark symmetry and its breaking
pattern provide a powerful tool to understand the nature of
X (5568). In the charm sector, Ds∗0(2317) has been suggested
to be a molecular state made from D K and Ds η interactions
in many studies; see, e.g., Refs. [23–28].1 Naively, X (5568)
might be a heavy-quark partner of the Ds∗0(2317). However,
in Ref. [18], the authors point out that if one assumes a
molecular picture for the X (5568), heavy-quark symmetry
then dictates that the charmed partner of X (5568) should be
located around 2.24 ± 0.15 GeV. So far, no signal has been
reported yet with the associated quantum numbers in this
energy region [18]. Note that the QCD sum rules predicted
that the charmed partner of X (5568) is located at much higher
energies about 2.5–2.6 GeV [4,29].
The unitary chiral perturbation theory (UChPT),
respecting both approximate heavy-quark symmetry and chiral
symmetry and their breaking pattern, has turned out to be a useful
tool in understanding the X Y Z states. In Ref. [28], using the
lattice QCD (lQCD) scattering lengths [27] to fix the
relevant low energy constants (LECs) of the covariant UChPT,
one found that the Ds∗0(2317) emerges naturally, similar to
Refs. [27,30,31]. When extended to the bottom sector guided
by heavy-quark symmetry, the covariant UChPT shows that
the interactions in the Bs π –B K¯ channel are rather weak and
do not support the existence of a low-lying resonance or
bound state, consistent with an explicit search on the
second Riemann sheet [28].
1 In Refs. [23–26,28], the bottom parter of Ds∗0(2317) was also pre
Nevertheless, in Ref. [32], the authors performed a fit to
the D0 invariant mass distribution [1] by employing the Bs π
and B K¯ coupled channel unitary chiral amplitudes and
treating the unknown subtraction constant as a free parameter.
The best fit yields a dynamically generated state consistent
with X (5568) [1]. Nevertheless, the authors noted that a large
cutoff , compared with a “natural” size of about 1 GeV, is
needed to describe the D0 data. The unusual size of the
cutoff points clearly to the presence of missing channels,
contributions of other sources, or existence of “non-molecular”
components, such as sizable tetraquark configurations, in the
framework of the UChPT. As a result, it was concluded that
a pure molecular state, dynamically generated by the unitary
loops, is disfavored. A similar conclusion was reached in a
later study utilizing p-wave coupled channel dynamics in the
UChPT [33]. It should be pointed out that irrespective of the
nature of X (5568), the UChPT of Ref. [32] provides a good
description of the D0 data. Therefore, it needs to be further
tested.
In the present work, we would like to formulate the UChPT
of Refs. [28,32] in finite volume and compute relevant
discrete energy levels and scattering lengths. A comparison with
lattice QCD simulations then allows one to distinguish these
two scenarios and provides more clues about the nature of
the X (5568). In the remaining of this paper, we denote the
UChPT of Ref. [32] by X -UChPT and that of Ref. [28] by X/
UChPT to indicate that one of them dynamically generates
X (5568) and the other does not.
This paper is organized as follows. In Sect. 2, we briefly
describe the UChPT of Ref. [28] and Ref. [32] and then in
Sect. 3 we point out that from the perspective of
heavyquark symmetry and chiral symmetry as implemented in
UChPT, the lQCD scattering lengths of D K and its
coupled channels imply that the Bs π and B K¯ interactions are
rather weak and do not support a molecular state that can be
identified as X (5568). As a result, a typical molecular
picture for X (5568) similar to that for the Ds∗0(2317) case is
not favored in UChPT, which is supported by the extended
Weinberg compositeness condition. In Sect. 4, we formulate
the UChPT of Refs. [28,32] in a finite box and calculate
the discrete energy levels that can be extracted in a lattice
QCD simulation. The results are contrasted with the latest
lQCD simulation of Ref. [34], followed by a short summary
in Sect. 5.
2 Unitary chiral perturbation theory
UChPT has two basic building blocks, a kernel potential
provided by chiral perturbation theory and a unitarization
procedure. The kernel potentials constrained by chiral symmetry
and other relevant symmetries, such as heavy-quark
symmetry in the present case, are standard in most cases, while the
unitarization procedures can differ in their treatment of
lefthand cuts or higher order effects, although they all satisfy
two-body elastic unitarity.
The leading order kernel potential employed in Refs. [28,
32] has the following form:
3s − (Mi2 + mi2 + M 2j + m2j ) −
1 2
s
where i = 1(2) denotes the Bs π (B K¯ ) channel, s is the
invariant mass squared of the system, f is the pseudoscalar
meson decay constant in the chiral limit, i = Mi2 − mi2
and M1(m1) and M2(m2) are the masses of Bs (π ) and B(K¯ )
mesons. The coefficients Ci j are C11 = C22 = 0, C12 =
C21 = 1.
In Refs. [28,32], the Bethe–Salpeter equation is adopted
to unitarize the chiral kernel obtained above. In the context
of the UChPT, the integral Bethe–Salpeter equation is often
simplified and approximated as an algebraic equation with
the use of the on-shell approximation.2 It reads
where a is the subtraction constant, and μ the regularization
scale and s = P2. The difference between Eq. (4) and its
counterpart in Ref. [32] is a constant and can be absorbed
into the subtraction constant.
It has been noted that the relativistic loop function, Eq. (4),
violates heavy-quark symmetry and the naive chiral power
counting. Several methods (see, e.g., Refs. [24,26,36,37])
2 For a recent study of off-shell effects, see Ref. [35] and the references
T = V + V G T ,
Gi = i
where T is the unitarized amplitude, V the potential, and G
the one-loop 2-point scalar function. In n dimensions, G has
the following simple form:
(2π )n [( P − q)2 − mi + i ][q2 − M 2
i + i ]
where P is the total center-of-mass momentum of the system.
The loop function G is divergent and needs to be
regularized. In the dimensional regularization scheme, it has the
following form:
m2 − M 2 + s log
have been proposed to deal with such a problem. In X/
UChPT [28], the minimal subtraction scheme is modified
to explicitly conserve heavy quark symmetry and the naive
chiral power counting. Confined to either the charm sector
or the bottom sector alone, the modified subtraction scheme,
termed the heavy-quark-symmetry (HQS) inspired scheme,
is equivalent to the MS scheme, but it can link both sectors
in a way that conserves heavy-quark symmetry up to order
1/MH , with MH the chiral limit of the heavy hadron mass.3
The loop function of the HQS scheme is related with that of
the MS scheme via
1
GHQS = GMS − 16π 2
− 2
where msub is the average mass of the Goldstone bosons, M˚
the chiral limit value of the bottom (charm) meson masses
and a the subtraction constant. In the HQS scheme, the
subtraction constant determined in the charm (bottom) sector is
the same as that determined in the bottom (charm) sector,
while this is not the case for the subtraction constant of the
MS scheme. However, one may use the same cutoff in the
cutoff scheme for both bottom and charm sectors related via
heavy-quark symmetry (for a different argument, see, e.g.,
Ref. [39]).
In Ref. [28], it was shown that the LO potential of Eq. (1)
cannot describe the lQCD scattering lengths of Ref. [27].
One needs go to the next-to-leading order (NLO). At NLO,
there are six more LECs, namely c0, c1, c24, c35, c4, c5.
Among them, c0 is determined by fitting to the light quark
mass dependence of lQCD D and Ds masses [27], and c1 is
determined by reproducing the experimental D and Ds mass
difference. Once c0 and c1 are fixed, the remaining LECs and
the subtraction constant are determined by fitting to the lQCD
scattering lengths of Ref. [27], yielding a χ 2/d.o.f. = 1.23.
With these LECs, Ds∗0(2317) appears naturally at 2317 ± 10
MeV. As a result, in the present work we employ the NLO
UChPT of Ref. [28].
3 Scattering lengths and compositeness
3.1 Scattering lengths
The scattering lengths of a D (Ds ) meson with a Nambu–
Goldstone pseudoscalar meson have been studied on a
lattice [27]. With these scattering lengths as inputs, various
groups have predicted the existence of Ds∗0(2317) and its
counterparts both in the charm sector and in the bottom
sector [28,30,31]. It is worth pointing out that the predicted
counterparts of the Ds∗0(2317) and Ds1(2416) are indeed
observed in a later lQCD simulation [40]. In the
following, we compare the scattering lengths of Bs π , B K¯ , Ds π ,
and D K obtained in X/ -UChPT [28] with those obtained in
X -UChPT [32], to check the consistency between the
constraint imposed by the existence of the X (5568) and that of
Ds∗0(2317) and the lQCD scattering lengths of Ref. [27].
The scattering length of channel i is defined as
Using the G function determined in Ref. [28] and Ref. [32],4
we obtain the scattering lengths of B K¯ and Bs π tabulated in
Table 1. Clearly, the results obtained in the two approaches
are quite different. The scattering lengths, particularly that of
Bs π obtained in X/ -UChPT [28], show clearly that the
interactions are rather weak in the Bs π and B K¯ coupled channels.
This implies that there is no bound state or resonant state,
consistent with a direct search on the second Riemann sheet.
For the sake of comparison, Table 1 also lists the scattering
lengths of D K and Ds π . We note that the scattering lengths
obtained in X -UChPT [32] are much larger than those of X/
UChPT [28], inconsistent with the lQCD results of Ref. [27].
lQCD simulations allow one to understand more physical
observables by studying their quark mass dependence. In
this regard, it is useful to study the mπ dependence of the
scattering lengths. For such a purpose in the UChPT, one
needs the pion mass dependence of the constituent hadrons,
namely, m K , m B and m Bs . Following Ref. [28], we take
m2B = m20 + 4c0(m2π + m2K ) − 4c1m2π ,
m2Bs = m20 + 4c0(m2π + m2K ) + 4c1(m2π − 2m2K ),
with aˆ = 0.317, bˆ = 0.487, c0 = 0.015, and c1 = −0.513.
These LECs are fixed using the experimental data and the
lattice QCD masses of Ref. [27] as explained above.
In Fig. 1, we show the scattering lengths aBs π and aB K¯ as a
function of the pion mass. One can see that aBs π shows some
“threshold” effects. These effects can easily be understood
from the amplitude expressed in terms of couplings and pole
positions,
Tii (s) = √
where gi is the coupling defined in Eq. (10), and √s0 the pole
position. To calculate the scattering lengths, √s = m Bs +
mπ . Once the trajectory of the threshold crosses that of the
3 For an application of the HQS scheme in the singly charmed (bottom)
baryon sector, see Ref. [38].
4 The loop function is always regularized in the dimensional regular
ization scheme, unless otherwise specified.
a For X -UChPT [32], the scattering lengths are calculated using the loop function regularized in the cutoff scheme with the cutoff fixed by fitting
to the D0 data. See Sect. 2 for details
Fig. 1 Real part of the scattering lengths aBsπ and aBK¯ as a function
of the pion mass mπ obtained in X/ -UChPT [28] and X -UChPT [32].
The uptriangle in the left panel denotes the lQCD result of Ref. [34],
which is an average of the six data obtained using different sets of gauge
configurations (see the bottom panel of Fig. 2 of Ref. [34]). The shaded
area indicates uncertainties originating from the lQCD data of Ref. [27]
pole as mπ varies, a singularity will emerge. In fact, similar
effects have already been observed in Ref. [41]. On the other
hand, as mπ increases, aB K¯ stays more or less constant in X
UChPT [32], while it increases slightly in X/ -UChPT [28].
For the sake of comparison, we also show the lQCD result of
Ref. [34]. We note that the lQCD result, obtained with a π
mass close to its physical value, lies in between the results of
Refs. [28,32], and thus cannot distinguish the two scenarios.
For such a purpose, lQCD simulations with quark masses
larger than their physical values will be more useful.
3.2 Compositeness of X (5568)
The large cutoff used in X -UChPT [32] indicates that the
dynamically generated X (5568) should contain rather large
non-Bs π and B K¯ components. This can be quantified using
the Weinberg composition condition and its extensions [42–
58].
Following Ref. [51] we define the weight of a hadron–
hadron component in a composite particle by
where √s0 is the pole position, GiII is the loop function
evaluated on the second Riemann sheet, and gi is the coupling
of the respective resonance or bound state to channel i
calculated as
(√s − √s0)TiIiI,
where TiIiI is the i i element of the T amplitude on the second
Riemann sheet.
The deviation of the sum of Xi from unity is related to the
energy dependence of the s-wave potential,
Xi = 1 − Z ,
Z = −
The quantity Z is often attributed to the weight of missing
channels.
Using X -UChPT [32], we obtain X B K¯ = 0.10 − 0.02i ,
X Bs π = 0.06 + 0.11i , and Z = 0.83 − 0.09i . The value of
Z is much larger than the typical size for a state dominated
by molecular components, which indicates the missing of
contributions of other components. Such a result is consistent
with the unusual size of and similar conclusions have been
drawn in Ref. [32].
One should note that the above defined compositeness and
the so-drawn conclusion are model dependent, which is
different from the original Weinberg criterion. In this formalism,
the compositeness and elementariness are defined as the
fractions of the contributions from the two-body scattering states
and one-body bare states to the normalization of the total
wave function within the particular model space, respectively
(see, e.g., Ref. [55]). In the present case, the small
compositeness 1 − Z simply indicates that in the model space of the X/
UChPT [32], the meson–meson components only account for
a small fraction of the total wave function, and thus X (5568)
cannot be categorized as a meson–meson molecule.
4 Bsπ and B K¯ interactions in finite volume
If X (5568) exists, one should be able to observe it in a lQCD
simulation, which can be anticipated in the near future, given
the fact that the LHCb result has cast doubts on the existence
of the X (5568). In view of such possibilities, in the following,
we predict the discrete energy levels that one would obtain in
a lattice QCD simulation. Such an exercise provides a highly
non-trivial test of X/ -UChPT [28] and X -UChPT [32].
In this work, we follow the method proposed in Ref. [59]
to calculate the loop function G in finite volume in the
dimensional regularization scheme. Introducing a finite-volume
correction, δG, G˜ can be written as
where G D is the loop function calculated in the dimensional
regularization scheme, either GHQS or GMS, and δG has the
following form [59]:
M2(s) = (x 2 − x )s + x M 2 + (1 − x )m2 − i .
For the case of √s > M + m, δr (M2(s)) can be written
as a sum of the following three parts [60,61]:
δr (M2(s)) = g1 − g2 + g3,
r r r
+∞ q2dq
0 2π 2 [q2 + M2(s)]r
1 r (x 2 − x )(s − ms2s )
,
− [q2 + M2(ms2s )]r + [q2 + M2(ms2s )]r+1
g3 = δr (M2(ms2s )) − r (x 2 − x )(s − ms2s )δr+1(M2(ms2s )),
r
and L is the spatial size of the lattice.5 The separation scale
mss needs to satisfy mss < M + m. In the case of √s <
M + m, δr (M2(s)) can be expressed as [62]
n=0
2−1/2−r (√M)3−2r
M2|n|)−3/2+r K3/2−r (L
n=0
nx =−∞ ny=−∞ nz=−∞
with n = (nx , n y , nz ). It should be mentioned that in actual
calculations the discrete summations in Eqs. (17)–(19) are
only taken up to a certain number, |n|max = L/(2a) with a
the lattice spacing.
The Bethe–Salpeter equation in finite volume reads
The discrete energy levels one would observe in a lattice
QCD simulation are determined via det(V −1 − G˜ ) = 0.
In Fig. 2, we show the so-obtained discrete energy levels
in both scenarios, X -UChPT [32] and X/ -UChPT [28]. From
the left panel, one can clearly identify an extra energy level,
namely, the second energy level, which can be associated
to X (5568). All the other discrete energy levels lie close
to one of the free energy levels, Bs π( ) or B K¯ ( ), where
denotes the energy of the corresponding discrete energy
wth k = 2Lπ ( = 0, 1, 2, . . .) and likewise for E [B K¯ ( )].
On the other hand, no extra energy level appears in the
right panel, consistent with the fact that the interactions
are weak and no resonance or bound state is found in
X/ -UChPT [28].
In a recent study [34], a lQCD simulation employing the
PACS-CS gauge configurations was performed [63]. It was
shown that no state corresponding to X (5568) exists in the
simulation, consistent with the LHCb result [21]. In addition,
the authors of Ref. [34] provided an analytic prediction based
on the Lüscher method. They included the X (5568)
explicitly via a resonant Breit–Wigner-type phase shift and then
related the phase shift with the discrete energy levels through
the Lüscher method. For more details we refer to subsection
II.A of Ref. [34]. In Fig. 3, we compare the lQCD discrete
energy levels with those obtained in our present study. Since
5 Throughout this paper, we assume a periodic boundary condition for
the lQCD setup and that the temporal size is much larger than the spatial
size and therefore can be taken as infinity.
Fig. 2 Discrete energy levels
of the Bs π –B K¯ system as a
function of the lattice size L.
Left panel obtained with
X -UChPT [32]; right panel
obtained with X/ -UChPT [28].
The solid lines are the energy
levels obtained by solving
Eq. (20), while the dashed and
dotted lines are the energy levels
of non-interacting Bs π and B K¯
pairs, respectively
Fig. 3 Discrete energy levels at L = 2.9 fm obtained in different
approaches: the solid points, up triangles, squares, and down triangles
correspond to the results of X -UChPT [32], X/ -UChPT [28], the analytic
prediction using the Lüscher method [34], and the lQCD results of
Ref. [34], respectively. The solid (dashed) lines refer to the energy
levels of non-interacting Bs π (B K¯ ) pairs
the quark masses in the lQCD simulation are not yet physical,
the non-interacting energy levels of the lQCD are slightly
shifted upward compared to those calculated theoretically
using physical meson masses. Apparently, the lQCD results
are consistent with X/ -UChPT [28], but not X -UChPT [32]
in which a large cutoff was used to reproduce the D0 data.
On the other hand, the analytic predictions based on the
Lüscher method [34] are consistent with X -UChPT [32], as
they should be since in Ref. [34], the D0 X (5568) mass and
width were employed in the Lüscher method as inputs and
in Ref. [32] the only parameter in X -UChPT, the
subtraction constant, was fixed by fitting to the D0 data. We should
note that, for a single channel, the Lüscher method is
consistent with the Jülich–Valencia approach adopted in the present
work (see, e.g., Ref. [59] for an explicit comparison in the
case of K K ∗ scattering).
From the above comparison, one can conclude that there
is indeed a tension among the D0 data, the lQCD results of
Ref. [34], and indirectly those of Ref. [27], provided that
heavy-quark symmetry and chiral symmetry are not
somehow strongly broken.
5 Summary
The recent D0 claim of the existence of X (5568) has aroused
a lot of interest. In the present paper, we showed explicitly the
tension between the D0 discovery and the lQCD results on
the charmed meson–pseduoscalar meson scattering,
including the scattering lengths and the existence of Ds∗0(2317),
from the perspective of approximate chiral symmetry and
heavy-quark symmetry as implemented in the unitary chiral
perturbation theory. We then formulated the unitary chiral
description of the coupled channel Bs π – B K¯ interactions in
finite volume. Our results, when compared with the latest
lattice QCD simulation, confirm the inconsistency and
disfavor the existence of X (5568) in unitary chiral perturbation
theory. We conclude that more experimental and theoretical
efforts are needed to clarify the current situation.
Acknowledgements We thank Eulogio Oset for a careful reading of
the first version of this manuscript and for his valuable comments.
This work is partly supported by the National Natural Science
Foundation of China under Grants Nos. 11375024, 11522539, 11335002, and
11411130147, the Fundamental Research Funds for the Central
Universities, the Research Fund for the Doctoral Program of Higher Education
under Grant No. 20110001110087, and the China Postdoctoral Science
Foundation under Grant No. 2016M600845.
Open Access This article is distributed under the terms of the Creative
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1. V.M. Abazov et al., D0 Collaboration, Phys. Rev. Lett . 117 , 022003 ( 2016 ). arXiv:1602.07588 [hep-ex]
2. H.X. Chen , W. Chen , X. Liu , S.L. Zhu , Phys. Rep. 639 , 1 ( 2016 ). arXiv:1601.02092 [hep-ph]
3. L. Tang , C.F. Qiao , Eur. Phys. J. C 76 , 558 ( 2016 ). arXiv: 1603 .04761 [hep-ph]
4. W. Chen , H.X. Chen , X. Liu , T.G. Steele , S.L. Zhu , Phys. Rev. Lett . 117 , 022002 ( 2016 ). arXiv: 1602 .08916 [hep-ph]
5. F. Stancu , J. Phys . G 43 , 105001 ( 2016 ). arXiv: 1603 .03322 [hepph]
6. Y.R. Liu , X. Liu , S.L. Zhu , Phys. Rev. D 93 , 074023 ( 2016 ). arXiv: 1603 .01131 [hep-ph]
7. C.M. Zanetti , M. Nielsen , K.P. Khemchandani , Phys. Rev. D 93 , 096011 ( 2016 ). arXiv: 1602 .09041 [hep-ph]
8. J.M. Dias , K.P. Khemchandani , A.A. Martínez Torres , M. Nielsen , C.M. Zanetti , Phys. Lett . B 758 , 235 ( 2016 ). arXiv: 1603 .02249 [hep-ph]
9. Z.G. Wang , Commun. Theor. Phys. 66 , 335 ( 2016 ). arXiv: 1602 .08711 [hep-ph]
10. Z.G. Wang , Eur. Phys. J. C 76 , 279 ( 2016 ). arXiv: 1603 .02498 [hepph]
11. S.S. Agaev , K. Azizi , H. Sundu , Phys. Rev. D 93 , 074024 ( 2016 ). arXiv: 1602 .08642 [hep-ph]
12. S.S. Agaev , K. Azizi , H. Sundu , Eur. Phys. J. Plus 131 , 351 ( 2016 ). arXiv: 1603 .02708 [hep-ph]
13. S.S. Agaev , K. Azizi , H. Sundu , Phys. Rev. D 93 , 114007 ( 2016 ). arXiv: 1603 .00290 [hep-ph]
14. Q.F. Lü , Y.B. Dong , Phys. Rev. D 94 , 094041 ( 2016 ). arXiv: 1603 .06417 [hep-ph]
15. X.H. Liu , G. Li , Eur. Phys. J. C 76 , 455 ( 2016 ). arXiv: 1603 .00708 [hep-ph]
16. C.J. Xiao , D.Y. Chen , arXiv:1603.00228 [hep-ph]
17. T.J. Burns , E.S. Swanson , Phys. Lett . B 760 , 627 ( 2016 ). arXiv: 1603 .04366 [hep-ph]
18. F.K. Guo, U.-G. Meißner , B.S. Zou , Commun. Theor. Phys. 65 , 593 ( 2016 ). arXiv: 1603 .06316 [hep-ph]
19. X. Chen , J. Ping , Eur. Phys. J. C 76 , 351 ( 2016 ). arXiv: 1604 .05651 [hep-ph]
20. W. Wang , R. Zhu , Chin. Phys . C 40 , 093101 ( 2016 ). arXiv: 1602 .08806 [hep-ph]
21. R. Aaij et al., LHCb Collaboration, Phys. Rev. Lett . 117 , 152003 ( 2016 ). arXiv:1608.00435 [hep-ex]
22. CMS Collaboration, CMS-PAS-BPH-16-002 (2016)
23. W.A. Bardeen , E.J. Eichten , C.T. Hill , Phys. Rev. D 68 , 054024 ( 2003 ). arXiv:hep-ph/0305049
24. E.E. Kolomeitsev , M.F.M. Lutz , Phys. Lett . B 582 , 39 ( 2004 ). arXiv:hep-ph/0307133
25. F.K. Guo , P.N. Shen , H.C. Chiang , R.G. Ping , B.S. Zou , Phys. Lett . B 641 , 278 ( 2006 ). arXiv:hep-ph/0603072
26. M. Cleven , F.K. Guo , C. Hanhart , U.-G. Meißner , Eur. Phys. J. A 47 , 19 ( 2011 ). arXiv: 1009 .3804 [hep-ph]
27. L. Liu , K. Orginos , F.K. Guo , C. Hanhart , U.-G. Meißner , Phys. Rev. D 87 , 014508 ( 2013 ). arXiv:1208.4535 [hep-lat]
28. M. Altenbuchinger , L.S. Geng , W. Weise , Phys. Rev. D 89 , 014026 ( 2014 ). arXiv: 1309 .4743 [hep-ph]
29. S.S. Agaev , K. Azizi , H. Sundu , Phys. Rev. D 93 , 094006 ( 2016 ). arXiv: 1603 .01471 [hep-ph]
30. Z.H. Guo , U.-G. Meißner , D.L. Yao , Phys. Rev. D 92 , 094008 ( 2015 ). arXiv: 1507 .03123 [hep-ph]
31. D.L. Yao , M.L. Du , F.K. Guo , U.-G. Meißner , JHEP 1511 , 058 ( 2015 ). arXiv: 1502 .05981 [hep-ph]
32. M. Albaladejo , J. Nieves , E. Oset , Z.F. Sun , X. Liu , Phys. Lett . B 757 , 515 ( 2016 ). arXiv: 1603 .09230 [hep-ph]
33. X.W. Kang , J.A. Oller , Phys. Rev. D 94 , 054010 ( 2016 ). arXiv: 1606 .06665 [hep-ph]
34. C.B. Lang , D. Mohler , S. Prelovsek , Phys. Rev. D 94 , 074509 ( 2016 ). arXiv:1607.03185 [hep-lat]
35. M. Altenbuchinger , L.S. Geng , Phys. Rev. D 89 , 054008 ( 2014 ). arXiv: 1310 .5224 [hep-ph]
36. D. Gamermann , E. Oset , D. Strottman , M.J. Vicente , Vacas. Phys. Rev. D 76 , 074016 ( 2007 ). arXiv:hep-ph/0612179
37. D. Gamermann , E. Oset , Eur. Phys. J. A 33 , 119 ( 2007 ). arXiv: 0704 .2314 [hep-ph]
38. J.X. Lu , Y. Zhou , H.X. Chen , J.J. Xie , L.S. Geng , Phys. Rev. D 92 , 014036 ( 2015 ). arXiv: 1409 .3133 [hep-ph]
39. A. Ozpineci , C.W. Xiao , E. Oset , Phys. Rev. D 88 , 034018 ( 2013 ). arXiv: 1306 .3154 [hep-ph]
40. C.B. Lang , D. Mohler , S. Prelovsek , R.M. Woloshyn , Phys. Lett . B 750 , 17 ( 2015 ). arXiv:1501.01646 [hep-lat]
41. Y. Zhou , X.L. Ren , H.X. Chen , L.S. Geng , Phys. Rev. D 90 , 014020 ( 2014 ). arXiv:1404.6847 [nucl-th]
42. S. Weinberg , Phys. Rev . 130 , 776 ( 1963 )
43. S. Weinberg , Phys. Rev . 137 , B672 ( 1965 )
44. C. Hanhart , Y.S. Kalashnikova , A.V. Nefediev , Phys. Rev. D 81 , 094028 ( 2010 ). arXiv: 1002 .4097 [hep-ph]
45. V. Baru , J. Haidenbauer , C. Hanhart , Y. Kalashnikova , A.E. Kudryavtsev , Phys. Lett . B 586 , 53 ( 2004 ). arXiv:hep-ph/0308129
46. M. Cleven , F.K. Guo , C. Hanhart , U.-G. Meißner , Eur. Phys. J. A 47 , 120 ( 2011 ). arXiv: 1107 .0254 [hep-ph]
47. D. Gamermann , J. Nieves , E. Oset , E. Ruiz Arriola , Phys. Rev. D 81 , 014029 ( 2010 ). arXiv: 0911 .4407 [hep-ph]
48. J. Yamagata-Sekihara , J. Nieves , E. Oset , Phys. Rev. D 83 , 014003 ( 2011 ). arXiv: 1007 .3923 [hep-ph]
49. F. Aceti , E. Oset , Phys. Rev. D 86 , 014012 ( 2012 ). arXiv: 1202 .4607 [hep-ph]
50. C.W. Xiao , F. Aceti , M. Bayar , Eur. Phys. J. A 49 , 22 ( 2013 ). arXiv: 1210 .7176 [hep-ph]
51. F. Aceti , L.R. Dai , L.S. Geng , E. Oset , Y. Zhang , Eur. Phys. J. A 50 , 57 ( 2014 ). arXiv:1301.2554
52. F. Aceti , E. Oset , L. Roca , Phys. Rev. C 90 , 025208 ( 2014 ). arXiv: 1404 .6128 [hep-ph]
53. T. Hyodo , D. Jido , A. Hosaka , Phys. Rev. C 85 , 015201 ( 2012 ). arXiv:1108.5524 [nucl-th]
54. T. Hyodo , Int. J. Mod. Phys. A 28 , 1330045 ( 2013 ). arXiv: 1310 .1176 [hep-ph]
55. I. Sekihara , I. Hyodo , D. Jido , PTEP 2015 , 063D04 ( 2015 ). arXiv: 1411 .2308 [hep-ph]
56. H. Nagahiro , A. Hosaka , Phys. Rev. C 90 , 065201 ( 2014 ). arXiv: 1406 .3684 [hep-ph]
57. C. Garcia-Recio , C. Hidalgo-Duque , J. Nieves , L.L. Salcedo , L. Tolos , Phys. Rev. D 92 , 034011 ( 2015 ). arXiv: 1506 .04235 [hepph]
58. Z.H. Guo , J.A. Oller , Phys. Rev. D 93 , 096001 ( 2016 ). arXiv: 1508 .06400 [hep-ph]
59. L.S. Geng , X.L. Ren , Y. Zhou , H.X. Chen , E. Oset , Phys. Rev. D 92 , 014029 ( 2015 ). arXiv: 1503 .06633 [hep-ph]
60. V. Bernard , U.-G. Meißner , A. Rusetsky , Nucl. Phys . B 788 , 1 ( 2008 ). arXiv:hep-lat/0702012
61. X.L. Ren , L.S. Geng , J. Meng , Phys. Rev. D 89 , 054034 ( 2014 ). arXiv:1307.1896 [nucl-th]
62. L.S. Geng , X.L. Ren , J. Martin-Camalich , W. Weise , Phys. Rev. D 84 , 074024 ( 2011 ). arXiv: 1108 .2231 [hep-ph]
63. S. Aoki et al., PACS-CS Collaboration, Phys. Rev. D 79 , 034503 ( 2009 ). arXiv:0807.1661 [hep-lat]