#### Estimating decay rate of \(X^{{\pm }}(5568)\rightarrow B_s\pi ^{{\pm }}\) while assuming them to be molecular states

Eur. Phys. J. C
Estimating decay rate of X ±(5568) → Bsπ ± while assuming them to be molecular states
Hong-Wei Ke 1
Xue-Qian Li 0
0 School of Physics, Nankai University , Tianjin 300071 , China
1 School of Science, Tianjin University , Tianjin 300072 , China
Discovery of X (5568) brings up a tremendous interest because it is very special, i.e. made of four different flavors. The D0 collaboration claimed that they observed this resonance through portal X (5568) → Bs π , but unfortunately, later the LHCb, CMS, CDF and ATLAS collaborations' reports indicate that no such state was found. Almost on the Eve of 2017, the D0 collaboration reconfirmed existence of X (5568) via the semileptonic decay of Bs . To further reveal the discrepancy, supposing X (5568) as a molecular state, we calculate the decay rate of X (5568) → Bs π + in an extended light front model. Numerically, the theoretically predicted decay width of (X (5568) → Bs π +) is 20.28 MeV which is consistent with the result of the D0 collaboration ( = 18.6+−76..91(stat)+−33..58(syst) MeV). Since the resonance is narrow, signals might be drowned in a messy background. In analog, two open-charm molecular states D K and B D named as Xa and Xb, could be in the same situation. The rates of Xa → Ds π 0 and Xb → Bcπ 0 are estimated as about 30 and 20 MeV respectively. We suggest the experimental collaborations round the world to search for these two modes and accurate measurements may provide us with valuable information.
1 Introduction
Following discovery of numbers of X, Y, Z particles [
1–10
],
whose exotic behaviors cannot be interpreted by the
regular qq¯ structures and must be attributed to a new type,
either four-quark states or hybrids structures, the discussion
on them becomes a hot topic of the hadron physics. For the
four-quark states, there are several possibilities: molecular
state which is made of two color-singlet mesons; tetraquark
which consists of a color-anti-triplet diquark and a
colortriplet anti-diquark, or a mixture of the previous two. All
the possibilities are under intensive discussions from various
angles.
Mostly, the observed exotic X, Y, Z states are composed
of hidden charm or bottom flavors. In 2016 the D0
collaboration declared to have observed a new resonance X (5568) at
the Bs π ± invariant mass spectrum with the mass and width
being (5567.8 ± 2.9−+10..99) and (21.9 ± 6.4−+25..05) MeV [
11
].
Since the decay rate of X (5568) → Bs π ± is much larger
than that determined by weak interactions, one can assure
that this is a decay caused by strong interaction. Since for
the strong interaction, flavor components do not change and
the final state includes Bs whose quark-component is (b¯s)
and π + made of ud¯, so in the final state there are four
different flavors which cannot be created from vacuum, thus one
can confirm that X (5568) is a four-quark state which
consists of b¯sud¯ ingredients. Analysis implies X (5568) to be an
exotic state (if it indeed exists), but whether it is a molecule
or a tetraquark would be another open question and need to
be answered by precise measurements combining with
careful theoretical studies. In this work, we investigate its inner
structure via studying its decay behavior.
Unfortunately, the LHCb collaboration [
12
], the CMS
collaboration of LHC [
13
], the CDF collaboration of
Fermilab [
14
] and the ATLAS Collaboration of LHC [
15
]
claimed that no such decay mode was detected. Of course,
all experimentalists are very careful, so that they only offered
upper bounds on the decay channel. Just on the Eve of new
year, the D0 collaboration declared that X (5568) was
reconfirmed in the portal X (5568) → Bs π ± via a sequent
semileptonic decay of Bs0 → μ± Ds∓ [
16
] and the result
is consistent with the previous data which were obtained
with Bs → J /ψ φ, but the measured width is slightly
shifted to 18.6+−76..91(stat)+−33..58(syst) MeV. The acute
discrepancy among the experimental groups stimulates a dispute.
Because X (5568) may be the first observed exotic state
possessing four different flavors, studies on it (both theoretical
and experimental) are of obvious significance for getting a
better understanding of the quark model.
In literature [
17–31
], there are different opinions which
originate from different considerations. In various models,
the spectrum of X (5568) was computed to be compared with
the measured value. Naively, by its decay width it seems to
be a molecular state of B K [
21,31
] and its binding energy
is about 205 MeV which is a bit too large for binding two
mesons into a hadronic molecule based on our intuition, thus
an alternative suggestion is that it is a tetraquark [
22–24
]. The
authors of Ref. [32] regard that neither a molecular state nor
a tetraquark can explain the data, so they consider that Bs π
is produced in an electroweak decay where an extra hadron
is also created, but evades detection.
In this work, accepting the D0 analysis that X (5568)
indeed exists, we would ask which structure is more
preferred by the nature, it should be answered by fitting more
data besides the mass spectrum, namely one needs to
investigate its decay behaviors. Thus a careful computation on its
decay rate is absolutely necessary even though such a
calculation is somehow model-dependent. In fact, a few groups
of authors assumed X (5568) as a tetraquark and computed
the rate of X (5568) → Bs π in terms of the QCD sum rules
[
33–35
].
Different inner structures may result in different decay
rates for a designated channel. Theoretically assigning the
molecular structure to X (5568), we can predict its decay
rate to Bs π . Since strong interaction is blind to quark flavors,
the running effective coefficients for b and c quarks do not
deviate much from each other. By the heavy flavor symmetry,
one believes that at the leading order, the binding energies for
B K and D K are the same and the symmetry breaking should
occur at ( m1c − m1b ) corrections. As noted, the binding energy
for B D might be different from that of B K . Even though the
SU(2) symmetry between c and s quarks is not a good one,
the deviation does not prevent us to make a rough estimate
on the binding energy. We will study their decays while they
are supposed to be molecular states and the results can be a
cross check for the mysterious X (5568).
In order to explore the decay rates of a molecular state,
we extend the light front quark model (LFQM) which has
been successfully applied for calculating decay rates of
regular mesons and baryons [
36–56
]. Using the method and the
parameters obtained by fitting well measured data, we deduce
the corresponding transition matrix element and estimate the
decay widths of X (5568) → Bs π +. Then, we further
estimate decay rates of Xa → Ds π 0 and Xb → Bcπ 0 in terms
of the same method where Xa and Xb are the molecular states
consisting of D K and B D constituents respectively.
After the introduction we derive the amplitude for
transition X (5568) → Bs π +, Xa → Ds π 0 and Xb → Bcπ 0 in
Sect. 2. Then we numerically evaluate their decay widths in
Sect. 3. In the last section we discuss the numerical results
and draw our conclusion. Some details about the approach
are collected in the Appendix.
2 The strong decays X (5568), Xa and Xb
2.1 The strong decays X (5568) → Bs π +
In this section we calculate the decay rate of X (5568)+ →
Bs π +, while assuming X (5568) as a B K¯ molecular state
whose quantum number I ( J P ) is 0(0+), in the light-front
model. Because of successful applications of the method to
study strong decay processes of molecular states [
57
] we
apply the framework to the present case. The configuration
of the concerned B K molecular state is √12 (B0 K + + B+ K 0)
[
19
]. The Feynman diagrams for X (5568) decaying into
Bs π + by exchanging B∗0 (B¯ ∗0) or K ∗+ (K ∗−) mesons are
shown in Fig. 1.
Following Ref. [
52
], the hadronic matrix element
corresponding to the diagrams in Fig. 1 is written as
1
A1 = i (2π )4
with
d4 p1
HA(S(a) + S(b))
N1 N1 N2
(1)
S(a) = −i gBB∗π gK B∗ Bs gαβ ( p1 + q)α(2 P − p1 − q)β
× F (m1, p1)F (m2, p2)F 2(m B∗ , q ),
S(b) = −2i gK K∗π gK∗ BBs gαβ ( p1 + q)α(2 P − p1 − q)β
where N1 = p12 − m21 + i ε, N1 = q 2 − mq2 + i ε, N2 =
p2 − m2
2 2 + i ε and P stands for the momentum of X (5568).
(mi + )2−m2
The form factor F (mi , k2) = (mi + )2−k2i is introduced to
compensate the off-shell effect caused by the intermediate
meson of mass mi and momentum k. The concerned
normalized wavefunction of the decaying meson with the assigned
quantum numbers is included in the vertex function H which
is invariant in the four-dimensional space-time. In fact, for
a practical computation their exact forms are not necessary,
because after integrating over d p1− the integral is reduced
into a three-dimensional one, and then H is replaced by h
whose explicit form is calculable in the light-front frame. In
that frame the momentum pi is written in terms of its
components as ( pi−, pi+, pi ⊥) and integrating out p1− with the
method given in Ref. [
50
] one has
X(5568)
X(5568)
B(p1)
K(p2)
where M is the mass of the decaying meson and M is the
mass of the heavier one of the two produced mesons. In the
expression, q is the four-momentum of the lighter meson
of the decay products, while calculating the hadronic
transition matrix element, we deliberately let q2 vary within a
reasonable range, then while obtaining the partial width of
X (5568) → Bs π , we set q2 to be the on-shell mass of the
produced pion as m2π . The factor √x1x2(M 2 − M02) in h A
was introduced in literature [
52
]. The explicit expressions of
the effective form factors h A are presented in the Appendix
for readers’ convenience.
Since we calculate the transition in the q+ = 0
reference frame the zero mode contributions which come from the
residues of virtual pair creation processes, were not included.
To involve them, p1μ and p1ν in sa must be replaced by
appropriate expressions as discussed in Ref. [
52
], that is
p1μ → Pμ A(11) + qμ A(21)
where P = P + P and q = P − P with P and P denoting
the momenta of the concerned mesons in the initial and final
states respectively.
For example, S(a) turns into a replaced form as
Sˆ(a) =
− m12 + (1 + A(11) + A(12)) M2 − M 2
+ 3 A(11) M 2 − A(12) M 2 − N 1 − A(11) q2 − A(12) q2
× −i gBmB∗Bπ∗g2K B∗ Bs F (m1, p1)F (m2, p2)F 2(m B∗ , q ).
(3)
(4)
Some notations such as Ai( j) and M0 can be found in Ref.
[
52
]. With the replacement the amplitude A can be calculated
numerically.
K(p1)
B(p2)
2.2 The decay rate of Xa → Ds π 0
Now we turn to study the decays of molecules with an open
charm. The formulas are similar to that in the case of
openbottom molecules.
Due to the quark structure, decay Xa → Ds π 0 realizes via
strong interaction. The supposed molecular state D K (Xa ) is
structured as √12 (D0 K + + D+ K 0). The Feynman diagrams
for Xa → Ds π 0 are shown in Fig. 2. The corresponding S(a)
and S(b) are
S(a) = −i gDD∗π gK D∗ Ds gαβ ( p1 + q)α(2 P − p1 − q)β
× F (m1, p1)F (m2, p2)F 2(m D∗ , q ),
S(b) = −2i gK K∗π gK∗ DDs gαβ ( p1 + q)α(2 P − p1 − q)β
× F (m1, p1)F (m2, p2)F 2(m K ∗ , q ),
2.3 The decay rate of Xb → Bcπ 0
The molecular state B D (Xb) is structured as √12 (D¯ 0 B− +
D− B0). The Feynman diagrams for the decay Xb → Bcπ 0
is shown in Fig. 3. The corresponding S(a) and S(b) should
be modified as
S(a) = −i gBB∗π gDB∗ Bc gαβ ( p1 + q)α(2 P − p1 − q)β
× F (m1, p1)F (m2, p2)F 2(m B∗ , q ),
S(b) = −i gDD∗π gD∗ BBc gαβ ( p1 + q)α(2 P − p1 − q)β
× F (m1, p1)F (m2, p2)F 2(m D∗ , q ).
3 Numerical results
3.1 For X (5568) → Bs π +
In this subsection we present our predictions on the decay
rate of X (5568) → Bs π + while all the input parameters are
taken from relevant literatures.
First, we need to calculate the corresponding amplitude
which was deduced in last section. The formula include some
parameters which need to be priori fixed. We use the central
Xa
Xb
value of the observed resonance peak 5.5678 GeV [
11
] as the
mass of X (5568). The masses of the involved mesons are set
as m B = 5.279 GeV, m Bs = 5.367 GeV, mπ = 0.139 GeV,
m B∗ = 5.325 GeV and mρ = 0.775 GeV according to the
data book [
58
]. The coupling constant gK∗ Kπ is 4.61 [
59
].
About the coupling constants gB∗ Bπ , gK∗ Bs B and gB∗ Bs K one
cannot fix them from the corresponding physical processes
at present but it is natural to conjecture that they would be
equal to gD∗ Dπ , gK∗ Ds D and gD∗ Ds K respectively under the
heavy quark limit and then they are set as 17.9 [
59
], 3.787
[
60
] and 2.02 [
61
] respectively. The cutoff parameter in
the vertex F was suggested to be 0.88 to 1.1 GeV [
62
]. In
our calculation we vary it from 0.88 to 1.1 GeV to study
how it affects the numerical results. β in the wavefunction
is a free parameter, even though so far it cannot be precisely
determined by phenomenological studies yet, its value can be
roughly estimated to fall within a certain range. We observe
that it should be close to the value for B meson which was
fixed as 0.5329 GeV.
Since the amplitude is derived in the reference frame of
q+ = 0 (q2 < 0) i.e. in the space-like region, we need to
extend it to the time-like region by means of a normal
procedure provided in literatures. In Ref. [
52
] a three-parameter
form factor as
A(q2) =
A(0)
1 − a
q2
MX2
− b
q2
MX2
2 ,
π(q)
Ds(p )
π(q)
Bc(p )
b
was employed in order to naturally extrapolate the formula
from the space-like region to the time-like (physical) region.
Let us turn to discuss the decays of an open-charm molecular
state via strong interaction.
K(p1)
D(p2)
D(p1)
B(p2)
D∗(q )
Table 1 The amplitude of X (5568) → Bs π+ with three parameters
( = 0.88 GeV,)
β (GeV−1)
The resultant form factors are listed in Table 1 and the
dependence of the corresponding decay width (X (5568) →
Bs π +) on β is illustrated in Fig. 4. By the results, we
notice that the model parameter β affects the numerical
results within a tolerable range. We also explore the change
of the decay width for different values when one sets
β = 0.5329 GeV. Since the channel X (5568) → Bs π +
is the dominant portal the theoretical estimation supports the
allegation that X (5568) is a molecular B K state, especially
when = 0.88GeV and β = 0.5329 GeV the estimated
decay width (X (5568) → Bs π +) is close to the
experimentally measured total width (Table 2).
(5)
3.2 Xa → Ds π 0
Fig. 4 The dependence of (X (5568) → Bs π +) on β
As the reduced mass of the D K system is slightly smaller
than that for the B K system, the corresponding kinematic
energy may be larger. Thus, with the same potential, naively,
one would expect a smaller binding energy than that for the
B K system (but not much because the reduced mass is closer
to the mass of the lighter constituent, i.e. the K -meson) where
the binding energy is determined to be 205 MeV as it is
considered as an X(5568) molecule. In our concrete numerical
computations, we let the adopted binding vary from 100 to
200 MeV.
The masses m D = 1.8696 GeV, m D∗ = 2.010 GeV and
m Ds = 1.968 GeV are taken from the Databook [
58
]. A naive
consideration suggests that the parameter β is close to that
for Ds which is 0.4395 GeV, meanwhile we set the cutoff
parameter to be 0.88 GeV which was obtained in
previous works. The mass variation covers a range from 2.164 to
2.264 GeV corresponding to the variation of binding energy
from 100 to 200 MeV. The results are shown in Table 3.
3.3 Xb → Bcπ 0
Since the D meson is heavier than K meson, assuming the
same arguments on the reduced mass, the binging energy of
the bound state of B D (Xb) might be larger than 205 MeV
which is the binding energy of B K . In our calculation
(Table 4) we let it vary from 160 to 240 MeV, which is a
typical energy range (close to QC D) for binding two mesons
into a compact system. m Bc = 6.2756 GeV is taken from Ref.
[
58
] and the parameter β adopted for a molecule with open
bottom and charm should be close to Bc. Although one
cannot fix it yet from a reliable source at present, we set it to be a
value between 0.631 and 1.257 GeV which are the β
parameters for J /ψ and ϒ respectively, namely we interpolate the
β value for Xb to be 0.944 GeV. The cutoff parameter is
set as 0.88 GeV. If the mass of B D molecular state is close
to 6.929 GeV its width is estimated to be around 20 MeV.
4 Conclusion and discussions
Supposing X (5568) to be a molecular state made by B
and K mesons (B K ), we calculate the decay rate of
X (5568) → Bs π + in the light front model. Inside the
fourquark molecule, the two constituents interact by exchanging
corresponding mesons (scalar and/or vector). In this
phenomenological study, the model parameters and β are not
fully determined yet at present, so we vary them within a
reasonable range in the numerical computations.
Numerically when = 0.88 GeV and β = 0.5329 GeV are chosen,
we obtain the rate of X (5568) → Bs π as 20.28 MeV which
is consistent with the new data measured by the D0
collaboration = 18.6−+67..19(stat)−+33..85 MeV. The consistency
somewhat supports the allegation that X (5568) is a molecular state
composing of B and K mesons.
As long as X (5568) is a molecular state of B K one can
expect two similar states of D K and B D which are named
as Xa and Xb in this work. The widths of Xa and Xb are
estimated in the same theoretical framework as roughly 30 and
20 MeV respectively. The results do not sensitively depend
on the choices of the binding energies. It is worth of putting
effort to search for Xa → Ds π 0 and Xb → Bcπ 0 reactions
in sensitive experimental facilities. It is of obvious
theoretical significance, namely a definite conclusion would help to
clarify if such molecular states are favored by the Nature.
X (5568) is indeed facing an eccentric situation, namely,
the D0 collaboration reconfirmed their observation of
X (5568) at the channel Bs π ± whereas LHCb, CMS, ATLAS
and CDF collaborations all gave negative reports. The sharp
discrepancy might be due to a wrong experimental treatment,
but there is still a slim possibility that both measurements are
reasonable because all the measurements with negative
conclusion only gave upper bounds of the rate. Actually, one
should make a theoretical investigation towards the
mysterious exotic hadron, i.e independent of the experimental data
anyway. As a matter of fact, from the theoretical aspect, there
is no rule to forbid existence of a four-quark state with four
different flavors such as X (5568). Following the lessons we
learned from the structures of X, Y, Z exotic states, it is
natural to assume a molecular state composed of b¯, s, u, d¯ whose
main decay portal is Bs π . In this work we used the LFQM to
calculate the decay rate of such a molecule (X (5568)) into
Bs π , while another group [
31
] has also calculated this rate
based on the molecule assumption in terms of the Bethe–
Salpeter equation. Their results are qualitatively consistent
with ours and the data measured by the D0 collaboration.
Interesting, some theoretical groups calculated the decay rate
based on the tetraquark assumption and obtained results of
the same order of magnitude. All these theoretical studies
indicate that X (5568) still may exist, i.e the possibility
cannot be simply negated. However, the discrepancy between
the D0 collaborations with the others persists and must be
taken serious, a reasonable interpretation might be needed.
We believe that this mist would be clarified by the efforts of
both theorists and experimentalists soon.
Acknowledgements This work is supported by the National
Natural Science Foundation of China (NNSFC) under the Contract nos.
11375128 and 11675082.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
Appendix A: The vertex function of molecular state
The wavefunction of a molecular state with total spin J and
momentum P is [
57
]
|X ( P, J, Jz ) =
{d3 p˜1}{d3 p˜2} 2(2π )3δ3( P˜ − p˜1 − p˜2)
SSz ( p˜1, p˜2, λ1, λ2)
×
λ1
× F | B( p1, λ1)K ( p2, λ2) .
For 0+ molecular state of B K
SSz ( p˜1, p˜2, λ1, λ2) = C0ϕ(x , p⊥) ≡ hC0
where C0 is the normalization constants which can be fixed
by normalizing the state [
52
]
X ( P , J , Jz )|X ( P, J, Jz )
= 2(2π )3 P+δ3( P˜ − P˜ )δJ J δJZ JZ ,
and let the normalization
p⊥) = δL,L δL Z ,L Z hold.
C0 is fixed by calculating Eq. (A3)
d2x(d2π2p)3⊥ ϕ ∗
L ,L Z (x , p⊥)ϕL,L Z (x ,
(A1)
(A2)
(A3)
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