cpcChinese Physics C1674-1137Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltdcpc_43_2_02310610.1088/1674-1137/43/2/02310643/2/023106PaperParticle and field theoryEstimating the production rates of D -wave charmed mesons via the semileptonic decays of bottom mesons *
Supported by the National Natural Science Foundation of China (11222547, 11175073, 11647301), the National Program for Support of Top-notch Young Professionals and the Fundamental Research Funds for the Central Universities
Using the covariant light-front approach with conventional vertex functions, we estimate the production rates of D -wave charmed/charmed-strange mesons via the
semileptonic decays. As the calculated production rates are significant, it seems possible to experimentally search for D -wave charmed/charmed-strange mesons via semileptonic decays, which may provide an additional approach for exploring D -wave charmed/charmed-strange mesons.
With accumulation of experimental data, more and more open-charm and open-bottom states are reported by the experiments (see the review paper [1] for more details). Among the observed states, there are abundant candidates for charmed and charmed-strange mesons including the famous
and
. In recent years, experimentalists have especially made great progress in observing D -wave charmed mesons, as well as D -wave charmed-strange mesons. For example, the observed
,
[2, 3],
and
[4, 5] are good candidates for 1D states in charmed and charmed-strange meson families [6-15]. In addition,
[16, 17] can be assigned to a 1D state of charmed-strange meson, although there exist other interpretations [18-25]. Readers can refer to Refs. [26, 27] for more information on D -wave charmed and charmed-strange mesons.
When examining the production processes involving D -wave charmed and charmed-strange mesons, we observe that these states are mainly produced via nonleptonic weak decays of bottom/bottom-strange mesons. However, as an important decay mode, semileptonic decays of
mesons are the ideal platform for producing D -wave
mesons because they can be estimated more accurately than nonleptonic decays. In order to estimate the branching ratios of these processes, we need to perform a serious theoretical study of the production of D -wave
mesons via the semileptonic decays of
mesons, which is the main task of the present work.
We adopt in this work the light-front quark model (LFQM) [28-32], which is a relativistic quark model. Since the involved light-front wave function is manifestly Lorentz invariant and the hadron spin is constructed by using the Melosh-Wigner rotation [33, 34], LFQM can be suitably applied to a study of semileptonic decays of
mesons. In Refs. [35-48], the production rates of S- and P-wave
mesons have been estimated through the decay processes of
in the covariant LFQM.
As of yet, there has been no study of the production of D -wave
mesons via the semileptonic decays of
mesons in the covariant light-front approach, which makes the present work, to our knowledge, the first paper on this issue. As shown in the following sections, the technical details relevant for the above processes are far more complicated than for S- and P-wave mesons. Thus, our work is not only an application of LFQM, but is also a development of this research field since the formulas presented can be helpful for studying other processes involving D -wave mesons. We consider this aspect valuable to the readers and provide the details of our analysis.
Finally, we hope that the present study will stimulate the interest of experiments in their search for D -wave
mesons via the semileptonic decays of
mesons, as it opens another window for exploring D -wave
mesons and contributes to gathering more experimental information.
This paper is organized as follows. In Section 2, we introduce the covariant light-front approach for D -wave mesons and their corresponding form factors. In Section 3, we give our numerical results including the form factors and the decay branching ratios. In Section 4, the relation between the light-front form factors and the requirements from the heavy quark symmetry are presented. The final section is devoted to a summary of our work. In Appendices A through E, we give the algebraic details related to the production of D -wave mesons via
semileptonic decay in LFQM, while Appendix F is devoted to proving the Lorentz invariance of the matrix elements in the toy model proposed in Ref. [49] with a multipole ansatz for vertex functions.
Covariant light-front quark model
In the conventional light-front quark model, the quark and antiquark inside a meson are required to be on their mass shells. One can then extract physical quantities by calculating the plus component of the corresponding matrix element. However, as discussed in Ref. [35], this approach may result in missing the so-called Z-diagram contribution, so that the matrix element depends on the choice of frame. A systematic way of incorporating the zero-mode effect was proposed in Ref. [49] by maintaining the associated current matrix elements frame independent, so that the physical quantities can be extracted.
In this work, we apply the covariant light-front approach to investigate the production of
mesons via the semileptonic decays of
mesons (see Fig. 1), where
denotes a general D -wave
meson. First, we briefly introduce how to deal with the transition amplitudes.
Diagram of the meson transition processes
.
is the momentum of the incoming (outgoing) B/B_{s} (D-wave D_{s}) meson.
denotes the momentum carried by the bottom (charm) quark, while p_{2} is the momentum of a light quark.
According to Ref. [49], the relevant form factors are calculated in terms of Feynman loop integrals, which are manifestly covariant. The constituent quarks inside a hadron are off-shell, i.e. the incoming (outgoing) meson has the momentum
, where
and
are the off-shell momenta of the quark and antiquark, respectively. These momenta can be expressed in terms of the appropriate internal variables
, defined by,
with
. In the light-front coordinates,
with
, which satisfy the relation
. One needs to point out that there exist different conventions for momentum conservation in the covariant light-front and conventional light-front approaches. In the covariant light-front approach, four components of a momentum are conserved at each vertex, where the quark and antiquark are off-shell. In the conventional light-front approach, the plus and transverse components of a momentum are conserved quantities, where the quark and antiquark are required to be on their mass shells. Thus, it is useful to define internal quantities for on-shell quarks
where
is the kinetic invariant mass squared of the incoming meson.
denotes the energy of quark
, while
and
are the masses of the quark and antiquark, respectively.
In Ref. [35], the form factors for semileptonic decays of bottom mesons into S -wave and P -wave charmed mesons were obtained within the framework of the covariant light-front quark model. In the following, we adopt the same approach to deduce the form factors for the production of D -wave charmed/charmed-strange mesons by semileptonic decays of bottom/bottom-strange mesons. Here, D -wave
mesons, denoted as
,
,
, and
, have quantum numbers
and
, respectively. In the following, we use this notation for simplicity.
In the heavy quark limit
, the heavy quark spin
decouples from the other degrees of freedom. Hence, a more convenient way to describe charmed/charmed-strange mesons is to use the
basis, where
denotes the total spin and
denotes the total angular momentum of the light quark. There exists a connection between the physical states
and the states described by
for
[50, 51], i.e.,
This relation shows that two physical states
and
with
are linear combinations of
and
states. When dealing with the transition amplitudes for the production of
and
states, we need to consider the mixing of states as in Eqs. (7) and (8).
One can write the general definition of the matrix elements for the production of D -wave
mesons via the semileptonic decays of
mesons, i.e.,
Here,
,
and
.
,
and
are the polarization vector (tensors). The details of the derivation are given in Appendix E. The Lorentz invariance has been assumed when deriving these form factors. One should note that the
transition occurs through a
current, where
denotes the general
-wave charmed (charmed-strange) meson. For the semileptonic decays involving
and
states, a
term arises in
, which corresponds to the contribution of the vector current. Contrary to the case of
and
states, for
and
states, the
term arises from the axial vector current. A minus sign is added in front of this term, so that we have
. When the sign of the
term is fixed, the signs of the other form factors can also be determined.
We now focus on the hadronic matrix elements given by Eqs. (9)-(12). Here, we show how to calculate them by taking the
transition as an example, where
denotes the
state of the charmed/charmed-strange meson. The corresponding matrix element for
can be written as
Following the calculations in Ref. [35], we first obtain the
transition form factors, and then calculate the processes involving the other D -wave charmed/charmed-strange states. The details for the other matrix elements are given in Appendix A. Here, one needs to introduce the vertex wave functions to describe
and
mesons. The expression for a vertex function for an initial
meson was obtained in Ref. [35]. In the following, we give a detailed discussion for the vertex function of the final state
meson.
The D -wave vertex function has been studied in Ref. [52]. We list all D -wave vertex functions in Appendix B; one may refer to Ref. [52] for more details. First, we use
vertex functions for calculating the
transition.
In the conventional LFQM,
and
are on their mass shell, while in the covariant [49] light-front approach, the quark and antiquark are off-shell, but the total momentum
is still the on-shell momentum of a meson, i.e.
, where
is the mass of the incoming meson. One needs to relate the vertex function deduced in the conventional LFQM to the vertex in the covariant light-front approach. A practical method for this process has been proposed in a covariant light-front approach in Ref. [49]. We obtain the corresponding covariant vertex function as
where
and
denote the corresponding scalar functions for
state.
The explicit expression for the matrix element
, which corresponds to the hadronic one-loop Feynman diagram of Fig. 2, reads
(color online) A hadronic one-loop Feynman diagram for the process shown in Fig. 1. The V-A current is attached to a blob in the upper middle of the circle.
where
is the number of colors,
,
.
is the vertex function of a pseudoscalar meson, and
One can integrate over
via a contour integration with
and the integration picks up a residue
, where the antiquark is set to be on-shell,
. The momentum of the quark is given by the momentum conservation,
. Consequently, after performing the
integration, we make the replacements:
where the explicit trace expansion of
, after integrating Eq. (15) over
, is presented in Appendix A. In addition,
is given in Ref. [35] as,
where
is the solid harmonic oscillator for
-wave and describes the momentum distribution of the initial
meson.
As noted in Ref. [52], after carrying out the contour integral over
, the quantities
,
and
are replaced by
,
and
, respectively. Here,
is related to
as,
which is derived in Appendix B, and
with
.
As pointed out in Refs. [35, 49],
can be expressed in terms of the external vectors,
and
:
where
[35, 49] is a light-like four vector in the light-front coordinate system. Since the constant vector
is not Lorentz covariant, the presence of
terms implies that the corresponding matrix elements are not Lorentz invariant. This
dependence also appears in the products of two
's. This spurious contribution is related to the so-call zero-mode effect and should be canceled when calculating physical quantities.
Initiated by the toy model proposed in Ref. [49], Jaus developed a method which allows calculating the zero-mode contributions associated with the corresponding matrix element.
as well as the products of a couple of
's can be decomposed into products of vectors
,
,
, and
, as shown in Appendix C, with functions
,
, and
, where
and
are related to
-dependent terms. Based on the toy model, the vertex function of a ground state pseudoscalar meson is described by a multipole ansatz,
which is different from our conventional vertex functions. Jaus has proven that at the toy model level, the spurious loop integrals of
,
and
,
vanish in the following integrals,
where
or
. This fact is a natural consequence of the Lorentz invariance of the theory. The
-dependent terms have been systematically eliminated in the toy model since the
and
give trivial contributions to the calculated form factors [49].
However, this method has a narrow scope of application. Note that Jaus proposed this method in a very simple multipole ansatz for the vertex function. One may get totally different contributions from the zero-mode effects once the form of a vertex function for a meson is changed. For instance, as indicated in Ref. [53], for the weak transition form factors between pseudoscalar and vector meson, the zero-mode contributions depend on the form of the vector meson vertex,
where the denominator
contains different types of terms. (Readers can also refer to Refs. [54-59] for more details.)
Beyond the toy model, the method of including the zero-mode contributions in Ref. [49] was further applied to study the decay constants and form-factors for
-wave and
-wave mesons [35]. In Ref. [35], Cheng et al. used the vertex functions for S- and P-wave mesons deduced from the conventional light-front quark model, which are different from the multipole ansatz proposed by Jaus. In [35] , they applied the method of the toy model to cancel the
functions. As for
functions, they have numerically checked that
give very small contributions to the corresponding form factors. That is, when the multipole vertices are replaced by conventional light-front vertex functions, and by setting
and
functions equal to 0, one can still obtain very good numerical results for the decay constants. Indeed, as indicated in Ref. [49], the numerical results obtained by applying conventional vertices are even better than those for vertices from the multipole ansatz.
It is natural to expect that this method can also be applied in our calculations of the form factors for the transition processes of D -wave mesons. In order to calculate the corresponding form factors, one also needs to eliminate the
and
functions introduced in the D -wave transition matrix elements. In the following, we introduce our analysis of the zero-mode contributions.
Following the discussion in Refs. [35, 49], to avoid the
dependence of
, as well as of the product of a couple of
's, one needs to do the following replacements:
where the
and
functions are disregarded at the toy model level, and their loop integrals vanish manifestly if conventional vertices are introduced. We give more details in the following discussion.
For the terms of products that are associated with
, the zero-mode contributions are introduced and the following replacements should be made to eliminate the
-dependent terms
where
and
and
are functions of
,
,
, and
. These functions have been obtained in Ref. [49]. Again, in the above replacements,
and
can be naturally disregarded at the toy model level and their loop integrals vanish manifestly with a standard meson vertex.
Let us take the second rank tensor decomposition
as an example how to effectively set the
and
functions to 0 and eliminate the
-dependent terms. There are two
-dependent functions in the leading order of
-decomposition, i.e.
and
.
From Ref. [49] we have
By introducing the explicit expression of
from Ref. [49], one can easily obtain
in the toy model. The loop integral of
naturally vanishes. On the other hand, beyond the toy model, this term should also be eliminated manifestly, i.e. we have the replacement
The same procedure can be applied to
, which gives,
We would like to emphasize that beyond the toy model, when the conventional light-front vertex functions are introduced, elimination of
is also necessary when calculating semileptonic form factors with
-wave and
-wave mesons as final states. However, one would obtain very small corrections since, as described earlier, these
functions give small contributions.
Expanding
, and replacing the
,
,
,
, and
terms with the replacements in Eqs. (23)-(24) and Eqs. (26)-(28), we can obtain the form factors for the
state by comparing with the general definition of a matrix element given in Eq. (9). We note that, since in the expansion of
there is no term with three
's, Eq. (25) is not used. This equation just helps to find the tensor decomposition of
. This procedure is identical to what was used for the tensor decomposition of
by analyzing the product of
.
After including the zero-mode effect introduced by the
and
functions, we get the explicit form factors
,
,
and
as
The same procedure can also be applied for the transitions in the case of
,
and
states, as given in Appendix A. In the following, we continue to discuss these states and focus on the new issues that need to be introduced when dealing with higher spin D -wave states.
By analogy to the conventional vertex functions obtained in Appendix B, we write the covariant vertex functions for
,
, and
in one loop Feynman diagrams as,
where
and
are functions of the associated states in momentum space.
In order to obtain the
,
transition form factors, the matrix elements are denoted as
It is straightforward to obtain explicit expressions for the corresponding one loop integrals as
By integrating over
as discussed in the case of
, the following replacements should be made,
where
in the subscript or superscript denotes
,
and
, so that the physical quantities corresponding to different transitions can be easily distinguished. The explicit forms of
are given by Eq. (B9) in Appendix B. We also present the trace expansions of
,
and
in Appendix A. After performing the contour integral over
, the quantities
,
and
replace
,
and
, respectively. The next step is to maintain the
independence, so that
and
vanish manifestly when the zero-mode effect is included.
Apart from decomposing the tensors as in Eqs. (23)-(27) for the
transition discussed above, for
states, one also needs to consider the product of four
's to obtain the reduction of
, which has been done in Ref. [35]^{①)}
We should mention that there is a typo in REf. [35] for the function, whose correct expression is given by .
, i.e.,
and the corresponding tensor decomposition of
is given by
Furthermore, in the
transition,
can be replaced by the product of five
's. The derivation of the explicit form for
is given in Appendix C. Accordingly, we obtain
One can refer to Ref. [35, 49] for explicit expressions for the
functions.
In fact, after expanding the products of two
's , and the products of several
's with
, to first order in
, we find that the conditions deduced from the
and
functions can be independently expressed in terms of the lower order of
functions. To illustrate this point, we give in Table 1 all replacements deduced from
and
. Strictly speaking, these equations hold only when loop integration of these functions has been performed. The relations presented in Table 1 have been applied to Eqs. (26)-(28) and Eqs. (47)-(48).
The replacements
corresponding to the conditions
and
.
Related
functions
Related
functions
The replacements presented in Table 1 can be proven for the toy model vertex, as given in Appendix F. This indicates that a generalization of Jaus's model to higher spin J states is possible. However, when a conventional D -wave vertex function is introduced in the loop integration of Eq. (21), it is difficult to prove these identities. We emphasize that for the conventional light-front vertex functions, the replacements listed in Table 1 work very well for obtaining the form factors and semileptonic decay widths. Besides, the vanishing of
-dependent terms
and
is not only the result of Jaus's model, but also a requirement for obtaining physical quantities, i.e. for keeping the Lorentz invariance. Hence, we continue to use the replacements listed in Table 1 to perform our analysis.
Numerical results5.Summary
In the past several years, considerable progress has been achieved in observing
-wave
mesons in various experiments [2-5]. These observations enrich the
meson families. Although all candidates for
-wave
are produced in nonleptonic weak decays of
mesons, we studied in this work the possibility of producing
and
meson families via semileptonic decays, although
states of
meson families have not yet been experimentally observed.
In order to get numerical results for semileptonic decays, we adopted the light-front quark model, which has been extensively applied in the studies of decay processes including semileptonic decays [35-48]. Our study in the framework of LFQM shows that the analysis of the production of the relevant D -wave
mesons via
mesons is much more complicated than the analysis of
-wave and
-wave
mesons [35-37]. We have given detailed derivation of many formulas necessary for obtaining the final semileptonic decay widths. The numerical results obtained show that the semileptonic decays of
mesons are suitable for searching for
-wave charmed and charmed-strange mesons. Furthermore, we have shown that our light-front form factors approximately satisfy HQS requirements.
Theoretical studies of the
semileptonic decays to D -wave charmed mesons were performed in the past using the QCD sum rule [65, 66] and the instantaneous Bethe-Salpeter method [51]. We have noted that different theoretical groups have given different results for the
semileptonic decays to D -wave charmed mesons. Thus, experimental search for semileptonic decays predicted by our results could provide a crucial test for the theoretical frameworks for studying
semileptonic decays.
As indicated by our numerical results, semileptonic decays of pseudoscalar
mesons could be the ideal platform for investigating D -wave charmed and charmed-strange mesons. With the LHCb continuing to take data at 13 TeV and the forthcoming run of Belle-II, we expect further experimental progress.
Kan Chen would like to thank Qi Huang and Hao Xu for helpful discussion. We also would like to thank Yu-Ming Wang for the suggestion of form factors adopted in this work.
Appendix A: Trace expansion and form factors
In this Appendix, we present the detailed expansions of
,
,
and
. Form factors associated with these expressions are also given.
When integrating over
, we need to do the following integrations
where the trace expansions of
,
,
and
are
The corresponding form factors for the
state are
The corresponding form factors for the
state are
The corresponding form factors for the
state are
Appendix B: Conventional vertex functions for <italic>D</italic>-wave mesons
In the conventional light-front approach, a meson with momentum
and spin
can be defined as
where
and
denote the quark and antiquark inside a meson, respectively, and
and
are the on-shell light-front momenta of the quark and antiquark, respectively. The symbol “~” means an operation on a momentum to extract only plus and transverse components, i.e.,
and
In the light-front coordinate, the definition of the light-front relative momentum
reads
The meson wave function
in momentum space is expressed as
where
describes the momentum distribution of the constituent quarks inside a meson with the orbital angular momentum
, and
is the corresponding Clebsch-Gordan (CG) coefficient. In Eq. (B4),
transforms a light-front helicity
eigenstate into a state with spin
where
is the momentum of the meson in the rest frame,
,
and
. Using the spinor representation of
and
from Appendix of Ref. [68], the explicit expression for
is calculated in Ref. [69].
With the potential model for a definite meson state with quantum numbers
, its mass and the corresponding numerical spatial wave function can be calculated. The spatial wave function can be obtained by expansion of a set of SHO wave functions (the number of basis SHO wave functions is
), where the expansion coefficients form the corresponding eigenvector.
The meson wave function
in momentum space reads
Here,
, where
are the expansion coefficients of the corresponding eigenvectors, and
denotes the corresponding vertex structure of
-wave mesons.
One can further simplify these wave functions by using the Dirac equations
and
. After this simplification, the wave function of
-wave mesons can be written as ^{②)}②)
We need to mention that a minus sign is needed in front of which is a typo in Ref. [52].
with
and
Appendix C: Tensor decomposition
The second-order tensor decomposition of
and the third-order of
are given in Ref. [49]. The fourth-order tensor decomposition of
is obtained in Ref. [35]^{③)}③)
When reproducing the coefficients , , and , we find a typo in Ref. [35] whose correct expression is given by .
. For a
state, we need the fifth-order tensor decomposition of
. Here, we just include the leading-order contribution from
, and get the following expression,
with
By contracting
with
,
, and
, and comparing with the explicit expression for
and
, we obtain the coefficients, i.e.,
Appendix D: Helicity amplitudes and decay widths
In this appendix, we give the explicit forms of helicity form factors for semileptonic decays. The decay widths can be easily obtained from the helicity form factors.
We study the production of D-wave charmed/charmed-strange mesons and their partners via the semileptonic decay of
mesons. The effective weak Hamiltonian for the
and
transitions is
where
is the Fermi coupling constant and
denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix element.
The explicit expression for the of decay width of the
matrix element can be obtained by using the helicity form factors as
Here,
. With the above equations, one can write the semileptonic decay width in terms of helicity amplitudes as
For the production of charmed/charmed-strange mesons with
via the semileptonic decay of bottom/bottom-strange mesons, we obtain the following total decay width
We should emphasize that in our calculations the mixing between
and
states has been taken into account. For the form factors
in the
transitions, the mixing is included in
where
.
In a similar way, we can obtain the decay width for the production of
states, i.e.,
Appendix E: Polarization tensor algebra
When considering the polarization vector of a massive vector boson, the four-momentum in any other inertial system can be obtained by a Lorentz transformation. Hence, it is sufficient to consider the four-momentum in the rest frame,
In the rest frame, there exist three possible helicities of a spin 1 particle, i.e. the three independent polarization vectors have the form,
which satisty
.
In the following, for convenience of the readers, we present the tensor algebra in the rectangular coordinate system. One can also do this in the light-front frame by adjusting the corresponding metric tensor, as the tensor algebra will lead to identical results. The normalization of polarization vectors is given by
Due to the Lorentz covariance, the sum over the polarization states is
where
, and
In general, the higher-rank polarization tensor must satisfy the following conditions
We take the CG coefficient of the
state to show how to rewrite a CG coefficient in the tensor contracted form. First, we take the tensor
. Due to the Lorentz covariance, the decomposition of this tensor must be a linear combination of
,
or
,
, i.e.,
Multiplying
on both sides of the above equation, introducing the transverse condition of Eq. (E7) and having
, we easily find
. The symmetry property of this CG coefficient results in
. Thus, the original tensor is reduced into a simpler form
Next, the traceless condition is applied by multiplying
on both sides of Eq. (E8), and we find
. We multiply
on both sides of Eq. (E8), and obtain,
Here, only one undetermined constant remains. We can assign a specific value to
,
and
on the left side of Eq. (E9), and introduce the expression for
,
and
on the right side to solve for
. Finally, we obtain
, and get
This example shows how a CG coefficient can be transformed into a tensor contracted form. In the same way, one can transform the other CG coefficients.
When calculating the semileptonic decay width, we use the second-order and third-order tensors. The polarization tensor of a higher spin state with angular momentum j and helicity
can be constructed using lower-rank polarization tensors and CG coefficients. A general relation reads
with
. One can generalize this equation to obtain the polarization tensors of higher spin states. Thus, we have
for
state, and
for
state.
Appendix F: Proof of the Lorentz invariance of the matrix elements with a multipole ansatz
In this Appendix, we show explicitly that loop integrals of
and
vanish analytically for the toy model, i.e. with the multipole ansatz for the vertex functions. This procedure can be extended to the other
and
. Let us consider the integral,
where
is the function related to
as well as
and
. By inserting the identities from Table 1, we can prove that the loop integrals of
and
vanish in the multipole ansatz.
Starting from Jaus's result [49], the complete momentum integral of
is given by
where
The loop integral of
is obtained as
Readers can refer to Ref. [49] for the definition of
. Thus, one obtains the replacement
, which is related to
.
Furthermore, the integral of
is given by
where
We take the integral
as an example and perform the integration.
is similar to
in Ref. [49]. It contains a
term, which is proportional to
. This means that this term vanishes. We only need to prove that the other terms in the integral vanish, i.e.,
By using Eq. (F3) and performing a partial integration, we have
and by introducing Eq. (F4), we directly obtain
which means that
and hence,
Equation
can also be obtained by using Eq. (F3) and Eq. (F4). For the loop integrals of
and
in which
, one can use the techniques from Ref. [49] and show that
.
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