# Current-Component Independent Transition Form Factors for Semileptonic and RareD⟶πK Decays in the Light-Front Quark Model

Authors: Ho-Meoyng Choi
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

We investigate the exclusive semileptonic and rareD⟶πK decays within the standard model together with the light-front quark model (LFQM) constrained by the variational principle for the QCD-motivated effective Hamiltonian. The form factors are obtained in the q+=0 frame and then analytically continue to the physical timelike region. Together with our recent analysis of the current-component independent form factors f±q2 for the semileptonic decays, we present the current-component independent tensor form factor fTq2 for the rare decays to make the complete set of hadronic matrix elements regulating the semileptonic and rare D⟶πK decays in our LFQM. The tensor form factor fTq2 are obtained from two independent sets JT+⊥,JT+− of the tensor current JTuv . As in our recent analysis of f−q2, we show that fTq2 obtained from the two different sets of the current components gives the identical result in the valence region of the q+=0 frame without involving the explicit zero modes and the instantaneous contributions. The implications of the zero modes and the instantaneous contributions are also discussed in comparison between the manifestly covariant model and the standard LFQM. In our numerical calculations, we obtain the q2-dependent form factors (f±,fT) for D⟶πK and branching ratios for the semileptonic D⟶πKℓvℓℓ=e,μ decays. Our results show in good agreement with the available experimental data as well as other theoretical model predictions.


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## Body

## 1. Introduction

The three flavors of charged leptons,e,μ,τ, are the same in many respects. In the standard model (SM), the couplings of leptons to gauge bosons are supposed to be independent of lepton flavors, which is known as lepton flavor universality (LFU) [1]. The experimental tests of LFU in various semileptonic B decays have been reported [2–6] by measuring the ratios of branching fractions RD∗=BrB⟶D∗τντ/BrB⟶D∗ℓνℓℓ=e,μ. Currently, the SM prediction is roughly three standard deviations away from the global average of results from the BABAR, Belle, and LHCb experiments. Many theoretical efforts have been made in resolving the issue of RD∗ anomaly and searching for new physics beyond the SM [7–10]. In view of this, tests of LFU in D decays are also intriguing complementary endeavors.Exclusive semileptonic and rareD decays provide rigorous tests of the SM in the charm sector including not only the LFU but also the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [11, 12], which describe the mixings among the quark flavors in the weak decays and hold the key to the CP violation issues in the quark sector. Compared to the semileptonic D⟶πKℓνℓ=e,μ decays induced by flavor-changing charged current, the rare D⟶πKℓℓ decays are induced by the flavor-changing neutral current (FCNC). Since the rare decays are loop-suppressed in the SM as they proceed through FCNC, they are also pertinent to test the SM and search for physics beyond the SM. Recent BES III measurements [13–20] for many exclusive semileptonic charm decays also allow one to test the SM in the charm sector more precisely.While the experimental measurements of exclusive decays are much easier than those of inclusive ones, the theoretical knowledge of exclusive decays is sophisticated essentially due to the hadronic form factors entered in the long-distance nonperturbative contributions. Along with new particle effects beyond the SM, which may amend the Wilson coefficients of the effective weak Hamiltonian that describes physics below the electroweak scale, the reliable and precise calculations of the hadronic form factors are very important to constrain the SM and search for new physics effects beyond the SM.The calculations of hadronic form factors for semileptonic and rareD decays have been made by various theoretical approaches, such as lattice QCD (LQCD) [21–24], QCD sum rules [25, 26], QCD light-cone sum rules [27–29], symmetry-preserving continuum approach to the SM strong-interaction bound-state problem [30], quark potential model [31–33], relativistic quark model(RQM) based on the quasipotential approach [34], covariant confining quark model (CCQM) [35], chiral quark model [36], and constituent quark model [37].Perhaps, one of the most apt formulations for the analysis of exclusive processes involving hadrons may be provided in the framework of light-front (LF) quantization [38]. The semileptonic and rare D decays have also been analyzed by the light-front quark model (LFQM) [39–47] based on the LF quantization.In the standard LFQM that we use in this work, the constituent quark and antiquark in a bound state are required to be on-mass shells, and the spin-orbit wave function (WF) is obtained by the interaction-independent Melosh transformation [48] from the ordinary equal-time static spin-orbit WF assigned by the quantum number JPC. For the radial part, we use the phenomenologically accessible Gaussian WF ϕx,k⊥. Since the standard LFQM itself is not amenable to pin down the zero modes, the exactly solvable manifestly covariant Bethe-Salpeter (BS) model with the simple multipole type qq¯ vertex was utilized [45, 46, 49, 50] to help identify the zero modes in a systematic way. On the other hand, this BS model is less realistic than the standard LFQM. Thus, as an attempt to apply the zero modes found in the BS model to the standard LFQM, the effective replacement [45, 46, 49] of the LF vertex function χx,k⊥ obtained in the BS model with the more realistic Gaussian WF ϕx,k⊥ in the standard LFQM has been made.However, we found [51–53] that the correspondence relation between χ and ϕ proposed in [45, 46, 49] encounters the self-consistency problem, e.g., the vector meson decay constant obtained in the standard LFQM was found to be different for different sets of the LF current components and polarization states of the vector meson [51]. We also resolved [51–53] this self-consistency problem by imposing the on-mass shell condition of the constituent quark and antiquark in addition to the original correspondence relation between χ and ϕ. Specifically, our new finding for the constraint of the on-mass shell condition corresponds to the replacement of physical meson mass M with the invariant mass M0 in the calculation of the matrix element. The remarkable feature of our new additional correspondence relation M⟶M0 between the two models in the calculations of the two-point functions such as the weak decay constants and the distribution amplitudes of mesons [51–53] was that the LF treacherous points such as the zero modes and the off-mass shell instantaneous contributions appeared in the BS model are absent in the standard LFQM. This prescription M⟶M0 can be regarded as an effective inclusion of the zero modes in the valence region of the LF calculations.As an extension our analysis of the two-point functions [51–53] to the three-point ones, in our very recent LFQM analysis [54] of the semileptonic B⟶Dℓνℓ decays, we presented the self-consistent descriptions of the weak transition form factors (TFFs) f+q2 and f−q2. Especially, f−q2 should be obtained by using least two components of the weak vector current JVμ while f+q2 can be obtained from the single and “good” component (JV+) of the current. Because of this, f−q2 has been known to receive the zero mode mainly due to the unavoidable usage of the so-called “bad” components of the current, i.e., J⊥V=Jx,Jy and JV−, many efforts have been made to obtain the Lorentz covariant form factors [45, 46, 49] within the standard LFQM by properly handling the zero-mode as well as the instantaneous contributions. Applying the same correspondence relations found in [51–53] to the B⟶Dℓνℓ decays, we found that the zero modes and instantaneous contributions to f−q2 are made to be absent in the standard LFQM while they exist in the BS model. In other words, we obtained the current-component independent form factor f−q2 in the standard LFQM, i.e., f−q2 obtained from J+,J−V is exactly the same as the one obtained from J+,J⊥V numerically, and both are expressed as the convolution of the initial and final state LFWFs in the valence region of the q+=0 frame. This verifies that our new correspondence relations found in the two-point functions are also applicable to the three-point functions.The purpose of this paper is to extend our previous analysis [54] of the form factors f±q2 for the semileptonic decays between the two pseudoscalar mesons to obtain the current-component independent tensor form factor fTq2 for the rare decays, which complete the set of hadronic matrix elements regulating the exclusive semileptonic and rare decays between the two pseudoscalar mesons. We then apply our Lorentz covariant form factors f±,fT for the analysis of the semileptonic and rare D⟶πK decays within the standard model and the light-front quark model (LFQM) constrained by the variational principle for the QCD-motivated effective Hamiltonian [49, 55–57].The paper is organized as follows: in Section2, we introduce three form factors f±,fT for the semileptonic and rare decays between two pseudoscalar mesons. In the q+=0 frame, we define the form factors extracted from the various combinations of vector and tensor currents. In Section 3, we set up the current matrix elements for the form factors in an exactly solvable model based on the covariant BS model of (3+1) dimensional fermion field theory. We then present our LF calculations of tensor form factor fT in the BS model using the two different sets (JT+⊥ and JT+−) of the tensor current JTμν. For completeness, we also present the results of the current-component independent form factors f±q2 found [54]. We note that while fTq2 obtained from JT+⊥ is immune to the zero mode and the instantaneous contribution, fTq2 obtained from JT+− cannot avoid those contributions in this BS model. Linking the covariant BS model to the standard LFQM with our new correspondence relations between the two models [51–53], however, we find that fTq2 obtained from JT+− in the standard LFQM is made to be free of the zero mode as well as the instantaneous contribution. In other words, we obtained the current-component independent tensor form factor fTq2 in the standard LFQM regardless of using JT+⊥ or JT+− as in the case of f−q2 calculation [54]. Finally, we present the current-component independent TFFs f±,fT in the q+=0 frame of the standard LFQM. In Section 4, we present our numerical results of the form factors for the semileptonic and rare D⟶πK decays as well as the branching ratios for the semileptonic D⟶πKℓνℓℓ=e,μ. Summary and discussion follow in Section 5.


## 2. Theoretical Framework

The matrix elements of the vectorJVμ=q¯γμc and the tensor JTμν=q¯σμνc currents for the weak c⟶qq=u,d,s transitions between the initial D meson and the final π or K meson can be parametrized by the following set of invariant form factors, (f+,f−,s) [33]:
(1)MVμ≡P2JVμP1=f+q2Pμ+f−q2qμ,(2)MTμν≡P2JTμνP1=isq2Pμqν−qμPν,whereP=P1+P2 and q=P1−P2 are the four-momentum transfer to the lepton pair (ℓνℓ) with mℓ2≤q2≤M1−M22 for the semileptonic decays or to the pair (ℓ+ℓ−) with 4mℓ2≤q2≤M1−M22 for the rare decays, respectively. The antisymmetric tensor in Eq. (2) is given by σμν=i/2γμ,γν.On many occasions, it is useful to express Eq. (1) in terms of the form factors f+q2 and f0q2, which are related to the transition amplitude with the exchange of a vector (1−) and a scalar (0+) boson in the t-channel, respectively, and satisfy
(3)f0q2=f+q2+q2M12−M22f−q2.Likewise, the tensor form factorsq2 in Eq. (2) can also be redefined by
(4)sq2=fTq2M1+M2,to makefTq2 dimensionless.Including the nonzero lepton mass (mℓ), the differential decay rate for the semileptonic P1⟶P2ℓνℓ process is given by [58, 59].
(5)dΓdq2=GF296π3p⟶∗VQ1Q¯22q2M121−mℓ2q22×1+mℓ22q2H+2+3mℓ22q2H02,whereGF=1.166×10−5 GeV−2 is the Fermi constant, VQ1Q¯2 is the relevant CKM mixing matrix element, and
(6)p⟶∗=12M1M12+M22−q22−4M12M22,is the modulus of the three-momentum of the daughter meson in the parent meson rest frame, and the helicity amplitudesH+ and H0 are given by
(7)H+=2M1p⟶∗q2f+q2,H0=M12−M22q2f0q2.We note thatq2=qmax2 corresponds to the zero-recoil of the final meson in the initial meson rest frame, and the q2=0 corresponds to the maximum recoil of the final meson recoiling with the maximum three momentum P⟶2=M12−M22/2M1.In the LF calculation of the form factors, we use the metric conventiona·b=1/2a+b−+a−b+−a⊥·b⊥. Performing the LF calculation in the q+=0 frame (i.e., q2=−q⊥2=−Q2<0) with P1=P1+,P1−,P1⊥=P1+,M12/P1+,0⊥, we utilize all three components (μ,ν=+,−,⊥) of the current JVμ and JTμν in Eqs. (1) and (2) to obtain f+q2, f−q2 [or f0q2], and fTq2. The form factors obtained in the spacelike region q2<0 are then analytically continued to the timelike region by changing q⊥2 to −q2 in the form factors as we show in our numerical calculations.While the form factorf+q2 can be obtained from the plus component (JV+) of the vector current, one cannot but use two different combinations of the current to obtain f−q2 such as J+,J⊥V or J+,J−V. That is, using those sets of the current components in the q+=0 frame, one obtains the relations between the weak form factors f±q2 and the current matrix elements in Eq. (1) as follows [54]:
(8)f+q2=MV+2P1+,(9)f−+⊥q2=MV+2P1++MV⊥·q⊥q⊥2,(10)f−+−q2=−MV+2P1+ΔM+2+q⊥2M−2−q⊥2+P1+MV−ΔM−2−q⊥2,whereΔM±2=M12±M22, and we denote f−q2 obtained from J+,J⊥V and J+,J−V as f−+⊥q2 and f−+−q2, respectively. It is prerequisite to show that f−+⊥q2=f−+−q2 to assert the Lorentz invariance of the form factor and the self-consistency of the model.Likewise, the tensor form factorsq2 can be obtained from using either JT+⊥ or JT+−. In this case, the relations between sq2 and the current matrix element in Eq. (2) are given by
(11)s+⊥q2=−iMT+⊥·q⊥2q⊥2P1+,(12)s+−q2=−iMT+−2ΔM−2−q⊥2,wheres+⊥q2 and s+−q2 represent the form factor sq2 obtained from JT+⊥ and JT+−, respectively. Of course, s+⊥q2=s+−q2 should be satisfied in the self-consistent model calculation.Our aim in this work is to shows+⊥q2=s+−q2 in addition to our previous verification of f−+⊥q2=f−+−q2 [54] in our LFQM, which completes the analysis of the exclusive semileptonic and rare decays between two pseudoscalar mesons. For this purpose, we start from the exactly solvable manifestly covariant BS model and then connect it to our phenomenologically accessible LFQM. Although we analyzed f±q2 in [54], we shall include them again in the next section for the completeness of the analysis.


## 3. Model Description

### 3.1. Manifestly Covariant Model

In the solvable model, based on the covariant BS model of (3+1)-dimensional fermion field theory [50, 54], the matrix elements M=MVμ,MTμν are given by
(13)M=iNc∫d4k2π4Hp1THp2Np1NkNp2,where the corresponding trace termsT=Sμ,Tμν are given by
(14)Sμ=Trγ5p1+m1γμp2+m2γ5−k+mq, for the vector current and
(15)Tμν=Trγ5p1+m1σμνp2+m2γ5−k+mq,for the tensor current, respectively.Nc is the number of colors, and pj=Pj−kj=1,2 and k are the internal momenta carried by the quark and antiquark propagators of mass mj and mq, respectively. The corresponding denominators are given by Npj=pj2−mj2+iε and Nk=k2−mq2+iε. We take the qq¯ bound-state vertex functions Hpjpj2,k2=gj/pj2−Λj2+iε of the initial (j=1) and final (j=2) state pseudoscalar mesons, where gj and Λj are constant parameters in this manifestly covariant model.Performing the LF calculation in theq+=0 frame, one obtains the following identity q=qon+1/2γ+Δq−, where Δq−=q−−qon− and the subscript (on) denotes the on-mass shell quark momentum, i.e., pjon2=mj2 and kon2=mq2. Using this identity, one can separate the trace terms into the “on”-mass shell propagating part and the “off”-mass shell instantaneous one, i.e., Sμ=Sonμ+Soffμ for the vector current and Tμν=Tonμν+Toffμν for the tensor current.The explicit LF calculation in parallel with the manifestly covariant calculation of Eq. (13) to compute f±q2 can be found in [49] where f−q2 was obtained from f−+⊥q2. The identical result for f−+⊥q2 was also obtained in [45, 46] using the so-called “covariant LFQM” analysis. As shown in Ref. [45, 46, 49], while f+q2 obtained from the plus current was immune to the zero mode, the form factor f−q2 received both instantaneous and zero-mode contributions. The same situation happens for f−+−q2 although the zero-mode and the instantaneous contributions may differ quantitatively from f−+⊥q2. However, as we have shown in [54], f−+⊥q2 and f−+−q2 obtained in the standard LFQM by using our new correspondence relations between the BS model and the standard LFQM show identical result in the valence region of the q+=0 frame without involving explicit zero modes and instantaneous contributions. In this work, we shall show that the tensor form factor sq2 in the standard LFQM is independent of the components of the current, i.e., s+⊥q2=s+−q2. We should note that all of those equalities, i.e., f−+⊥q2=f−+−q2 and s+⊥q2=s+−q2are derived from the constraint of the on-mass shellness of the quark and antiquark propagators together with the zero-binding energy limit (i.e., M=M0) used in the standard LFQM.Therefore, from now on, we discuss only for the on-mass shell contributions in the valence region of theq+=0 frame between the manifestly covariant BS model and the standard LFQM. The on-shell contributions to Sμ and Tμν are given by
(16)Sonμ=4p1onμp2on·kon+m2mq+p2onμp1on·kon+m1mq+konμm1m2−p1on·p2on,and(17)Tonμν=4ip1onμm2konν+mqp2onν−p2onμm1konν+mqp1onν+konμm1p2onν−m2p1onν,respectively. The LF four-momenta of the on-mass shell quark and antiquark propagators in theq+=0 (i.e. P1+=P2+) frame are given by
(18)p1on=xP1+,m12+k⊥2xP1+,−k⊥,(19)p2on=xP1+,m22+k⊥+q⊥2xP1+,−k⊥−q⊥,(20)kon=1−xP1+,mq2+k⊥21−xP1+,k⊥,wherex=p1+/P1+ and x¯=k+/P1+ are the LF longitudinal momentum fractions of the quark and antiquark, which satisfy x+x¯=1.By the integration overk− in Eq. (13) and closing the contour in the lower half of the complex k− plane, one picks up the residue at k−=kon− in the valence region of 0<k+<P2+ (or 0<x<1). We denote the on-mass shell contribution to MVμ and MTμν in the valence region as MVμonBS and MTμνonBS, respectively. The explicit forms of MVμonBS and MTμνonBS are obtained as [54].
(21)MonBS=Nc∫01dxx¯∫d2k⊥16π3χ1x,k⊥χ2x,k⊥′Ton,whereM=MVμ,MTμν pairs with T=Sμ,Tμν. The LF quark-meson vertex function χ12 of the initial (final) state is given by
(22)χ12x,k′⊥=g12x2M122−M0′2M122−MΛ1Λ2′2,wherek⊥′=k⊥+1−xq⊥ and
(23)M02=k⊥2+m12x+k⊥2+mq21−x,(24)M0′2=k⊥′2+m22x+k⊥′2+mq21−x,are the invariant masses of the initial and final states, respectively. Likewise,MΛ12 is obtained as MΛ1=M0m1⟶Λ1and MΛ2′=M0′m2⟶Λ2.For the traceT=Sμ,Tμν calculations relevant to the form factors, the on-mass shell contributions Sonμ obtained from all three components μ=+,⊥,− of the vector current JVμ are given by [54].
(25)Son+=4P1+x¯k⊥·k⊥′+A1A2,(26)S⊥on=−2k⊥xx¯2k⊥·k⊥′+x¯q⊥2+m12+m22+2x2mq2+2xx¯m1mq+m2mq−m1m2−2q⊥xx¯k⊥2+A12,(27)Son−=4x2x¯P1+x¯m1A1+k⊥2m22+k⊥+q⊥2+x2x¯M02k⊥2+k⊥·q⊥+x2m1m2mq2+k⊥2+xx¯m2mqm12+k⊥2,whereAj=1−xmj+xmqj=1,2. Likewise, the on-shell contributions Tonμν obtained from the two sets of the tensor current JTμν, i.e., μ,ν=+,⊥ and +,−, are given by
(28)Ton+⊥=−4iP1+m1−m2k⊥+A1q⊥,(29)Ton+−=4ixx¯1−2xm1−m2k⊥2+2x¯A1k⊥·q⊥+x¯A1q⊥2+m2−m1A1A2.Using Eqs. (8), (9), (10), (11), (12), and (21), one obtains the on-mass shell contributions to the weak form factors f+,f−,s as follows
(30)FonBS=Nc∫01dxx¯∫d2k⊥16π3χ1x,k⊥OonBSχ2x,k⊥′,where the form factorsF=f+,f−+⊥,f−+−,s+⊥,s+− obtained from different combinations of the vector and tensor currents pair with the following corresponding operators OonBS=O+,O−+⊥,O−+−,Os+⊥,Os+− including the spin and external momenta factors:
(31)O+=Son+2P1+,(32)O−+⊥=Son+2P1++S⊥on·q⊥q⊥2,(33)O−+−=−Son+2P1+ΔM+2+q⊥2ΔM−2−q⊥2+P1+Son−ΔM−2−q⊥2,(34)Os+⊥=−iTon+⊥·q⊥2q⊥2P1+,(35)Os+−=−iTon+−2ΔM−2−q⊥2.In the manifestly covariant BS model given by Eq. (13), we note that only the two form factors f+q2 and s+⊥q2 defined by Eqs. (31) and (34), respectively, are immune to the zero modes as well as the instantaneous contributions and thus exactly equal to the full exact solution (i.e., manifestly covariant solution), i.e., f+onBS=f+Cov and s+⊥onBS=sCov, without involving such LF treacherous points. However, since the other three form factors f−+⊥, f−+−, and s+− are contaminated by the zero modes as well as the instantaneous contributions, the on-mass shell contributions f−+⊥onBS, f−+−onBS, and s+⊥onBS themselves can never be the same as the exact solutions unless the zero modes and the instantaneous contributions are taken into account. Furthermore, one can easily check that f−+⊥onBS≠f−+−onBS and s+−onBS≠s+⊥onBS.However, in the following subsection, we shall show in the standard LFQM (denoted by SLF) thatf−SLF=f−+⊥onSLF=f−+−onSLF and sSLF=s+−onSLF=s+⊥onSLF without involving explicit zero-mode and instantaneous contributions, which comes after using our new correspondence relations between the BS model and the standard LFQM.


### 3.2. The Standard LFQM

In the standard LFQM [39, 41, 42, 55–57, 60, 61], the LF wave function (LFWF) of a ground state pseudoscalar meson as a qq¯ bound state is given by
(36)Ψλλ¯x,k⊥=ϕx,k⊥Rλλ¯x,k⊥,whereRλλ¯x,k⊥ is the spin-orbit WF that is obtained by the interaction-independent Melosh transformation from the ordinary spin-orbit WF assigned by the quantum number JPC. The covariant form of Rλλ¯ with the definite spin S,Sz=0,0 constructed out of the LF helicity λλ¯ of a quark (antiquark) is given by
(37)Rλλ¯=u¯λpqγ5vλ¯pq¯2M02−m1−mq21/2,which satisfies the unitarity condition,∑λλ¯Rλλ¯†Rλλ¯=1. Its explicit matrix form is given by
(38)Rλλ¯=12k⊥2+A12−kLA1−A1−kR,wherekR=kx+iky and kL=kx−iky.For the radial WFϕx,k⊥ in Eq. (36), we use the Gaussian WF
(39)ϕx,k⊥=4π3/4β3/2∂kz∂xexp−k⟶2/2β2,wherek⟶2=k⊥2+kz2 and β is the variational parameter. The longitudinal component kz is defined by kz=x−1/2M0+mq2−m12/2M0, and the Jacobian of the variable transformation x,k⊥⟶k⟶=k⊥,kz is given by ∂kz/∂x=M0/4x1−x1−m12−mq22/M04. The normalization of our Gaussian radial WF is then given by
(40)∫01dx∫d2k⊥16π3ϕx,k⊥2=1.In particular, the key idea in our LFQM [49, 55–57] for mesons is to treat ϕx,k⊥ as a trial function for the variational principle to the QCD-motivated effective Hamiltonian saturating the Fock state expansion by the constituent quark and antiquark. Using this Hamiltonian, we analyze the meson mass spectra and various wave-function-related observables, such as decay constants, electromagnetic form factors of mesons in a spacelike region, and the weak form factors for the exclusive semileptonic and rare decays of pseudoscalar mesons in the timelike region [49, 51–57].In this standard LFQM, the matrix elements of the vector and tensor currents in Eqs. (1) and (2) are obtained by the convolution formula of the initial and final state LFWFs in the q+=0 frame as follows:
(41)MonSLF=∑λ′s∫01dx∫d2k⊥16π3ϕ1x,k⊥ϕ2x,k⊥′×Rλ2λ¯†u¯λ2p2p2+Γuλ1p1p1+Rλ1λ¯,whereM=MVμ,MTμν pairs with Γ=γμ,σμν.Then, we first compute the zero-mode free form factors, i.e.f+onSLF and s+⊥onSLF, in the SLF formulation as follows
(42)FonSLF=∫01x¯dx∫d2k⊥32π3ϕ1x,k⊥A12+k⊥2OonSLFϕ2x,k⊥′A22+k⊥′2,where the form factorsF=f+,s+⊥ pair with the following corresponding operators OonSLF=OonBS=O+,Os+⊥ given by Eqs. (31) and (34).We should note that the main differences between the covariant BS model and the standard LFQM are attributed to the different spin structures of theqq¯ system (i.e., off-shellness in the BS model vs. on-shellness in the standard LFQM) and the different meson-quark vertex functions (χ vs. ϕ). In other words, while the results of the covariant BS model allow the nonzero binding energy EB.E=M2−M02, the SLF result is obtained from the condition of on-mass shell quark and antiquark (i.e., M⟶M0).Comparing these two form factorsF=f+,s+⊥ defined in Eq. (30) in the BS model and Eq. (42) in the standard LFQM, one can easily find the correspondence relation between the two models as follows:
(43)2Ncχ12x,k⊥′1−x⟶ϕ12x,k⊥′A122+k⊥′2.In many previous LFQM analyses [40, 45, 46, 49], the correspondence in Eq. (43) has also been used for the mapping of other physical observables contaminated by the zero modes and/or the instantaneous contributions. However, applying Eq. (43) together with the same operators given by Eqs. (32), (33), and (35) to the other form factors F=f−+⊥,f−+−,s+− obtained from the only on-mass shell contributions, one encounters the same problems as the BS model, i.e., f−+⊥onSLF≠f−+−onSLF and s+−onSLF≠s+⊥onSLF implying that the same physical quantities obtained from different components of the current yield different results.In our previous analysis [51–53], however, we found that the correspondence relation including only LF vertex functions given by Eq. (43) brings about the self-consistency problem, i.e., the same physical quantity obtained from different components of the current and/or the polarization vectors yields different results in the standard LFQM. Furthermore, we also discovered the additional requirement for the correct correspondence relation between the two models to obtain the current-component independent physical observables in the standard LFQM.Our new correspondence relation (denoted by “CJ-scheme” for convenience) to restore the self-consistency in the standard LFQM is given by [51–54]:
(44)2Ncχ12x,k⊥′1−x⟶ϕ12x,k⊥′A122+k⊥′2,M12⟶M′0,that is, the physical massM12 included in the integrand of the BS amplitude, e.g., the operators OonBS in Eqs. (31) (32), (33), (34), and (35) should be replaced by the invariant mass M0′ as all constituent quark and antiquarks are required to be on their respective mass shell in the standard LFQM. We should note that this “CJ-scheme” has been verified through our previous analyses for the decay constants and the twist-2 and-3 DAs of pseudoscalar and vector mesons [51–53] and the form factor f−q2 for the semileptonic B decays [54].We now show in this work that the “CJ-scheme” is also valid to obtain the current-component independent tensor form factorsq2 in addition to f−q2 [54]. That is, applying Eq. (44) to the form factors F=f−+−,s+− defined in Eq. (30) implies the replacements of the current operators OonBS=O−+−,Os+− in the BS model with OonSLF=O~−+−,O~s+− in the standard LFQM, i.e.,
(45)O~−+−=−Son+2P1+ΔM0+2+q⊥2ΔM0−2−q⊥2+P1+Son−ΔM0−2−q⊥2,(46)O~s+−=−iTon+−2ΔM0−2−q⊥2,whereΔM0±2=M02±M0′2. Then, we obtain from Eqs. (42), (45), and (46) the current-component independent form factors, i.e., f−+⊥onSLF≐f−+−onSLF and s+−onSLF≐s+⊥onSLF in the standard LFQM, where “≐” represents the equality of both sides numerically. The additional requirement in the “CJ-scheme”, i.e., M12⟶M0′, can therefore be regarded as the effective inclusion of the zero modes in the valence region of the q+=0 frame in the standard LFQM. This replacement M12⟶M0′ is not possible in the BS model due to the form of the LF vertex function χ given by Eq. (22).The final results forf+, f−=f−+⊥≐f−+−, and s=s+⊥≐s+− in the standard LFQM are given by
(47)f+q2=∫01dx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2A1A2+k⊥·k⊥′,(48)f−+⊥q2=∫01x¯dx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2−x¯M02+m2−mqA1−mqm1−mq+k⊥·q⊥q2M02+M′02−2m1−mqm2−mq,(49)f−+−q2=∫01dxx2∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2a0x2x¯M02k⊥2+k⊥·q⊥+x¯m1A1+k⊥2m22+k⊥+q⊥2+x2m1m2mq2+k⊥2+xx¯m2mqm12+k⊥2−x2b0k⊥·k⊥′+A1A2, for the vector current, and
(50)s+⊥q2=−∫011−xdx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2m1−m2k⊥·q⊥q⊥2+A1,(51)s+−q2=∫01dxx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2a021−2xm1−m2k⊥2+21−xA1k⊥·q⊥+1−xA1q⊥2+m2−m1A1A2,for the tensor current, wherea0=2/M02−M0′2−q⊥2 and b0=M02+M0′2+q⊥2/M02−M0′2−q⊥2. Indeed, our prescription M12⟶M0′ is applied through the two terms a0,b0 in f−+− and s+−. Finally, we confirm from the numerical calculations the current independencies of the form factors, i.e., f−q2=f−+⊥≐f−+− and sq2=s+⊥≐s+−, which supports the universality of the “CJ-scheme” given by Eq. (44) and the self-consistency of our standard LFQM.For our numerical calculations in the following section, we use the tensor form factorfTq2=sq2M1+M2 as defined in Eq. (4). We should emphasize that the physical masses M12 used in defining fT is nothing to do with our correspondence relations. Only the physical masses M12 appeared as a result from the choice of minus component (μ,ν=−) of the vector and tensor currents given by Eqs. (1) and (2) are eligible for the transformation into the corresponding invariant masses M0′ as shown in Eq. (10).


## 3.1. Manifestly Covariant Model

In the solvable model, based on the covariant BS model of (3+1)-dimensional fermion field theory [50, 54], the matrix elements M=MVμ,MTμν are given by
(13)M=iNc∫d4k2π4Hp1THp2Np1NkNp2,where the corresponding trace termsT=Sμ,Tμν are given by
(14)Sμ=Trγ5p1+m1γμp2+m2γ5−k+mq, for the vector current and
(15)Tμν=Trγ5p1+m1σμνp2+m2γ5−k+mq,for the tensor current, respectively.Nc is the number of colors, and pj=Pj−kj=1,2 and k are the internal momenta carried by the quark and antiquark propagators of mass mj and mq, respectively. The corresponding denominators are given by Npj=pj2−mj2+iε and Nk=k2−mq2+iε. We take the qq¯ bound-state vertex functions Hpjpj2,k2=gj/pj2−Λj2+iε of the initial (j=1) and final (j=2) state pseudoscalar mesons, where gj and Λj are constant parameters in this manifestly covariant model.Performing the LF calculation in theq+=0 frame, one obtains the following identity q=qon+1/2γ+Δq−, where Δq−=q−−qon− and the subscript (on) denotes the on-mass shell quark momentum, i.e., pjon2=mj2 and kon2=mq2. Using this identity, one can separate the trace terms into the “on”-mass shell propagating part and the “off”-mass shell instantaneous one, i.e., Sμ=Sonμ+Soffμ for the vector current and Tμν=Tonμν+Toffμν for the tensor current.The explicit LF calculation in parallel with the manifestly covariant calculation of Eq. (13) to compute f±q2 can be found in [49] where f−q2 was obtained from f−+⊥q2. The identical result for f−+⊥q2 was also obtained in [45, 46] using the so-called “covariant LFQM” analysis. As shown in Ref. [45, 46, 49], while f+q2 obtained from the plus current was immune to the zero mode, the form factor f−q2 received both instantaneous and zero-mode contributions. The same situation happens for f−+−q2 although the zero-mode and the instantaneous contributions may differ quantitatively from f−+⊥q2. However, as we have shown in [54], f−+⊥q2 and f−+−q2 obtained in the standard LFQM by using our new correspondence relations between the BS model and the standard LFQM show identical result in the valence region of the q+=0 frame without involving explicit zero modes and instantaneous contributions. In this work, we shall show that the tensor form factor sq2 in the standard LFQM is independent of the components of the current, i.e., s+⊥q2=s+−q2. We should note that all of those equalities, i.e., f−+⊥q2=f−+−q2 and s+⊥q2=s+−q2are derived from the constraint of the on-mass shellness of the quark and antiquark propagators together with the zero-binding energy limit (i.e., M=M0) used in the standard LFQM.Therefore, from now on, we discuss only for the on-mass shell contributions in the valence region of theq+=0 frame between the manifestly covariant BS model and the standard LFQM. The on-shell contributions to Sμ and Tμν are given by
(16)Sonμ=4p1onμp2on·kon+m2mq+p2onμp1on·kon+m1mq+konμm1m2−p1on·p2on,and(17)Tonμν=4ip1onμm2konν+mqp2onν−p2onμm1konν+mqp1onν+konμm1p2onν−m2p1onν,respectively. The LF four-momenta of the on-mass shell quark and antiquark propagators in theq+=0 (i.e. P1+=P2+) frame are given by
(18)p1on=xP1+,m12+k⊥2xP1+,−k⊥,(19)p2on=xP1+,m22+k⊥+q⊥2xP1+,−k⊥−q⊥,(20)kon=1−xP1+,mq2+k⊥21−xP1+,k⊥,wherex=p1+/P1+ and x¯=k+/P1+ are the LF longitudinal momentum fractions of the quark and antiquark, which satisfy x+x¯=1.By the integration overk− in Eq. (13) and closing the contour in the lower half of the complex k− plane, one picks up the residue at k−=kon− in the valence region of 0<k+<P2+ (or 0<x<1). We denote the on-mass shell contribution to MVμ and MTμν in the valence region as MVμonBS and MTμνonBS, respectively. The explicit forms of MVμonBS and MTμνonBS are obtained as [54].
(21)MonBS=Nc∫01dxx¯∫d2k⊥16π3χ1x,k⊥χ2x,k⊥′Ton,whereM=MVμ,MTμν pairs with T=Sμ,Tμν. The LF quark-meson vertex function χ12 of the initial (final) state is given by
(22)χ12x,k′⊥=g12x2M122−M0′2M122−MΛ1Λ2′2,wherek⊥′=k⊥+1−xq⊥ and
(23)M02=k⊥2+m12x+k⊥2+mq21−x,(24)M0′2=k⊥′2+m22x+k⊥′2+mq21−x,are the invariant masses of the initial and final states, respectively. Likewise,MΛ12 is obtained as MΛ1=M0m1⟶Λ1and MΛ2′=M0′m2⟶Λ2.For the traceT=Sμ,Tμν calculations relevant to the form factors, the on-mass shell contributions Sonμ obtained from all three components μ=+,⊥,− of the vector current JVμ are given by [54].
(25)Son+=4P1+x¯k⊥·k⊥′+A1A2,(26)S⊥on=−2k⊥xx¯2k⊥·k⊥′+x¯q⊥2+m12+m22+2x2mq2+2xx¯m1mq+m2mq−m1m2−2q⊥xx¯k⊥2+A12,(27)Son−=4x2x¯P1+x¯m1A1+k⊥2m22+k⊥+q⊥2+x2x¯M02k⊥2+k⊥·q⊥+x2m1m2mq2+k⊥2+xx¯m2mqm12+k⊥2,whereAj=1−xmj+xmqj=1,2. Likewise, the on-shell contributions Tonμν obtained from the two sets of the tensor current JTμν, i.e., μ,ν=+,⊥ and +,−, are given by
(28)Ton+⊥=−4iP1+m1−m2k⊥+A1q⊥,(29)Ton+−=4ixx¯1−2xm1−m2k⊥2+2x¯A1k⊥·q⊥+x¯A1q⊥2+m2−m1A1A2.Using Eqs. (8), (9), (10), (11), (12), and (21), one obtains the on-mass shell contributions to the weak form factors f+,f−,s as follows
(30)FonBS=Nc∫01dxx¯∫d2k⊥16π3χ1x,k⊥OonBSχ2x,k⊥′,where the form factorsF=f+,f−+⊥,f−+−,s+⊥,s+− obtained from different combinations of the vector and tensor currents pair with the following corresponding operators OonBS=O+,O−+⊥,O−+−,Os+⊥,Os+− including the spin and external momenta factors:
(31)O+=Son+2P1+,(32)O−+⊥=Son+2P1++S⊥on·q⊥q⊥2,(33)O−+−=−Son+2P1+ΔM+2+q⊥2ΔM−2−q⊥2+P1+Son−ΔM−2−q⊥2,(34)Os+⊥=−iTon+⊥·q⊥2q⊥2P1+,(35)Os+−=−iTon+−2ΔM−2−q⊥2.In the manifestly covariant BS model given by Eq. (13), we note that only the two form factors f+q2 and s+⊥q2 defined by Eqs. (31) and (34), respectively, are immune to the zero modes as well as the instantaneous contributions and thus exactly equal to the full exact solution (i.e., manifestly covariant solution), i.e., f+onBS=f+Cov and s+⊥onBS=sCov, without involving such LF treacherous points. However, since the other three form factors f−+⊥, f−+−, and s+− are contaminated by the zero modes as well as the instantaneous contributions, the on-mass shell contributions f−+⊥onBS, f−+−onBS, and s+⊥onBS themselves can never be the same as the exact solutions unless the zero modes and the instantaneous contributions are taken into account. Furthermore, one can easily check that f−+⊥onBS≠f−+−onBS and s+−onBS≠s+⊥onBS.However, in the following subsection, we shall show in the standard LFQM (denoted by SLF) thatf−SLF=f−+⊥onSLF=f−+−onSLF and sSLF=s+−onSLF=s+⊥onSLF without involving explicit zero-mode and instantaneous contributions, which comes after using our new correspondence relations between the BS model and the standard LFQM.


## 3.2. The Standard LFQM

In the standard LFQM [39, 41, 42, 55–57, 60, 61], the LF wave function (LFWF) of a ground state pseudoscalar meson as a qq¯ bound state is given by
(36)Ψλλ¯x,k⊥=ϕx,k⊥Rλλ¯x,k⊥,whereRλλ¯x,k⊥ is the spin-orbit WF that is obtained by the interaction-independent Melosh transformation from the ordinary spin-orbit WF assigned by the quantum number JPC. The covariant form of Rλλ¯ with the definite spin S,Sz=0,0 constructed out of the LF helicity λλ¯ of a quark (antiquark) is given by
(37)Rλλ¯=u¯λpqγ5vλ¯pq¯2M02−m1−mq21/2,which satisfies the unitarity condition,∑λλ¯Rλλ¯†Rλλ¯=1. Its explicit matrix form is given by
(38)Rλλ¯=12k⊥2+A12−kLA1−A1−kR,wherekR=kx+iky and kL=kx−iky.For the radial WFϕx,k⊥ in Eq. (36), we use the Gaussian WF
(39)ϕx,k⊥=4π3/4β3/2∂kz∂xexp−k⟶2/2β2,wherek⟶2=k⊥2+kz2 and β is the variational parameter. The longitudinal component kz is defined by kz=x−1/2M0+mq2−m12/2M0, and the Jacobian of the variable transformation x,k⊥⟶k⟶=k⊥,kz is given by ∂kz/∂x=M0/4x1−x1−m12−mq22/M04. The normalization of our Gaussian radial WF is then given by
(40)∫01dx∫d2k⊥16π3ϕx,k⊥2=1.In particular, the key idea in our LFQM [49, 55–57] for mesons is to treat ϕx,k⊥ as a trial function for the variational principle to the QCD-motivated effective Hamiltonian saturating the Fock state expansion by the constituent quark and antiquark. Using this Hamiltonian, we analyze the meson mass spectra and various wave-function-related observables, such as decay constants, electromagnetic form factors of mesons in a spacelike region, and the weak form factors for the exclusive semileptonic and rare decays of pseudoscalar mesons in the timelike region [49, 51–57].In this standard LFQM, the matrix elements of the vector and tensor currents in Eqs. (1) and (2) are obtained by the convolution formula of the initial and final state LFWFs in the q+=0 frame as follows:
(41)MonSLF=∑λ′s∫01dx∫d2k⊥16π3ϕ1x,k⊥ϕ2x,k⊥′×Rλ2λ¯†u¯λ2p2p2+Γuλ1p1p1+Rλ1λ¯,whereM=MVμ,MTμν pairs with Γ=γμ,σμν.Then, we first compute the zero-mode free form factors, i.e.f+onSLF and s+⊥onSLF, in the SLF formulation as follows
(42)FonSLF=∫01x¯dx∫d2k⊥32π3ϕ1x,k⊥A12+k⊥2OonSLFϕ2x,k⊥′A22+k⊥′2,where the form factorsF=f+,s+⊥ pair with the following corresponding operators OonSLF=OonBS=O+,Os+⊥ given by Eqs. (31) and (34).We should note that the main differences between the covariant BS model and the standard LFQM are attributed to the different spin structures of theqq¯ system (i.e., off-shellness in the BS model vs. on-shellness in the standard LFQM) and the different meson-quark vertex functions (χ vs. ϕ). In other words, while the results of the covariant BS model allow the nonzero binding energy EB.E=M2−M02, the SLF result is obtained from the condition of on-mass shell quark and antiquark (i.e., M⟶M0).Comparing these two form factorsF=f+,s+⊥ defined in Eq. (30) in the BS model and Eq. (42) in the standard LFQM, one can easily find the correspondence relation between the two models as follows:
(43)2Ncχ12x,k⊥′1−x⟶ϕ12x,k⊥′A122+k⊥′2.In many previous LFQM analyses [40, 45, 46, 49], the correspondence in Eq. (43) has also been used for the mapping of other physical observables contaminated by the zero modes and/or the instantaneous contributions. However, applying Eq. (43) together with the same operators given by Eqs. (32), (33), and (35) to the other form factors F=f−+⊥,f−+−,s+− obtained from the only on-mass shell contributions, one encounters the same problems as the BS model, i.e., f−+⊥onSLF≠f−+−onSLF and s+−onSLF≠s+⊥onSLF implying that the same physical quantities obtained from different components of the current yield different results.In our previous analysis [51–53], however, we found that the correspondence relation including only LF vertex functions given by Eq. (43) brings about the self-consistency problem, i.e., the same physical quantity obtained from different components of the current and/or the polarization vectors yields different results in the standard LFQM. Furthermore, we also discovered the additional requirement for the correct correspondence relation between the two models to obtain the current-component independent physical observables in the standard LFQM.Our new correspondence relation (denoted by “CJ-scheme” for convenience) to restore the self-consistency in the standard LFQM is given by [51–54]:
(44)2Ncχ12x,k⊥′1−x⟶ϕ12x,k⊥′A122+k⊥′2,M12⟶M′0,that is, the physical massM12 included in the integrand of the BS amplitude, e.g., the operators OonBS in Eqs. (31) (32), (33), (34), and (35) should be replaced by the invariant mass M0′ as all constituent quark and antiquarks are required to be on their respective mass shell in the standard LFQM. We should note that this “CJ-scheme” has been verified through our previous analyses for the decay constants and the twist-2 and-3 DAs of pseudoscalar and vector mesons [51–53] and the form factor f−q2 for the semileptonic B decays [54].We now show in this work that the “CJ-scheme” is also valid to obtain the current-component independent tensor form factorsq2 in addition to f−q2 [54]. That is, applying Eq. (44) to the form factors F=f−+−,s+− defined in Eq. (30) implies the replacements of the current operators OonBS=O−+−,Os+− in the BS model with OonSLF=O~−+−,O~s+− in the standard LFQM, i.e.,
(45)O~−+−=−Son+2P1+ΔM0+2+q⊥2ΔM0−2−q⊥2+P1+Son−ΔM0−2−q⊥2,(46)O~s+−=−iTon+−2ΔM0−2−q⊥2,whereΔM0±2=M02±M0′2. Then, we obtain from Eqs. (42), (45), and (46) the current-component independent form factors, i.e., f−+⊥onSLF≐f−+−onSLF and s+−onSLF≐s+⊥onSLF in the standard LFQM, where “≐” represents the equality of both sides numerically. The additional requirement in the “CJ-scheme”, i.e., M12⟶M0′, can therefore be regarded as the effective inclusion of the zero modes in the valence region of the q+=0 frame in the standard LFQM. This replacement M12⟶M0′ is not possible in the BS model due to the form of the LF vertex function χ given by Eq. (22).The final results forf+, f−=f−+⊥≐f−+−, and s=s+⊥≐s+− in the standard LFQM are given by
(47)f+q2=∫01dx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2A1A2+k⊥·k⊥′,(48)f−+⊥q2=∫01x¯dx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2−x¯M02+m2−mqA1−mqm1−mq+k⊥·q⊥q2M02+M′02−2m1−mqm2−mq,(49)f−+−q2=∫01dxx2∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2a0x2x¯M02k⊥2+k⊥·q⊥+x¯m1A1+k⊥2m22+k⊥+q⊥2+x2m1m2mq2+k⊥2+xx¯m2mqm12+k⊥2−x2b0k⊥·k⊥′+A1A2, for the vector current, and
(50)s+⊥q2=−∫011−xdx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2m1−m2k⊥·q⊥q⊥2+A1,(51)s+−q2=∫01dxx∫d2k⊥16π3ϕ1x,k⊥A12+k⊥2ϕ2x,k⊥′A22+k⊥′2a021−2xm1−m2k⊥2+21−xA1k⊥·q⊥+1−xA1q⊥2+m2−m1A1A2,for the tensor current, wherea0=2/M02−M0′2−q⊥2 and b0=M02+M0′2+q⊥2/M02−M0′2−q⊥2. Indeed, our prescription M12⟶M0′ is applied through the two terms a0,b0 in f−+− and s+−. Finally, we confirm from the numerical calculations the current independencies of the form factors, i.e., f−q2=f−+⊥≐f−+− and sq2=s+⊥≐s+−, which supports the universality of the “CJ-scheme” given by Eq. (44) and the self-consistency of our standard LFQM.For our numerical calculations in the following section, we use the tensor form factorfTq2=sq2M1+M2 as defined in Eq. (4). We should emphasize that the physical masses M12 used in defining fT is nothing to do with our correspondence relations. Only the physical masses M12 appeared as a result from the choice of minus component (μ,ν=−) of the vector and tensor currents given by Eqs. (1) and (2) are eligible for the transformation into the corresponding invariant masses M0′ as shown in Eq. (10).


## 4. Numerical Results

In our numerical calculations for the semileptonic and rareD⟶π,K decays, we use the model parameters (mqq¯,βqq¯) for the harmonic oscillator (HO) confining potential given in Table 1 obtained from the calculation of the ground state meson mass spectra [49, 57]. The decay constants of π,K,D mesons obtained from the HO parameters are given by fπ,fK,fD=131,155,197 MeV compared to the experimental data [62], fπexp.,fKexp.,fDexp.=130.21.2,155.73,212.67 MeV. While the decay constant of D meson is not quite sensitive to the quark mass variation, e.g., fD=199+1−2 MeV for mc=1.7−0.1+0.1 GeV, we find that the form factors are somewhat sensitive to mc. Thus, as a sensitivity check of our LFQM, we use this charm quark mass variation for the calculations of the form factors and the branching ratios. For the physical D,K,π meson masses, we use the central values quoted by the Particle Data Group (PDG) [62].Table 1
The constituent quark mass (GeV) and the Gaussian parametersβ (GeV) for the HO potential obtained by the variational principle [49, 57]. q=u and d.
mqmsmcβqqβqcβsc0.250.481.80.31940.42160.4686In principle, it is possible to use theq+≠0 frame satisfying q2=q+q−−q⊥2>0 for this timelike semileptonic and rare decays. However, in this q+≠0 frame, it is inevitable to confront the particle-number-nonconserving Fock state (or nonvalence) contribution [63, 64]. The main source of difficulty in the LFQM phenomenology is the paucity of information on the nonwave-function vertex [50] in the nonvalence diagram arising from the quark-antiquark pair creation/annihilation. This should contrast with the usual LFWF used in the valence region. Contrary to the q+≠0 frame, the q+=0 frame does not suffer from the nonvalence contribution although one needs to be cautious about the zero-mode problem as we discussed already. Once the zero-mode issue is resolved as we proved in this work, it is straightforward to analytically continue the form factors given by Eqs. (47), (48), (49), (50), and (51) obtained in the spacelike region to the timelike physical region.Our results of the form factorsf±,f0,fT obtained from Eqs. (47), (48), (49), (50), and (51) can also be compared with several parametric forms. Among several forms, a more systematic and model-independent parametrization of semileptonic form factors, often referred to as “z-expansion” or “z-parametrization” [65, 66], has been developed based on general properties of analyticity, unitarity, and crossing symmetries. Especially, this z-parametrization provides better control of theoretical uncertainties in LQCD calculations [23, 24].Our direct LFQM results for the form factorsfiq2i=±,0,T are also well described by the “z-parametrization,” which takes the form [23, 24]
(52)fjq2=fj0+cjz−z01+z+z0/21−bjq2,where
(53)z=t+−q2−t+−t0t+−q2+t+−t0,andz0=zq2=0 with t±=M1±M22 and t0=t+1−1−t−/t+.The fitted parametersbj,cjj=+,0,T for the D⟶π and D⟶K TFFs f+,f0,fT are summarized in Tables 2 and 3, respectively, where the errors occur due to the choice of mc=1.7−0.1+0.1 GeV. In Table 4, we also compare the form factors f+0 and ∣fT0∣ for D⟶π,K transitions at q2=0 with those obtained from various theoretical model predictions and experimental data [13, 67, 68].Table 2
Fitted parametersbj,cj in Eq. (52) for the D⟶π TFFs with mc=1.7−0.1+0.1 GeV. bj is in unit of (GeV−2).
f+,00b+c+b0c0fT0bTcT0.613+22−210.1899+233−208−0.8200−317+2750.2986+11.8745−0.08821.9051+107.077−0.5986−0.501−39+360.1957+242−2160.6290+481−417Table 3
Fitted parametersbj,cj in Eq. (52) for the D⟶K decay with mc=1.7−0.1+0.1 GeV. bj is in unit of (GeV−2).
f+,00b+c+b0c0fT0bTcT0.744+23−220.1787+176−157−1.1711−618+571−0.0563+0.1426+5.0348−2.3039+1.8289+78.3509−0.660−45+420.1826+182−1630.9893+897−794Table 4
Form factorsf+0 and fT0 for D⟶π,K transitions at q2=0 compared with various model predictions and experimental data.
F0This work[23, 24][27][34][35][44][47]BES III [13]BABAR [67, 68]f+Dπ00.613+22−210.612 (35)0.63 (11)0.6400.63 (9)−0.66 (1)0.637 (24)0.610 (25)fTDπ00.501−39+360.506 (79)−−−0.84−13+16−−−f+DK00.744+23−220.765 (31)0.75 (12)0.7160.77 (11)−0.79 (1)0.737 (4)0.727 (11)fTDK00.660−45+420.687 (54)−−−0.96−15+17−−−In Figure1, we show the q2 dependences of f+Dπq2 (black lines), f0Dπq2 (blue lines), and fTDπq2 (red lines) for D⟶π decay, where the solid and dashed lines represent the results obtained from mc=1.8 GeV and 1.6 GeV, respectively. That is, the bands correspond to the sensitivity coming from the charm quark mass variation, mc=1.7−0.1+0.1 GeV in our LFQM. We should note that the form factors are displayed not only for the whole timelike kinematic region [0≤q2≤MD−Mπ2] (in unit of GeV2) but also for the spacelike region (−0.5≤q2≤0) (in unit of GeV2) to demonstrate the validity of our analytic continuation from spacelike region to the timelike one by changing q⊥2 to −q2 in the form factors. For comparison, the data (circles) of the form factors f+,f0,fT from the LQCD (for the ETM Collaboration) [23, 24] and the data of f+ (squares) extracted from the BABAR [67] are shown. Our results for f+Dπ0=0.613+22−21 and ∣fTDπ0∣=0.501−39+36 are in good agreement with f+Dπ0=0.61025 from the BABAR [67] and f+Dπ0=0.63724 from the BES III [13], as well as f+Dπ0=0.61235 and ∣fTDπ0∣=0.50679 from the LQCD [23, 24]. As one can see from Figure 1, the sensitivity to the charm quark mass is more pronounced at the zero-recoil (q2=qmax2) of the final meson than the maximum recoil (q2=0). Especially, the q2-dependent behaviors of our results show better agreement with the data from the BABAR [67] and LQCD [23, 24] when we use mc≃1.6 GeV rather than 1.8 GeV.Figure 1
(Color online): theq2 dependent form factors (f+,f0,fT) of the D⟶π decay for both spacelike and the kinematic timelike regions, −0.5≤q2≤MD−Mπ2 GeV2. For comparison, the data taken from the LQCD (circles) [23, 24] and BABAR [67] (squares) are shown.In Figure2, we show the q2 dependences of f+DK,f0DK,fTDK for D⟶K decay, compared with the results from the LQCD [23, 24]. The same line codes are used as in Figure 1. Our predictions of f+DK0=0.744+23−22 and ∣fTDK0∣=0.660−45+42 agree with f+DK0=0.7374 from the BES III [13] and f+DK0=0.72711 from the BABAR [68], as well as f+DK0=0.76431 and ∣fTDK0∣=0.68754 from the LQCD [23, 24] within the error bars. As in the case of D⟶π decay, the q2-dependent behaviors of our results show better agreement with the data from the LQCD [23, 24] when we use mc≃1.6 GeV rather than 1.8 GeV.Figure 2
(Color online): theq2 dependent form factors (f+,f0,fT) of the D⟶K decay for both spacelike and the kinematic timelike regions, −0.5≤q2≤MD−MK2 GeV2. For comparison, the data taken from the LQCD (circles) [23, 24] are shown.Figures3 and 4 show our predictions for the differential decay rates of D⟶πeνe and D⟶Keνe decays, respectively, compared with the experimental data from the BABAR [67, 68] (black circles), CLEO [69] (blue squares), and BES III [13, 14] for neutral D0 (red diamonds) and charged D+ with the account of isospin factor (green triangles). In our numerical calculations of the branching ratios, we use the CKM matrix elements Vcd=0.221±0.004 and Vcs=0.987±0.011 quoted by the PDG [62]. Considering uncertainties coming from the CKM elements and the constituent charm quark mass mc=1.7−0.1+0.1 GeV, we made band plots, i.e., the solid (dashed) lines represent the results obtained from mc=1.81.6 GeV with lower (upper) limits of the CKM elements. Our results are shown to be consistent with the current available experimental data within those uncertainties.Figure 3
Differential decay rate for theD⟶πeνe decay compared with the experimental data from the BABAR [67, 68] (black circles), CLEO [69] (blue squares), and BES III [13, 14] for neutral D0 (red diamonds) and charged D+ with the account of isospin factor (green triangles).Figure 4
Differential decay rate for theD⟶Keνe decay compared with the experimental data from the BABAR [67, 68] (black circles), CLEO [69] (blue squares), and BES III [13, 14] for neutral D0 (red diamonds) and charged D+ with the account of isospin factor (green triangles).In Tables5 and 6, we summarize our results for the branching ratios for D⟶πℓνℓ and D⟶Kℓνℓ (ℓ=e,μ), respectively, and compare with the experimental data from PDG [62]. Our results for Br(D⟶π) are best fit to the data with mc=1.6 GeV but those for Br(D⟶K) prefers mc=1.7 GeV.Table 5
Branching ratios (in10−3) for D⟶πℓνℓ (ℓ=e,μ) obtained from using mc=1.7−0.1+0.1 GeV together with ∣Vcd∣=0.221±0.004 [62].
ChannelOursPDG [62]D+⟶π0e+νe3.03−0.44+0.573.72±0.17D+⟶π0μ+νμ2.97−0.44+0.583.50±0.15D0⟶π−e+νe2.37−0.34+0.452.91±0.04D0⟶π−μ+νμ2.33−0.34+0.442.67±0.12Table 6
Branching ratios (in %) forD⟶Kℓνℓℓ=e,μ decays obtained from using mc=1.7−0.1+0.1 GeV together with Vcs=0.987±0.011 [62].
ChannelOursPDG [62]D+⟶K¯0e+νe8.88−0.80+1.108.73±0.10D+⟶K¯0μ+νμ8.64−0.79+1.078.76±0.19D0⟶K−e+νe3.50−0.31+0.433.54±0.034D0⟶K−μ+νμ3.41−0.30+0.413.41±0.004Finally, as a test for the LFU, theR ratios of the semileptonic D⟶π,K decays are defined by
(54)RP=BrD⟶PμνμBrD⟶Peνe,whereP=π,K. Our predictions for RP obtained from using mc=1.7−0.1+0.1 GeV are as follows: Rπ0,Rπ−=0.980−0.003+0.165,0.983−0.003−0.001, RK−,RK¯0=0.974+0.001−0.002,0.973−0.001−0.001. Our results are consistent with the recent measurements from the BES III, Rπ0,Rπ−=0.942±0.046,0.905±0.035 [15], and RK−=0.974±0.014 [18], as well as other theoretical predictions such as Rπ=0.9852 and RK=0.9751 from the LQCD [70], Rπ=0.985 and RK=0.980 from the RQM [34], and Rπ=0.98 and RK=0.97 from the CCQM [35].


## 5. Summary and Discussion

In this work, we discussed the self-consistence description on the weak form factorsf+, f− (or f0), and fT for the exclusive semileptonic D⟶π,Kℓνℓℓ=e,μ,τ and rare D⟶π,Kℓℓ decays in the standard LFQM. It has been known in the LF formulation that while the plus component (J+) of the LF current Jμ in the matrix element can be regarded as the “good” current, the perpendicular (J⊥) and the minus (J−) components of the current were known as the “bad” currents since J⊥,J− is easily contaminated by the treacherous points such as the LF zero mode and the off-mass shell instantaneous contributions.For a systematic analysis of such treacherous points in case one cannot avoid the use ofJ⊥ or J−, we utilized the exactly solvable manifestly covariant BS model to carry out the LF calculations for three form factors, f+,f−,fT. In particular, we obtained f− from two sets of the vector current, J+,J⊥V and J+,J−V, and fT from two sets of tensor current, JT+⊥ and JT+−. In this BS model, we found that while f+ obtained from J+ and fT obtained from JT+⊥ are free from the zero modes, f− obtained from both J+,J⊥V and J+,J−V sets and fT obtained from JT+− receive the zero-mode contributions as well as the instantaneous ones. We then linked the BS model to the standard LFQM using the “CJ-scheme” [51–54] given by Eq. (44) for the correspondences between the two models and replaced the LF vertex function in the BS model with the more phenomenologically accessible Gaussian wave function provided by the LFQM analysis of meson mass spectra [55, 56]. As in the case of previous analysis [51–54], it is astonishing to discover that the zero modes and the instantaneous contributions present in the BS model become absent in the standard LFQM. In other words, our LFQM results of f−,fT are shown to be independent of the components of the current without involving any of those treacherous contributions. Since the absence of the zero mode found in the standard LFQM is mainly due to the replacement of the physical mass M12 with the invariant mass M0′ in the course of linking the two models, this replacement could be regarded as an effective treatment of the zero mode in the standard LFQM.In the standard LFQM, the constituent quark and antiquark in a bound state are required to be on-mass shell, which is different from the covariant formalism, in which the constituents are off-mass shell. The common feature of the standard LFQM is thus to use the sum of the LF energy of the constituent quark and antiquark for the meson mass in the spin-orbit wave function, which is obtained by the interaction-independent Melosh transformation from the ordinary equal-time static spin-orbit wave function assigned by the quantum numberJPC. Under these circumstances, it is natural to apply the replacement M12⟶M0′ in the calculation of the physical observables in the standard LFQM. Indeed, we have shown explicitly that this correspondence relation for the calculations of the decay constants and weak transition form factors between two pseudoscalar mesons provide the current-component independent predictions in the standard LFQM.We then apply our current-component independent form factorsf±,fT for the self-consistent analysis of semileptonic and rare D⟶π,K decays using our LFQM constrained by the variational principle for the QCD-motivated effective Hamiltonian with the HO plus Coulomb interaction [49, 57]. The form factors f±,fT obtained in the q+=0 frame (q2=−q⊥2<0) are then analytically continued to the timelike region by changing q⊥2 to −q2 in the form factors. In our numerical calculations, we also checked the sensitivity of the constituent charm quark mass mc=1.7−0.1+0.1 GeV through the analysis of the form factors f±,fT for D⟶π,K decays. Our results for the form factors and branching ratios for D⟶π,K decays show in good agreement with the available experimental data as well as other theoretical predictions. Especially, the smaller charm quark mass mc≃1.6 GeV seems preferable to larger mc=1.8 GeV for D⟶π,K decays while they are not much different for the analysis of the decay constant of the D meson. Finally, we obtained the the R ratios of the semileptonic D⟶π,K decays as a test for the LFU, and our results are consistent with the recent measurements from the BES III [15, 18] as well as other theoretical results [34, 35, 70].While the rare decay analyses including the tensor form factor can in principle be made, I just focused on the extraction of the current-component independent weak transition form factors as well as the comparison with the available experimental data in the present work. More complete phenomenological analyses regarding on the rare decays of heavyD and B mesons are also under consideration.



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