#### \(B_c\rightarrow B_{sJ}\) form factors and \(B_c\) decays into \(B_{sJ}\) in covariant light-front approach

Eur. Phys. J. C
Bc → Bs J form factors and Bc decays into Bs J in covariant light-front approach
Yu-Ji Shi 1
Wei Wang 0 1
Zhen-Xing Zhao 1
0 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190 , China
1 INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao-Tong University , Shanghai 200240 , China
We suggest to study the Bs and its excitations Bs J in the Bc decays. We calculate the Bc → Bs J and Bc → BJ form factors within the covariant light-front quark model, where the Bs J and BJ denote an s-wave or p-wave b¯s and b¯d meson, respectively. The form factors at q2 = 0 are directly computed while their q2-distributions are obtained by extrapolation. The derived form factors are then used to study semileptonic Bc → (Bs J , BJ ) ¯ν decays, and nonleptonic Bc → Bs J π . Branching fractions and polarizations are predicted in the standard model. We find that the branching fractions are sizable and might be accessible at the LHC experiment and future high-energy e+e− colliders with a high luminosity at the Z -pole. The future experimental measurements are helpful to study the nonperturbative QCD dynamics in the presence of a heavy spectator and also of great value for the study of spectroscopy.
1 Introduction
In the past decades, there has been a lot of progress in
hadron spectroscopy, thanks to the well-operating
experiments including the e+e− colliders and hadron colliders.
The immense interest in spectroscopy is not only due to
the fact that one is able to find many missing hadrons to
complete the quark model, but more importantly due to the
observations of states that are unexpected in the simple quark
model. The latter ones are generally called hadron exotics.
A milestone in the exploration of the exotics is the
discovery of X (3872), first in B decays by Belle
Collaboration [1] and subsequently confirmed in many distinct
processes in different experiments [2–4]. It was found the
propa e-mail:
b e-mail:
c e-mail:
erties of this meson are peculiar. Since then the identification
of multiquark hadrons has become a focus topic in hadron
physics. Inspired by the discovery of X (3872), a number
of new interesting structures were discovered in the mass
region of heavy quarkonium. Refer to Refs. [5–8] for recent
reviews.
On the theoretical side, deciphering the underlying
dynamics of these multiquark states is a formidable challenge, and
is often based on explicit and distinct assumptions. In many
assumptions, the quarkonium-like states are usually
composed of a pair of heavy constituents, which makes it vital
to study first the heavy–light hadron. In the system with one
heavy charm quark, a series of important results have started
with the discoveries of the narrow states Ds (2317) in the
Ds+π 0 final state and Ds (2460) in the Ds∗π 0 and Ds γ final
state [9,10]. Along this line, a few other new states, such
as Ds1(2536), Ds2(2573), Ds (2710), have been observed at
the B factory and other facilities [11].
Bottomed hadrons are related to charmed mesons by heavy
quark symmetry. But compared to the charm sector, there is
less progress for the bottomed hadrons. In experiment, only
a few bottom–strange mesons are observed, most of which
are believed to be in agreement with the quark model. In this
paper, we propose to use the Bc decays and study the
spectrum of the Bs J . We observe a few advantages. First of all, the
large production rate of the Bc is in expectation, in particular
the LHCb will produce a number of Bc events and thus the
Bc → Bs J decays will have a large potential to be observed.
Second, the scale over the mW can be computed in the
perturbation theory and the QCD evolution between the mW and the
low energy scale mc is well organized by making use of the
renormalization group improved perturbation theory.
Consequently, the Bc decays into Bs and other excited states have
received some theoretical attention [12–30]. In the following
we will address the investigation of the production rates of
Bs J meson (an s-wave or p-wave b¯s hadron) in semileptonic
and nonleptonic Bc meson decays in the framework of the
covariant light-front quark model (LFQM) [31].
In the Bc → Bs J decays, the quark level transition is
c → s in which the heavy bottom quark acts as a spectator.
Since most of the momentum of the hadron is carried by
the spectator, there is no large momentum transfer and the
transition is dominated by the soft mechanism. A form factor
can then be expressed as an overlap of the wave functions of
the initial and final state hadrons. Treatments in quark models
like the LFQM are of this type.
As pointed out in Ref. [32], the light-front approach shows
some unique features which are suitable to handle a hadronic
bound state. The LFQM [33–36] can provide a
relativistic treatment of moving hadrons and give a full treatment
of hadron spins in terms of the Melosh rotation.
Lightfront wave functions, which characterize the hadron in terms
of their fundamental quark and gluon degrees of freedom,
are independent of hadron momentum and thus are Lorentz
invariant. Moreover, in covariant LFQM [31], the spurious
contribution which depends on the orientation of the light
front is elegantly eliminated by including zero-mode
contributions. This covariant model has been successfully extended
to study the decay constants and form factors of various
mesons [37–49]. Through this study of Bc → Bs J in LFQM,
we believe that one will not only obtain the information as
regards the decay dynamics in the presence of a heavy
spectator but will also provide a side-check for the classification
of the heavy–light mesons. It is also helpful toward the
establishment of a global picture of the heavy–light spectroscopy
including the exotic spectrum.
The rest of this paper is organized as follows. In Sect. 2,
we will give a brief description of the parametrization of form
factors, the framework of covariant LFQM, and the form
factor calculation in this model. We present our numerical results
for various transitions in Sect. 3. In Sect. 4, we use the form
factors to study semileptonic and nonleptonic Bc decays. In
this section, we will present our predictions for branching
fractions and polarizations. The last section contains a brief
summary.
2 Transition form factors in the covariant LFQM
2.1 Bc → Bs J form factors
The effective electroweak Hamiltonian for the Bc → Bs J l¯ν
reads
where G F and Vcs are the Fermi constant and Cabibbo–
Kobayashi–Maskawa matrix element, respectively. Leptonic
parts can be computed in perturbation theory while hadronic
contributions are parametrized in terms of form factors.
An s-wave meson corresponds to a pseudoscalar meson
or a vector meson, abbreviated as P and V , respectively. For
a p-wave meson, the involved state is a scalar S, an
axialvector A or a tensor meson T . In the following we introduce
the abbreviations P = P + P , q = P − P , and we adopt
the convention of 0123 = 1. The Bc → P, V form factors
can be defined as follows:
1
V ( P , ε )|Vμ|Bc( P ) = − m Bc + mV
× μναβ ε ∗ν Pαqβ V BcV (q2),
V ( P , ε )| Aμ|Bc( P ) = 2i mV
+i (m Bc + mV ) A1BcV (q2) εμ∗ − ε ∗ · q qμ
In analogy with Bc → V form factors, we parametrize the
Bc → T form factors as
2V BcT (q2) μνρσ (εT∗ )ν ( P )ρ ( P )σ ,
= − m Bc + mT
T ( P , ε )| Aμ|Bc( P ) = 2i mT εT∗q 2· q qμ A0BcT (q2)
+i (m Bc + mT ) A1BcT (q2) εT∗ μ − εT∗ · q qμ
The Bc → S, A form factors can be defined by exchanging
the vector and axial-vector current:
S( P )| Aμ|Bc( P ) = −i
A( P , ε )|Vμ|Bc( P ) = −2m A
−(m Bc + m A)V1Bc A(q2) εμ∗ − ε ∗ · q qμ
V2Bc A(q2) Pμ − m2B − m2A qμ ,
1
A( P , ε )| Aμ|Bc( P ) = − m Bc − m A
i
× μναβ ε ∗ν Pαqβ ABc A(q2).
The spin-2 polarization tensor can be constructed using the
standard polarization vector ε:
εμν ( P , ±2) = εμ(±)εν (±),
1
εμν ( P , ±1) = √ [εμ(±)εν (0) + εν (±)εμ(0)],
2
It is symmetric and traceless, and εμν P ν = 0. If the
recoiling meson is moving on the plus direction of the z axis, their
explicit structures are chosen as
where ET and | pT | are the energy and the momentum
magnitude of the tensor meson in the Bc rest frame, respectively.
2.2 Covariant light-front approach
In the covariant LFQM, it is convenient to use the light-front
decomposition of the momentum P = ( P −, P +, P ), with
P ± = P 0 ± P 3, and thus P 2 = P + P − − P⊥⊥2. The
incoming (outgoing) meson has the momentum P = p1+ p2
( P = p1 + p2) and the mass M (M ). The quark and
antiquark inside the incoming (outgoing) meson have the
mass m ( ) and m2, respectively. Their momenta are denoted
1
as p1( ) and p2 respectively. In particular these momenta can
be written in terms of the internal variables (xi , p ) by
⊥
with the momentum fractions x1 + x2 = 1. With these
internal variables, one can define some useful quantities for both
incoming and outgoing mesons:
M02 = (e1 + e2)2 =
M0 =
mi( )2 + p⊥2 + pz2, pz =
p⊥2 + m12
p⊥2 + m22 ,
− 2x2 M0
Table 1 Meson–quark–antiquark vertices used in the covariant LFQM.
In the case of the outgoing meson, one should use instead i(γ0ΓM†γ0)
for the corresponding vertices
Pseudoscalar ( 1 S0)
Scalar ( 3 P0)
Vector ( 3 S1)
Axial ( 3 P1)
Axial ( 1 P1)
Tensor ( 3 P2)
i 21 HT [γμ − W1T ( p1 − p2)μ]( p1 − p2)ν
−p2
Fig. 1 Feynman diagram for transition form factors, where the cross
symbol in the diagram denotes the transition current
The Feynman rules for meson–quark–antiquark vertices
can be derived using the conventional light-front approach,
whose forms for the s-wave and p-wave states are collected
in Table 1 [31,37]. An extension to the d-wave vertices has
been conducted in Ref. [50]. In the following we will take
the Bc → Bs transition as the example and illustrate the
calculation. To do so, we will consider the matrix element
P( P )|Vμ| P( P ) ≡ BμP P ,
whose Feynman diagram is shown in Fig. 1. It is
straightforward to obtain
1 N1 N1 N2
SVPμP = Tr[γ5( /p1 + m1)γμ( /p1 + m1)γ5( /p2 − m2)],
N1( ) = p1( )2 − m1( )2 + i ,
Here we consider the q+ = 0 frame. The p1− integration
picks up the pole p2 = pˆ2 = [( p22⊥ + m22)/ p2+, p2+, p2⊥]
and leads to
M0 2 =
p⊥2 + m12
p⊥2 + m22 ,
with p⊥ = p⊥ − x2q⊥. The explicit form of h P has been
derived in Refs. [31,37]
h P = (M 2 − M02)
Nc √2M0
where ϕ is the light-front momentum distribution amplitude
for s-wave meson. In practice, the following Gaussian-type
wave function can be adopted [31,37]:
ϕ = ϕ (x2, p⊥) = 4
As shown in Refs. [31,37], the inclusion of the so-called
zero-mode contribution in the above matrix elements in
practice amounts to the replacements
pˆ1μ =. Pμ A(11) + qμ A(21),
.
where the symbol = in the above equation reminds us that it
is true only in the BμP P integration. A(ji) and Z2, which are
functions of x1,2, p 2, p⊥ · q⊥, and q2, are listed in Appendix
⊥
A. After the replacements, we arrive at
−x1(m1 − m2)2 − x1(m1 − m2)2 ,
⊥ x2 Nˆ 1 Nˆ 1
×(x2m1 + x1m2) + 2 q · P
q2
M 2 − x2(q2 + q · P) − (x2 − x1)M 2
+2x1 M02 − 2(m1 − m2)(m1 + m1) .
Finally we get the form factors through the relations
Similarly, one can derive the other form factors, whose
expressions are collected in Appendix A.
Before closing this section, it is worth mentioning that the
axial-vector mesons may not be classified as 3 P1 or 1 P1 state.
In the quark limit with m Q → ∞, the QCD interaction is
independent of the heavy quark spin and thus it will decouple
with the light system. A consequence of this decoupling is
that heavy mesons are classified into multiplets labeled by
the total angular momentum of the light degrees of freedom.
The s-wave pseudoscalar and vector states are in the same
multiplets denoted as sl = 1/2. For the p-wave states, two
kinds of axial-vector mesons P13/2 and P1/2 are mixtures of
1
3 P1 or 1 P1:
Since the form factors involving P3/2 and P1/2 can be
1 1
straightforwardly obtained by the linear combination for
those given above, we shall calculate the form factors using
the 2S+1 L J basis in the following analysis.
3 Numerical results for form factors
3.1 Input parameters
In the covariant LFQM, the constituent quark masses are used
as (in units of GeV):
which have been widely used in various B and Bc decays [42–
49]. The masses of the Bc and Bs J are taken from the PDG
(in units of GeV) [11]:
m Bs2 = 5.840,
while for the Bs0 and Bs1, we quote the results [51,52] (see
also estimates in Refs. [53–55]):
Since the masses of the P1/2 and P3/2 are close to the
1 1
observed state Bs1(5830) [11],
m Bs1(5830) = 5.829 GeV,
we use the same value for both the 3 P1 and the 1 P1 state.
The parameter β, characterizing the momentum
distribution, is usually determined by fitting the meson decay
constant. For instance, in this approach the pseudoscalar and
vector meson’s decay constants read
F (q2) =
F (q2) =
1 − a mq22H + b( mq22H )2
1.04∗ 0.37∗
0.75∗ 0.95∗
3.24∗ 9.56∗
BJ form factors, for
Table 2 Bc → Bs , Bs∗, Bs0, Bs1, Bs1, and Bs2 form factors in the
lightfront quark model, which are fitted using Eq. (32), while for the form
factors with an asterisk, the parametrization in Eq. (36) is adopted
A1Bc Bs∗
V Bc Bs2 18.60
2.28∗ 2.08∗
A2Bc Bs∗
We will also calculate the Bc →
which we use the masses [51,52]
The above result is derived from the decay constant result
[60],
f B = (193 ± 6) MeV.
3.2 Form factors and momentum transfer distribution
With the inputs in the previous subsection, we can predict
the Bc → Bs , Bs∗, Bs0, Bs1, Bs1, and Bs2 form factors in the
LFQM and we show our results in Table 2. In order to access
the q2 distribution, one may adopt the fit formula:
h P
⊥ x1x2(M 2 − M02) 4(m1x2 +m2x1),
× x1 M02 − m1(m1 − m2) − p 2
⊥ +
For the Bc meson, the decay constant can in principle be
determined by leptonic and radiative-leptonic decays [56–
59], both of which show lack of experimental data yet.
Twoloop contributions in the NRQCD framework have been
calculated in Ref. [58] and the authors have found
f Bc = 398 MeV.
We will adopt this result, but it is necessary to note that
the above value is smaller than the lattice QCD result by
approximately 2σ : f Bc = (434 ± 15) MeV. We use the
recent lattice QCD result for the Bs decay constant with
N f = 2 + 1 + 1 [60],
f Bs = (229 ± 5) MeV.
This is close to the previous lattice QCD result [61,62]:
f Bs = (224 ± 5) MeV. Using the decay constants, the shape
parameters are fixed as
βBc = 0.886 GeV, βBs = 0.623 GeV,
and we assume that the values of β for the other Bs J mesons
are approximately equal to that for the Bs , that is,
βBs∗ = βBs0 = βBs1 = βBs1 = βBs2 = 0.623 GeV.
In the literature, the dipole form has been used to
parametrize the q2 distribution:
with the m H = m D for D decays and m H = m B for B
decays. This parametrization is inspired by analyticity.
Taking the F1B→π as an example, we consider the timelike matrix
element
Fig. 2 The Bc → Bs form
factors and their q2-dependence
parameterized by Eqs. (32) and
(36)
0|u¯γ μb|π(− pπ )B( pB ) ∼
For the ABc Bs∗ , F0Bc Bs0 , V0B,1c Bs1 , and ABc Bs2 , we found that
2 2
the fitted values for the m2fit are negative, and thus we use the
following formula:
where the one-particle contribution has been singled out. The
lowest resonance that can contribute is the vector B∗. This
leads to the pole structure at large q2:
F (q2) =
F1(0)
F1B→π (q2) ∼ 1 − q2/m2B∗ .
Except the pole at m B∗ , there are residual dependences on
q2, which can be effectively incorporated into the a, b of the
dipole parametrization as shown in Eq. (33). However, for
the Bc → Bs J transition, one cannot simply apply Eq. (33),
since the contributing states are the Ds resonances. Using
the m H = m Bc will not only disguise the genuine poles, but
also lead to irrationally large results for parameters a and
b. So in order to avoid this problem, we have adopted the
parametrization in Eq. (32). From the results in Table 2, one
can see that the mfit for most form factors is between 1.5
and 2.0 GeV, close to the mass of a Ds J resonance. This has
validated our parametrization.
The q2-dependent form factors of Bc → Bs are shown
in Fig. 2. From this figure, we can see that except for the
Bc → Bs1 transition, most form factors are rather stable
against the variation of q2. This is partly because of the
limited phase space. This will also lead to a reliable prediction
for the branching fractions given in the next section.
4 Phenomenological applications
4.1 Semileptonic Bc decays
The decay width for semileptonic decays of Bc → Ml¯ν,
where M = P, V , S, A, T , can be derived by dividing the
= e
1.51 × 10−2
1.96 × 10−2
6.58 × 10−4
8.31 × 10−5
5.38 × 10−4
2.98 × 10−5
1.43 × 10−2
1.83 × 10−2
5.23 × 10−4
6.33 × 10−5
3.98 × 10−4
1.97 × 10−5
τBc = (0.452 × 10−12) s,
decay amplitude into hadronic part and leptonic part, both of
which are Lorentz invariant so that can be readily evaluated.
Then the differential decay widths for Bc → Pl¯ν and Bc →
V l¯ν turn out to be
The differential decay widths for Bc → Sl¯ν and Bc →
Al¯ν can be obtained by making the following replacements
in the above expressions for Bc → Pl¯ν and Bc → V l¯ν:
= 1−mˆ l2 2
+3mˆ l2(m2Bc − m2P )2 F02(q2) ,
F Bc P (q2) → F Bc S(q2), i = 0, 1,
m Bc + mV → m Bc − m A,
V BcV (q2) → ABc A(q2),
AiBcV (q2) → ViBc A(q2), i = 0, 1, 2,
= 1−mˆ l2 2
(m2Bc − m2V − q2)(m Bc + mV ) A1(q2)
= 1−mˆ l2 2
× m Bc + mV ∓
where the superscript +(−) denotes the right-handed
(lefthanded) polarizations of the vector mesons. We have λ(m2Bc ,
mi2, q2) = (m2Bc + m2
i − q2)2 − 4m2Bc mi2 with i = P, V .
mˆ l = ml / q2. The combined transverse and total
differential decay widths are given by
is given by Eq. (38) multiplied by ( 23 |mpTT| )2 and Eq. (39)
multiplied by ( √1 | pT | )2, respectively. Here the pT denotes
2 mT
the momentum of the tensor meson in the Bc rest frame and
mT is mass of the tensor meson.
For the Bs J final state, the inputs are form factors given
in Table 2 and the masses of Bc and Bs J s given in Eq. (21).
The other input parameters are given as follows [11]:
G F = 1.166 × 10−5 GeV−2, |Vcs | = 0.973,
and our predictions for the branching fractions are given in
Table 3. It should be mentioned that in the above
calculation we have considered Bs1(Bs1) and B1(B1) to be in the
3 P1(1 P1) eigenstates.
For the BJ final state, we need to evaluate the form factors
for Bc → BJ by following the same method, and our results
are given in Table 4. The masses of Bc and BJ s are also
given in Eqs. (21) and (29). The other inputs are the same as
Eq. (44) but with |Vcs | = 0.973 replaced by |Vcd | = 0.225
[11]. With these inputs, our results for the branching fractions
and ratios are given in Table 5.
From these tables, we can see that the branching fractions
for Bc → Bs ¯ν and Bc → Bs∗ ¯ν are at the percent level,
while those for the Bc → B ¯ν and Bc → B∗ ¯ν are
suppressed by one order of magnitude. This is consistent with
the results in the literature [12–20]. Branching fractions for
channels with p-wave bottomed mesons in the final state
Table 4 Bc → B, B∗, B0, B1, B1, and B2 form factors in the
covariant LFQM fitted through Eq. (32), except for the form factors with an
asterisk, which are fitted using Eq. (36)
3.14∗ 6.49∗
2.48∗ 51.50∗
range from 10−4 to 10−6. In decays with large phase space,
the electron and muon masses can introduce about a few
percents to branching ratios. While for those with limited phase
space like Bc → Bs2 ¯ν, the effects due to the lepton mass
difference can reach 30 %. We hope these predictions can be
examined in the future on the experimental side.
4.2 Nonleptonic Bc decays
Since the main purpose of this work is to investigate the
production of Bs J , we will focus on the decay modes which can
be controlled under the factorization approach. Such decay
modes are usually dominated by tree operators with effective
Hamiltonian
G F
Heff (c → sud¯) = √2 Vc∗s Vud
× C1[s¯αγ μ(1 − γ5)cβ ][u¯β γμ(1 − γ5)dα]
+C2[s¯αγ μ(1 − γ5)cα][u¯β γμ(1 − γ5)dβ ] ,
where C1 and C2 are the Wilson coefficients, and α and β
denote the color indices.
With the definitions of the decay constants,
π +( p)|u¯γμγ5d|0 = −i fπ pμ,
one can expect the factorization formula to have the following
forms:
i M(Bc+ → Bs π +) = N m2Bc (1 − r B2s )F0Bc Bs (m2π ),
where N = =
C1/Nc(Nc = 3).
The particle decay width for Bc → Bs J π is given as
G F /√2Vc∗s Vud a1 fπ , with a1
with | p1| being the magnitude of three-momentum of Bs J or
π meson in the final state in the Bc rest frame.
We use the transition form factors given in Table 2 and the
masses of Bc and Bs J s given in Eqs. (21), (22), and (23) and
the other inputs which are given as follows [11,44]:
τBc = (0.452 × 10−12) s, mπ = 0.140 GeV,
|Vcs | = 0.973, |Vud | = 0.974,
fπ = 130.4 MeV, a1 = 1.07,
where fπ can be extracted from π − → −ν¯ data and a1
is evaluated at the typical factorization scale μ ∼ mc [63].
Then our theoretical results for the Bc → Bs J π branching
ratios turn out to be as follows:
→ Bs1π +) = 0.36 %,
→ Bs2π +) = 0.023 %.
Using the 1 fb−1 data of proton–proton collisions
collected at the center-of-mass energy of 7 TeV and 2 fb−1 data
accumulated at 8 TeV, the LHCb Collaboration has observed
the decay Bc → Bs π + [64]:
= (2.37 ± 0.31 ± 0.11+−00..1173) × 10−3.
The first uncertainty is statistical, the second is systematic,
and the third arises from the uncertainty on the Bc+ lifetime.
The ratio of cross sections σ ( Bc+)/σ ( Bs0) depends
significantly on the kinematics, and a rough estimate has lead to
the branching ratio for Bc+ → Bs0π + of about 10 % [64].
The estimated branching fraction is somewhat larger than
but still at the same magnitude as our result. Moreover, our
results have indicated that the LHCb Collaboration might be
able to discover other channels with similar branching
fractions like the Bc → Bs∗π .
5 Conclusions
To understand the structure of the heavy–light mesons,
especially the newly observed states, and to establish an overview
of the spectroscopy, a lot of effort is required on both the
experiment and the theory sides. One particular remark is
the classification of these states. In the heavy quark limit,
the charm quark will decouple with the light degree of
freedom and acts as a static color source. Strong interactions
will be independent of the heavy flavor and spin. In this case,
heavy mesons, the eigenstates of the QCD Lagrangian in
the heavy quark limit, can be labeled according to the total
angular momentum sl of the light degree of freedom. The
heavy mesons with the same angular momentum sl but
different orientations of the heavy quark spin degenerate. One
consequence is that heavy mesons can be classified by the
multiplets characterized by sl instead of the usual scheme
using the 2S+1 L J .
In this work, we have suggested to study the Bs and its
excitations Bs J in the Bc decays. We have calculated the
Bc → Bs J and Bc → BJ form factors within the covariant
light-front quark model, where the Bs J and BJ denotes an
swave or p-wave b¯s and b¯d meson, respectively. The form
factors at q2 = 0 are directly calculated while the q2-distribution
is obtained by the extrapolation. The derived form factors are
then used to study semileptonic Bc → ( Bs J , BJ ) ¯ν decays,
and nonleptonic Bc → Bs J π decays. Branching fractions
and polarizations are predicted, through which we find that
the predicted branching fractions are sizable, especially at
the LHC experiment and future high-energy e+e− colliders
with a high luminosity at the Z -pole. The future experimental
measurements are helpful to study the nonperturbative QCD
dynamics in the presence of a heavy spectator and also they
are of great value for the study of spectroscopy.
Acknowledgments This work is supported in part by National
Natural Science Foundation of China under Grant No. 11575110, Natural
Science Foundation of Shanghai under Grant No. 15DZ2272100 and
No. 15ZR1423100, by the Open Project Program of State Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, China (No. Y5KF111CJ1), and by Scientific
Research Foundation for Returned Overseas Chinese Scholars, State
Education Ministry.
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Appendix A: Explicit expressions for form factors
In the LFQM, it is more convenient to adopt a new set of
parametrization of form factors, with the relations
V Bc V (q2) = −(m Bc + m V )g(q2),
= − m Bc + m V
− 2m V a−(q2),
− q · P
u−(q2),
The analytic expressions for P → P transition form
factorsinthecovariantLFQMhavebeengiveninEq.(17),while
for the P → V transition, they are given as follows [31,37]:
−m2BmcqT2b−(q2).
[ 3A,1A(q2),q 3A,1A(q2),c±3A,1A(q2)]
+2(m1 − m2)(m1 + m1)2
−2(m1 − m1)(m1 − m2) ,
+(x2 − x1)m1] p⊥q·2q⊥ − 2x2q2 + p⊥ · q⊥[p⊥ · p⊥
x2q2wV
+(x1m2 + x2m1)(x1m2 − x2m1)] ,
a−(q2) = 16Nπc3 dx2d2p⊥ x2hNPˆ1hNVˆ1 2(2x1 − 3)
p 2
×(x2m1 + x1m2) − 8(m1 − m2) q⊥2 + 2(p⊥q·4q⊥)2
4
+wV [M 2 + M 2 − q2 + 2(m1 − m1)(m1 − m2)]
−2m1(m1 + m2) − 2m2(m1 − m2)](A(11) + A(21) − 1)
+q · P pq⊥22 + (p⊥q·4q⊥) (4A(21) − 3) .
2
The explicit expressions for P → S and P → A
transitions can be readily obtained by making the following
replacements [37]:
where only the 1/W terms in P → 1A form factors are
kept. It should be cautious that the replacement of m1 →
−m1 should not be applied to m1 in w and h . The P → T
transition form factors are calculated [37]
Nc
h(q2) = −g(q2)|hV→hT + 16π3 dx2d2p
⊥ x2Nˆ1Nˆ1
× (m1 − m1)(A(32) + A(42))
×[m2(q2 − Nˆ1 − Nˆ1 − m12 − m12)
−m1(M 2 − Nˆ1 − m12 − m22)
×(2A(13) + 2A(23) − A(12))
q · P
−4 A(23)Z2 + 3q2 (A(12))2 + 2A(12)Z2 , (A10)
× 8(m2 − m1)( A(33) + 2 A(43) + A(53))
−2m1( A(11) + A(21))( A2(2) + A(32))
+2(m1 + m1)( A(22) + 2 A(32) + A(42))
×( A(33) + 2 A(43) + A(3) 2 − A(32))
5 − A(2)
+[q2 − Nˆ 1 − Nˆ 1 − (m1 + m1)2]
×( A(22) + 2 A(32) + A(2) 1 − A(21))] ,
4 − A(1)
Nc
b−(q2) = −a−(q2)|hV →hT + 16π 3
⊥ x2 Nˆ 1 Nˆ 1
× 8(m2 − m1)( A(43) + 2 A(53) + A(63))
−2 A(63) − A(21))
×( A(43) + 2 A(53) + A(3) 3 − A(42)) + 2Z2(3 A(42)
6 − A(2)
6 A(21) A(2) 2
×( A(22) + 2 A(32) + A(2) 1 − A(21)) .
4 − A(1)
The A(ji) in the above equations are given as follows:
p⊥ · q⊥ ,
A(22) = ( A(11))2, A(32) = A(11) A(21),
A(2) 1
4 = ( A(21))2 − q2 A(12),
A(33) = A(11) A(22), A(43) = A(21) A(22),
A(3) 2
6 = A(21) A(2)
4 − q2 A(21) A(12),
Z2 = Nˆ 1 + m12 − m22 + (1 − 2x1)M 2
+(q2 + q · P) p⊥ · q⊥ .
q2
The explicit forms of h M and wM are given by [37]
h P = hV = (M 2 − M02)
h S =
h 3 A = (M 2 − M02)
h 1 A = hT = (M 2 − M02)
1 − m2
Nc √
1 M˜ 02
, w 1 A = 2,
where ϕ and ϕ p are the light-front momentum distribution
amplitudes for s-wave and p-wave mesons, respectively [37]:
ϕ p = ϕ p(x2, p⊥) =
dx2 = x1x2 M0
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