Spectroscopic parameters and decays of the resonance \(Z_b(10610)\)
Eur. Phys. J. C
Spectroscopic parameters and decays of the resonance Zb(10610)
S. S. Agaev 2
K. Azizi 0 1
H. Sundu 3
0 School of Physics, Institute for Research in Fundamental Sciences (IPM) , P. O. Box 193955531, Tehran , Iran
1 Department of Physics, Dogˇus ̧ University , AcibademKadiköy, 34722 Istanbul , Turkey
2 Institute for Physical Problems, Baku State University , 1148 Baku , Azerbaijan
3 Department of Physics, Kocaeli University , 41380 Izmit , Turkey
The resonance Zb(10610) is investigated as the diquarkantidiquark Zb = [bu][bd] state with spinparity J P = 1+. The mass and current coupling of the resonance Zb(10610) are evaluated using QCD twopoint sum rule and taking into account the vacuum condensates up to ten dimensions. We study the vertices Zbϒ (n S)π (n = 1, 2, 3) by applying the QCD lightcone sum rule to compute the corresponding strong couplings gZbϒ(nS)π and widths of the decays Zb → ϒ (n S)π . We explore also the vertices Zbhb(m P)π (m = 1, 2) and calculate the couplings gZbhb(m P)π and the widths of the decay channels Zb → hb(m P)π . To this end, we calculate the mass and decay constants of the hb(1 P) and hb(2 P) mesons. The results obtained are compared with experimental data of the Belle Collaboration.

1 Introduction
Discovery of the charged resonances which cannot be
explained as c¯c or b¯b states has opened a new page in the
physics of exotic multiquark systems. The first tetraquarks
of this family are Z ±(4430) states which were observed by
the Belle Collaboration in B meson decays B → K ψ π ±
as resonances in the ψ π ± invariant mass distributions
[
1
]. The masses and widths of these states were
repeatedly measured and refined. Recently, the LHCb
Collaboration confirmed the existence of the Z −(4430) structure
in the decay B0 → K +ψ π − and unambiguously
determined that its spin–parity is J P = 1+ [
2,3
]. They also
measured the mass and width of Z −(4430) resonance and
updated the existing experimental data. Two
charmoniumlike resonances, Z1(4050) and Z2(4250), were discovered
by the Belle Collaboration in the decay B¯ 0 → K −π +χc1,
emerging as broad peaks in the χc1π invariant mass
distribution [4].
Famous members of the charged tetraquark family Zc±
(3900) were observed by the BESIII Collaboration in the
process e+e− → J /ψ π +π − as resonances with J P = 1+ in the
J /ψ π ± mass distribution [
5
]. The charged state Zc(4020)
was also found by the BESIII Collaboration in two different
processes, e+e− → hcπ +π − and e+e− → (D D¯ )±π ∓
(see Refs. [
6,7
]).
There is another charged state, namely the Zc(4200)
resonance which was detected and announced by Belle [
8
].
All aforementioned resonances belong to the class of the
charmoniumlike tetraquarks, and contain a c¯c pair and
light quarks (antiquarks). They were mainly interpreted as
diquark–antidiquark systems or bound states of D and/or D
mesons.
It is remarkable that the bcounterparts of the
charmoniumlike states, i.e. charged resonances composed of a b¯b pair
and light quarks were found as well. Thus, the Belle
Collaboration discovered the resonances Zb(10610) and
Zb(10650) (hereafter, Zb and Zb, respectively) in the decays
ϒ (5S) → ϒ (n S)π +π −, n = 1, 2, 3 and ϒ (5S) →
hb(m P)π +π −, m = 1, 2 [
9,10
]. These two states with
favored spin–parity J P = 1+ appear as resonances in the
ϒ (n S)π ± and hb(m P)π ± mass distributions. The masses
of the Zb and Zb resonances are
m = (10607.2 ± 2.0) MeV,
m = (10652.2 ± 1.5) MeV,
respectively. The width of the Zb state averaged over five
decay channels equals = (18.4 ± 2.4) MeV, whereas
the average width of Zb is = (11.5 ± 2.2) MeV.
Recently, the dominant decay channel of Zb, namely the
Zb → B+ B¯ 0 + B¯ 0 B + process, was also observed [11].
In this work fractions of different channels of Zb and Zb
resonances were reported as well. Further information on the
(1)
experimental status of the Zb and Zb states and other heavy
exotic mesons and baryons can be found in Ref. [
12
].
The existence of hiddenbottom states, i.e. of the Zb
resonances, was foreseen before their experimental
observation. Thus, in Ref. [
13
] the authors suggested to look for
the diquark–antidiquark systems with bb¯ud¯ content as peaks
in the invariant mass of the ϒ (1S)π and ϒ (2S)π systems.
The existence of the molecular state B B¯ was predicted in
Ref. [
14
].
After discovery of the Zb resonances theoretical studies
of the charged hiddenbottom states became more intensive
and fruitful. In fact, articles devoted to the structures and
decay channels of the Zb states encompass all existing
models and computational schemes suitable to study the
multiquark systems. Thus, in Refs. [
15,16
] the spectroscopic and
decay properties of Zb and Zb were explored using the heavy
quark symmetry by modeling them as J = 1 Swave
molecular B B¯ –B B¯ states and B B¯ , respectively. The existence
of similar states with quantum numbers 0+, 1+, 2+ was
predicted as well. The diquark–antidiquark interpretation of the
Zb states was proposed in Refs. [
17,18
]. It was demonstrated
that Belle results on the decays ϒ (5S) → ϒ (n S)π +π −
and ϒ (5S) → hb(m P)π +π − support Zb resonances as
diquark–antidiquark states. This analysis is based on a
scheme for the spin–spin quark interactions inside diquarks
originally suggested and successfully used to explore
hiddencharm resonances [
19
].
The Zb resonance was considered in Ref. [
20
] as a B B¯
molecular state, where its mass was computed in the
context of the QCD sum rule method. The prediction for the
mass m B B¯ = 10.54 ± 0.22 GeV obtained there allowed the
authors to conclude that Zb could be a B B¯ molecular state.
Similar conclusions were also drawn in the framework of the
chiral quark model. Indeed, in Ref. [
21
] the B B¯ and B B¯
bound states with J PC = 1+− were studied in the chiral
quark model, and found to be good candidates for the Zb and
Zb resonances. Moreover, the existence of molecular states
B B¯ with J PC = 1++, and B B¯ with J PC = 0++, 2++
was predicted. Explorations performed using the one
bosonexchange model also led to the molecular interpretations of
the Zb and Zb resonances [
22
]. However, an analysis carried
out in the framework of the Bethe–Salpeter approach
demonstrated that two heavy mesons can form an isospin singlet
bound state but cannot form an isotriplet compound. Hence,
the Zb resonance presumably is a diquark–antidiquark, but
not a molecular state [
23
].
Both the diquark–antidiquark and the molecular pictures
for the internal organization of Zb and Zb within the QCD
sum rules method were examined in Ref. [
24
]. In this work
the authors constructed different interpolating currents with
I G J P = 1+1+ to explore the Zb and Zb states and
evaluate their masses. Among alternative interpretations of the
Zb states it is worth to note Refs. [
25,26
], where the peaks
observed by the Belle Collaboration were explained as cusp
and coupling channel effects, respectively.
Theoretical studies that address problems of the Zb states
are numerous (see Refs. [
27–36
]). An analysis of these and
other investigations can be found in Refs. [
37,38
].
As is seen, the theoretical status of the resonances Zb and
Zb remains controversial and deserves further and detailed
explorations. In the present work we are going to calculate the
spectroscopic parameters of Zb = [bu][bd] state by
assuming that it is a tetraquark state with a diquark–antidiquark
structure and positive charge. We use QCD twopoint sum
rules to evaluate its mass and current coupling by taking
into account vacuum condensates up to ten dimensions. We
also investigate five observed decay channels of the Zb
resonance employing QCD sum rules on the lightcone. As a
byproduct, we derive the mass and decay constant of the
hb(m P), m = 1, 2 mesons.
This work has the following structure: In Sect. 2 we
calculate the mass and current coupling of the Zb
resonance. In Sect. 3 we analyze the decay channels Zb →
ϒ (n S)π, n = 1, 2, 3, and we calculate their widths.
Section 4 is devoted to an investigation of the decay modes
Zb → hb(m P)π, m = 1, 2, and it consists of two
subsections. In the first subsection we calculate the mass and
decay constant of the hb(1 P) and hb(2 P) mesons. To this
end, we employ the twopoint sum rule approach by
including into the analysis condensates up to eight dimensions.
In the next subsection using the parameters of the hb(m P)
mesons we evaluate the widths of the decays under
investigation. The last section is reserved for an analysis of the
obtained results and a discussion of possible interpretations
of the Zb resonance.
2 Mass and current coupling of the Zb state: QCD
twopoint sum rule predictions
In this section we derive QCD sum rules to calculate the mass
and current coupling of the Zb state by suggesting that it
has a diquark–antidiquark structure with quantum numbers
I G J P = 1+1+. To this end, we begin from the twopoint
correlation function
μν ( p) = i
d4x ei px 0T { JμZb (x ) JνZb†(0)}0 ,
(2)
where JμZb (x ) is the interpolating current for the Zb state
with the required quark content and quantum numbers.
It is possible to construct various currents to interpolate
the Zb and Zb resonances. One of them is a [ub]S=0[db]S=1–
[ub]S=1[db]S=0 type diquark–antidiquark current that is
used to consider the Zb state,
uaT (x )C γ5bb(x )
dd (x )γμC beT (x )
− uaT (x )C γμbb(x )
dd (x )γ5C beT (x )
.
(3)
The current for Zb can be defined in the form
Z
Jμ b (x ) = √˜ εμναβ uaT (x )C γ ν bb(x )
2
×Dα dd (x )γ β C beT (x ) ,
where Dα = ∂α − i gs Aα(x ) [
24
]. In Eqs. (3) and (4) we
have introduced the notation = abc and ˜ = dec. In above
expressions a, b, c, d and e are color indices, and C is the
charge conjugation matrix.
By choosing different currents to interpolate the Zb and Zb
resonances one treats both of them as groundstate particles
in the corresponding sum rules. We also follow this approach
and use the current JμZb (x ) to calculate the mass and current
coupling of the Zb state. To find the QCD sum rules we
first have to calculate the correlation function in terms of the
physical degrees of freedom. To this end, we saturate μν ( p)
with a complete set of states with the quantum numbers of
the Zb resonance and perform in Eq. (2) an integration over
x to get
Pμhνys( p) =
0 JμZb Zb( p) Zb( p) JνZb†0
m2Zb − p2
where m Zb is the mass of the Zb state, and dots indicate the
contributions of higher resonances and continuum states. We
define the current coupling f Zb through the matrix element
0 JμZb Zb( p) = f Zb m Zb εμ,
with εμ being the polarization vector of Zb state. Then in
terms of m Zb and f Zb , the correlation function can be written
in the following form:
m2Zb f Z2b
Pμhνys( p) = m2Zb − p2
−gμν +
(9)
(10)
(11)
interpolating current JμZb and quark propagators. After
contracting in Eq. (2) the bquark and light quark fields we get
×γ5 Sbbb (x ) Tr γμ Sbe e(−x )γν Sdd d (−x )
−Tr γμ Sbe e(−x )γ5 Sdd d (−x ) Tr γν Suaa (x )
× γ5 Sbbb (x ) − Tr γ5 Sua a(x )γμ Sbb b(x )
×Tr γ5 Sbe e(−x )γν Sdd d (−x ) + Tr γν Suaa (x )
× γμ Sbbb (x ) Tr γ5 Sbe e(−x )γ5 Sdd d (−x ) ,
where
Sb(q)(x ) = C Sbij(Tq)(x )C.
i j
In the expressions above Sqab(x ) and Sbab(x ) are the light u,
and d, and the heavy bquark propagators, respectively. We
choose the light quark propagator Sqab(x ) in the form
x/ mq qq
Sqab(x ) = i δab 2π 2x 4 − δab 4π 2x 2 − δab 12
+i δab
x/mq qq
48
x 2
− δab 192 q gsσ Gq
x 2x/mq q gsσ Gq
+i δab 1152
αβ
i gsGab
− 32π 2x 2 x/σαβ + σαβ x/ − i δab
−δab
x 4 qq gs2G2
27648
For the bquark propagator Sbab(x ) we employ the expression
Sbab(x) = i
d4k
(2π)4 e−ikx δab (k/ + mb)
k2 − mb2
−
gs Gaαβb σαβ (k/ + mb) + (k/ + mb) σαβ
4 (k2 − m2b)2
+ gs21G22 δabmb (k2 − mb2)4 +
k2 + mbk/
gs3G3 (k/ + mb)
48 δab (k2 − mb2)6
× k/ k2 − 3m2b + 2mb 2k2 − mb2 (k/ + mb) + . . . .
pμ pν
m2Zb
At the next stage we derive the theoretical expression for
the correlation function μQνCD( p) in terms of the quark–
gluon degrees of freedom. It can be determined using the
where A, B, C = 1, 2 . . . 8. In Eq. (11) t A = λA/2, λA are
the GellMann matrices, and the gluon field strength tensor
GαAβ ≡ GαAβ (0) is fixed at x = 0.
The QCD sum rule can be obtained by choosing the same
Lorentz structures in both Pμhνys( p) and μQνCD( p). We work
with terms ∼ gμν , which do not contain the effects of spin0
particles. The invariant amplitude QCD( p2) corresponding
to this structure can be written down as the dispersion integral
QCD( p2) =
∞ ρQCD(s)
4m2b s − p2 ds + . . . ,
where ρQCD(s) is the corresponding spectral density. It is
a key ingredient of sum rules for m2Zb and f Z2b and can be
obtained using the imaginary part of the invariant amplitude
QCD( p2). Methods of such calculations are well known and
presented numerously in the literature. Therefore, we omit
further details, emphasizing only that ρQCD(s) in the present
work is calculated by including into the analysis quark, gluon
and mixed condensates up to ten dimensions.
After applying the Borel transformation on the
variable p2 to QCD( p2), equating the expression obtained to
B Phys( p), and subtracting the continuum contribution, we
obtain the required sum rules. Thus, the mass of the Zb state
can be evaluated from the sum rule
m2Zb =
4sm02b dssρQCD(s)e−s/M2
4sm02b dsρQCD(s)e−s/M2 ,
whereas for the current coupling f Zb we employ the formula
1
f 2
Zb = m2Zb 4m2b
s0
dsρQCD(s)e(m2Zb −s)/M2 .
The sum rules for m Zb and f Zb depend on different
vacuum condensates stemming from the quark propagators, on
the mass of the bquark, and on the Borel variable M 2 and
continuum threshold s0, which are auxiliary parameters of
the numerical computations. The vacuum condensates are
(12)
(13)
(14)
parameters that do not depend on the problem under
consideration: their numerical values extracted once from some
processes are applicable in all sum rule computations. For
quark and mixed condensates in the present work we employ
q¯q = −(0.24 ± 0.01)3 GeV3, q gsσ Gq = m20 q¯q ,
where m2
0 = (0.8 ± 0.1) GeV2, whereas for the gluon
condensates we utilize αsG2/π = (0.012 ± 0.004) GeV4,
gs3G3 = (0.57 ± 0.29) GeV6. The mass of the b−quark
can be found in Ref. [
46
]: it is equal to mb = 4.18+−00..0043 GeV.
The choice of the Borel parameter M 2 and continuum
threshold s0 should obey some restrictions of sum rule
calculations. Thus, the limits within of which M 2 can be varied
(working window) are determined from convergence of the
operator product expansion and dominance of the pole
contribution. In the working window of the threshold parameter
s0 the dependence of the quantities on M 2 should be
minimal. In real calculations, however, the quantities of interest
depend on the parameters M 2 and s0, which affects the
accuracy of the extracted numerical values. Theoretical errors in
sum rule calculations may amount to 30% of the predictions
obtained, and a considerable part of these ambiguities are
connected namely with the choice of M 2 and s0.
An analysis performed in accordance with these
requirements allows us to fix the working windows for M 2 and s0:
M 2 = 9 − 12 GeV2, s0 = 123 − 127 GeV2.
(15)
In Figs. 1 and 2 we demonstrate the results of the numerical
computations of the mass m Zb and the current coupling f Zb
as functions of the parameters M 2 and s0. As is seen, m Zb
and f Zb are rather stable within the working windows of the
auxiliary parameters, but there is still a dependence on them
in the plotted figures. Our results for m Zb and f Zb read
m Zb = 10581+−114624 MeV, f Zb = (2.79+−00..5655) × 10−2 GeV4.
(16)
9.5
10.0
10.5
11.0
11.5
12.0
124
126
127
10.0
9.0
11.0
10.8
V10.6
e
G
Zb10.4
m
10.2
10.0
123
Within theoretical errors m Zb is in agreement with
experimental measurements of the Belle Collaboration (1). The
mass and current coupling Zb given by Eq. (16) will be used
as input parameters in the next sections to find the widths of
the decays Zb → ϒ (n S)π and Zb → hb(m P)π .
3 Decay channels Zb → ϒ(nS)π, n = 1, 2, 3.
This section is devoted to the calculation of the width of
the Zb → ϒ (n S)π, n = 1, 2, 3, decays. To this end we
determine the strong couplings gZbϒnπ , n = 1, 2, 3 (in the
formulas we utilize ϒn ≡ ϒ (n S)) using QCD sum rules on
the lightcone in conjunction with the ideas of the softmeson
approximation.
We start from an analysis of the vertices Zbϒnπ aiming to
calculate gZbϒnπ , and therefore we consider the correlation
function
μν ( p, q) = i
d4x ei px π(q) T { Jμϒ (x ) JνZb†(0)} 0 ,
where
Jμϒ (x ) = bi (x )γμbi (x )
is the interpolating current for mesons ϒ (n S). Here p, q
and p = p + q are the momenta of ϒ (n S), π and Zb,
respectively.
To derive the sum rules for the couplings gZbϒnπ , we
calculate μν ( p, q) in terms of the physical degrees of freedom.
It is not difficult to obtain
(17)
(18)
Pμhνys( p, q) =
3
n=1
×
0 J ϒ ϒn ( p)
μ
p2 − m2ϒ(nS)
Zb( p ) JνZb†0
For calculation of the strong couplings we choose to work
with the structure ∼ gμν . To this end, we have to isolate
the invariant function Phys( p2, p 2) corresponding to this
structure and find its double Borel transformation. But it is
well known that in the case of vertices involving a tetraquark
and two conventional mesons one has to set q = 0 [
39
]. This
is connected with the fact that the interpolating current for
the tetraquark is composed of four quark fields and, after
contracting two of them in the correlation function μν ( p, q)
with the relevant quark fields from the heavy meson’s
current, we encounter a situation where the remaining quarks are
located at the same spacetime point. These quarks fields,
sandwiched between a light meson and vacuum instead of
generating light meson’s distribution amplitudes, create the
local matrix elements. Then, in accordance with the
fourmomentum conservation at such vertices, we have to set
q = 0. In QCD lightcone sum rules the limit q → 0, when
the lightcone expansion reduces to a shortdistant
expansion over local matrix elements, is known as the “softmeson
approximation”. The mathematical methods to handle the
softmeson limit were elaborated in Refs. [
40,41
] and were
successfully applied to tetraquark vertices in our previous
articles [
42–45
]. In the soft limit p = p and the relevant
invariant amplitudes in the correlation function depend only
on one variable: p2. In the present work we use this approach,
which implies calculation of the correlation function with
equal initial and final momenta, p = p, and dealing with
the double pole terms obtained.
In fact, in the limit p = p we replace in Eq. (21)
1
,
where α and β are the spinor indices. We continue and use
the expansion
ua dd 1 j
α β → 4 βα u
a j dd ,
where m2n = (m2Zb + m2ϒn )/2, and we carry out the Borel
transformation over p2. Then for the Borel transformation
of Phys( p2) we get
B
Phys( p2) =
Now one has to derive the correlation function in terms of
the quark–gluon degrees of freedom and find the QCD side
of the sum rules. Contracting of the heavy quark fields in Eq.
(17) yields
μQνCD( p, q) =
d4x ei px √2
γ5 Sbib(x )γμ
×Sbei (−x )γν + γν Sbib(x )γμ Sbei (−x )γ5
× π(q)uaα(0)dβd (0)0 ,
(22)
αβ
(23)
(24)
where j is the full set of Dirac matrices
j = 1, γ5, γλ, i γ5γλ, σλρ /√2.
Replacing ua dβd in Eq. (23 ) by this expansion and
performα
ing summations over color indices it is not difficult to
determine local matrix elements of the pion which contribute to
μQνCD( p, q) (see Ref. [
39
] for details). It turns out that in the
soft limit the pion’s local matrix element which contributes
to Im μQνCD( p, q = 0) is
(25)
(26)
(27)
(28)
0d(0)i γ5u(0)π(q) = fπ μπ ,
where
m2π
μπ = mu + md .
After fixing in Im μQνCD( p, q = 0) the structure ∼ gμν it is
straightforward to extract ρϒQCD(s) as a sum of the
perturbative and nonperturbative components:
fπ μπ
ρQCD(s) = 12√2
ϒ
ρpert.(s) + ρn.−pert.(s) .
The ρQCD(s) can be obtained after the replacement mc →
ϒ
mb from the spectral density of Zc → J /ψ π decay
calculated in Refs. [
39,45
]. Its perturbative component ρpert.(s)
has a simple form and reads
The nonperturbative contribution ρn.−pert.(s) depends on the
vacuum expectation values of the gluon operators and
contains terms of four, six and eight dimensions. Its explicit
expression was presented in the appendix of Ref. [
45
].
The continuum subtraction in the case under consideration
can be done using quark–hadron duality, which leads to the
desired sum rule for the strong couplings. We get
3
n=1
=
s0
4m2b
gZbϒnπ fϒn f Zb m Zb mϒn mn2 e−mn2/M2
M 2
dse−s/M2 ρQCD(s).
ϒ
Here some comments are in order on the expression obtained,
Eq. (28). It is well known that the soft limit considerably
simplifies the QCD side of lightcone sum rule expressions [
40
].
At the same time, in the limit q → 0 the phenomenological
side of the sum rules gains contributions which are not
suppressed relative to a main term. In our case the main term
corresponds to vertex Zbϒ (1S)π , where the tetraquark and
mesons are groundstate particles. Additional contributions
emerge due to vertices Zbϒ π where some of particles (or all
of them) are on their excited states. In Eq. (28) terms
corresponding to vertices Zbϒ (2S)π and Zbϒ (3S)π belong to
this class of contributions. When we are interested in
extraction of parameters of a vertex built of only groundstate
particles, these additional contributions are undesired
contaminations which may affect the accuracy of the calculations.
A technique to eliminate them from sum rules is also well
known [
40,41
]. To this end, in accordance with elaborated
recipes one has to act by the operator
P(M 2, mn2) =
1 − M 2 d
d M 2
M 2emn2/M2
(29)
to Eq. (28). In the present work we are going to evaluate three
strong couplings gZbϒnπ and therefore we use the original
form of the sum rule given by Eq. (28). But it provides only
one equality for three unknown quantities. In order to get two
additional equations we act by operators d/d(−1/M 2) and
d2/d(−1/M 2)2 to both sides of Eq. (28) and solve obtained
equations to find gZbϒnπ .
The widths of the decays Zb → ϒ (n S)π, n = 1, 2, 3
can be calculated applying the standard methods and have
the same form as in the case of the decay Zc → J /ψ π .
After evident replacements in the corresponding formula we
get
(Zb → ϒnπ ) =
g2 m2
Zbϒnπ ϒn λ m Zb , mϒn , mπ
24π
× 3 +
2λ2 m Zb , mϒn , mπ
m2
ϒn
,
(30)
where
a4 + b4 + c4 − 2 a2b2 + a2c2 + b2c2
λ(a, b, c) =
2a
The key component in Eq. (30) is the strong coupling
gZbϒnπ . Relevant sum rules contain spectroscopic
parameters of the tetraquark Zb and the mesons ϒ (n S) and π . The
mass and current coupling of the Zb resonance have been
calculated in the previous section. For numerical computations
we take masses mϒn and decay constants fϒn of the mesons
ϒ (n S) from Ref. [
46
]. The relevant information is shown in
Table 1.
In our calculations the Borel parameter M 2 and continuum
threshold s0 are varied within the regions
.
M 2 = 10 − 13 GeV2, s0 = 124 − 128 GeV2,
(31)
which are almost identical to similar working windows in the
mass and current coupling calculations being slightly shifted
towards larger values.
Parameters
(32)
(33)
For the couplings gZbϒnπ we obtain (in GeV−1):
gZbϒ1π = 0.019 ± 0.005, gZbϒ2π = 0.090 ± 0.031,
For the widths of the decays Zb → ϒ (n S)π these couplings
lead to the predictions
The predictions obtained for the widths of the decays
(Zb → ϒ (n S)π ) are the final results of this section and
will be used for comparison with the experimental data.
4 Zb → hb(1 P )π and Zb → hb(2 P )π decays
The second class of decays which we consider contains two
processes Zb → hb(m P)π, m = 1, 2. We follow the same
prescriptions as in the case of Zb → ϒ (n P)π decays and
derive sum rules for the strong couplings gZbhbπ and gZbhbπ
(hereafter we employ the shorthand notations hb ≡ hb(1 P)
and hb ≡ hb(2 P)). From the analysis performed in the
previous section it is clear that the corresponding sum rules will
depend on numerous input parameters including mass and
decay constant of the mesons hb(1 P) and hb(2 P).
Information on the spectroscopic parameters of hb(1 P) is available
in the literature. Indeed, in the context of the QCD sum rule
method mass and decay constant of h(1 P) were calculated
in Ref. [
47
]. But the decay constant fhb of the meson hb(2 P)
was not evaluated; therefore, in the present work, we have
first to find the parameters mhb and fhb , and we shall turn
after that to our main task.
4.1 Spectroscopic parameters of the mesons hb(1 P) and
hb(2 P)
The meson h(1 P) is the spinsinglet Pwave bottomonium
with quantum numbers J PC = 1+−, whereas h(2 P) is its
first radial excitation. The parameters of the hb(1 P) and
hb(2 P) mesons in the framework of QCD twopoint sum
rule method can be extracted from the correlation function
μναβ ( p) = i
d4x ei px 0 T
Jμhν (x ) Jαhβ†(0) 0 , (34)
where the interpolating current for hb(m P) mesons is chosen
as
Jμhν (x ) = bi (x )σμν γ5bi (x ).
It couples both to hb(1 P) and hb(2 P), and is convenient for
the analysis of J PC = 1+− mesons (see Ref. [
47
]).
In order to find the required sum rules we use the
“groundstate+first radial excitation+continuum” scheme. Then the
physical side of the sum rule,
Phys
μναβ ( p) =
0 Jμhν hb( p) hb( p) Jαhβ†(0) 0
m2hb − p2
0 Jμhν hb( p) hb( p) Jαhβ†(0) 0
contains two terms of interest and also a contribution of
higher resonances and continuum states, denoted by dots.
We continue by introducing the matrix elements
0 Jμν h(b )( p) = fh( ) (εμ() pν − εν( ) pμ),
h
b
and we recast the correlation function
form
Phys hb
μναβ ( p) = m2hb − p2
f 2
gμα pν pβ − gμβ pν pα
Phys
μναβ ( p) into the
(35)
(36)
(37)
+ m2
hb − p2
−gνα pμ pβ + gνβ pμ pα
f 2
hb
gμα pν pβ
−gμβ pν pα − gνα pμ pβ + gνβ pμ pα ,
(38)
where
gμ(α) = −gμα +
pμ pα
m2h( )
b
.
The Borel transformation of Pμhνyαsβ ( p) can be obtained by
simple replacements in Eq. (38):
fh2( )
B m2h( ) − p2 = fh2(b ) e
b
b
−m2h( ) /M2
b
.
The expression obtained in this way contains numerous
Lorentz structures which, in general, may be employed to
derive the sum rules for masses and decay constants: We
choose a structure ∼ gμα pν pβ to extract sum rules. The
term with the same structure should be isolated in the Borel
transformation of μQνCαDβ ( p), i.e. in the expression of the
correlation function calculated using the quark–gluon degrees
of freedom.
After simple computations for
QCD
μναβ ( p) we get
QCD
μναβ ( p) = i
d4x ei px Tr γ5σαβ Sbji (−x )
×σμν γ5 Sbij (x ) .
The following operations are standard manipulations; they
imply the Borel transforming of μQνCαDβ ( p), equating the
structures ∼ gμα pν pβ in both the physical and the QCD
sides of the equality obtained, and subtracting the continuum
contribution. We obtain the second sum rule by acting on the
first one by d/d(−1/M 2). These two sum rules allow us to
evaluate the masses and decay constants of the hb(1 P) and
hb(2 P) mesons. At the first stage we employ the
“groundstate +continuum” scheme, which is commonly used in sum
rule computations. This means that we include the excited
hb(2 P) meson into the “higher resonances and continuum”
part of sum rules and fix working windows for M 2 and s0.
From these sum rules we extract spectroscopic parameters of
the hb(1 P) meson mhb and fhb . At the next step we employ
the same sum rules with s0∗ > s0 to include the contribution
arising from hb(2 P), and we treat mhb and fhb evaluated at
the first stage as fixed parameters.
Numerical analysis restricts the variation of the
parameters M 2 and s0 within the regions
M 2 = 10 − 12 GeV2, s0 = 103 − 105 GeV2,
and we find
mhb = 9886+−8718 MeV, fhb = 325+−6517 MeV.
At the next step we use
The parameters of the hb(2 P) meson are among essentially
new results of the present work; therefore, in Figs. 3 and 4
we demonstrate mhb(2P) and fhb(2P) as functions of the Borel
parameter M 2 and the continuum threshold s0.
10.6
M 2 10 GeV2
M 2 11 GeV2
M 2 12 GeV2
M 2 10 GeV2
M 2 11 GeV2
M 2 12 GeV2
we see a reasonable agreement between them.
4.2 Widths of decays Zb → hb(1 P)π and Zb → hb(2 P)π
Analysis of the vertices Zbhb(m P)π does not differ from the
analogous investigation carried out in the previous section.
We start here from the correlator
μνλ( p, q) = i
d4x ei px π(q)T { Jμhν (x ) JλZb†(0)}0 ,
and for its phenomenological representation get
(42)
. . . .
(43)
(44)
Pμhνyλs( p, q) =
0 Jμhν hb ( p)
p2 − m2hb
×
Zb( p ) JλZb†0
Pμhνyλs( p, q) contains two terms of interest and contributions
coming from higher resonances and continuum shown above
as dots. Using matrix elements of the currents Jμhν and JλZb
and introducing the vertex
h(b ) ( p) π(q)Zb( p ) = gZbh(b )π αβγ δεα∗ ( p)εβ ( p ) pγ pδ,
we find
The same correlation function expressed in terms of quark
propagators takes the following form:
μQνCλD( p, q) =
Expanding uaαdβd in accordance with Eq. (24) and substituting
into Eq. (46) the local matrix elements of the pion we obtain
μQνCλD( p, q), which can be matched to Pμhνyλs( p, q) to fix
the same tensor structures. In order to derive the sum rule
we use the structures ∼ μλγ δ pγ pδ pν from both sides of
the equality. The pion matrix element that contributes to this
structure is
0d(0)γ5γμu(0)π(q) = i fπ qμ.
In fact, it can be included into the chosen structure after
replacement qμ = pμ − pμ.
In the equality obtained we apply the soft limit q → 0
( p = p ) and perform the Borel transformation on the
variable p2. This operation leads to a sum rule for the two
strong couplings gZbhbπ and gZbhbπ . The second
expression is obtained from the first one by applying the operator
d/d(−1/M 2).
The principal output of these calculations, i.e. the spectral
density ρhQCD(s), reads
fπ
ρQCD(s) = 12√2
h
ρpert.(s) + ρn.−pert.(s) ,
where its perturbative part is given by the formula
.
The nonperturbative component of ρhQCD(s) includes
contributions up to eight dimensions and has the form
ρn.−pert.(s) =
+ gs3G3
m2b
0
Here the functions fk (z, s) are
.
In the expressions above the Dirac delta function δ(n)(s −
is defined in accordance with
)
δ(n)(s −
dn
) = dsn δ(s −
).
The widths of the decays Zb → hb(1 P)π and Zb →
hb(2 P)π are calculated using the formula
2
(Zb → hb(m P)π ) = gZbhb(m P)π
λ m Zb , mh(m P), mπ
12π
In our numerical computations we employ the parameters of
the hb(m P) mesons obtained in the previous subsection. The
working regions of the Borel parameter M 2 and continuum
threshold s0 are the same as in the analysis of the Zb →
ϒ (n S)π decays. Below we provide our results for the strong
couplings (in units GeV−1)
gZbhbπ = 0.94 ± 0.27, gZbhbπ = 3.43 ± 0.93.
(51)
In Fig. 5 we plot the coupling gZbhbπ as a function of the Borel
parameter and continuum threshold to show its dependence
on these auxiliary parameters. It is easy to see that theoretical
errors are within the limits accepted in sum rule calculations.
Using Eq. (51) it is not difficult we evaluate the widths of
the decays:
(Zb → hb(1 P)π ) = 6.30 ± 1.76 MeV,
(Zb → hb(2 P)π ) = 7.35 ± 2.13 MeV.
(50)
3
(52)
M 2 10.0 GeV2
M 2 11.5 GeV2
M 2 13.0 GeV2
0
10.0
The experimental data on the decay channels of the Zb(10610)
resonance were studied and presented in a rather detailed
form in Refs. [
9–11
]. Its full width was estimated as =
18.4 ± 2.4 MeV, an essential part of which, i.e.
approximately 86% of , is due to the decay Zb → B+ B∗0+ B∗+ B0.
The remaining part of the full width is formed by five decay
channels investigated in the present work. It is clear that our
results for the widths of the decays Zb → ϒ (n S)π and
Zb → hb(n P)π overshoot the experimental data.
Therefore, in the light of the present studies we refrain from an
interpretation of the Zb(10610) resonance as a pure diquark–
antidiquark [bu][bd] state.
Nevertheless, theoretical predictions are encouraging for
the ratios
R(n) =
(Zb → ϒ (n S)π )
(Zb → ϒ (1S)π ) , R(m) =
(Zb → hb(m S)π )
,
(Zb → ϒ (1S)π )
(53)
where we normalize the widths of the different decay
channels to (Zb → ϒ (1S)π ). The ratio R can be extracted from
the available experimental data and can be calculated from
the decay widths obtained in the present work. In order to fix
existing similarities and differences between theoretical and
experimental information on R we provide two sets of
corresponding values in Table 2. It is worth to note that we use
the latest available experimental information from Ref. [
46
].
Table 2 Experimental values and theoretical predictions for R
R
n = 2
n = 3
m = 1
m = 2
Exp. [
46
] 6.67+−32..1317 3.89−+12..5052 6.48−+23..1485 8.70−+34..4319
This work 12.63 ± 5.43 6.08 ± 2.76 4.63 ± 1.95 5.40 ± 2.32
It is seen that theoretical predictions follow the pattern of
the experimental data: we observe the same hierarchy of
theoretical and experimental decay widths. At the time, numerical
differences between them are noticeable. Nevertheless, as a
result of the large errors in both sets, there are sizable
overlap regions for each pair of Rs, which demonstrates not only
qualitative agreement between them but also the quantitative
compatibility of the two sets.
These observations may help one to understand the nature
of the Zb resonance. The Belle Collaboration discovered two
Zb and Zb resonances with very close masses. We have
calculated the parameters of an axialvector diquark–antidiquark
state [bu][bd], and we interpreted it as Zb. It is possible to
model the second Zb resonance using an alternative
interpolating current, as has been emphasized in Sect. 2, and we
explore its properties. The current with the same quantum
numbers but different color organization may also play a
role of such alternative (see, for example, Ref. [
44
]). One of
the possible scenarios implies that the observed resonances
are admixtures of these tetraquarks, which may fit measured
decay widths.
The diquark–antidiquark interpolating current used in the
present work can be rewritten as a sum of moleculartype
terms. In other words, some of the moleculartype currents
effectively contribute to our predictions, and by enhancing
these components (i.e. by adding them to the
interpolating current with some coefficients) better agreement with
the experimental data may be achieved. In other words, the
resonances Zb and Zb may “contain” both the diquark–
antidiquark and the molecular components.
Finally, the Zb and Zb states may have pure molecular
structures. But pure moleculartype bound states of mesons
are usually broader than diquark–antidiquarks with the same
quantum numbers and quark contents. In any case, all these
suggestions require additional and detailed investigations.
In the present study we have fulfilled only a part of this
program. In the framework of QCD sum rule methods we have
calculated the spectroscopic parameters of the Zb state by
modeling it as a diquark–antidiquark state, and we found the
widths of five of its observed decay channels. We have also
evaluated the mass and decay constant of te hb(2 P) meson,
which are necessary for analysis of the Zb → hb(2 P)π
decay. Calculation of the Zb resonance’s dominant decay
channel may be performed, for example, using the QCD
threepoint sum rule approach, which is beyond the scope
of the present work. The decays considered here involve the
excited mesons ϒ (n S) and h(m P), the parameters of which
require detailed analysis in the future. More precise
measurements of Zb and Zb partial decay widths can also help
in making a choice between the scenarios outlined.
Acknowledgements S. S. A. thanks T. M. Aliev for helpful
discussions. K. A. thanks TÜ BITAK for the partial financial support provided
under Grant No. 115F183.
Open Access This article is distributed under the terms of the Creative
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