Tensor form factors of $P\to P,\,S,\,V$ and $A$ transitions within the standard and the covariant light-front approaches

In this paper, we investigate the tensor form factors of $P\to P,\,S,\,V$ and $A$ transitions within the standard light-front (SLF) and the covariant light-front (CLF) quark models (QMs). The self-consistency and Lorentz covariance of CLF QM are analyzed via these quantities, and the effects of zero-mode are discussed. For the $P\to V$ and $A$ transitions, besides the inconsistence between the results extracted via longitudinal and transverse polarization states, which is caused by the residual $\omega$-dependent spurious contributions, we find and analyze a"new"self-consistence problem of the traditional CLF QM, which is caused by the different strategies for dealing deal with the trace term in CLF matrix element. A possible solution to the problems of traditional CLF QM is discussed and confirmed numerically. Finally, the theoretical predictions for the tensor form factors of some $c\to q,\,s$ and $b\to q,\,s\,,c$ ($q=u,d$) induced $P\to P,\,S,\,V$ and $A$ transitions are updated within the CLF QM with a self-consistent scheme.

1 Introduction addition, it should be noted that above-mentioned works are performed within the traditional CLF QM [9], which however has covariance and self-consistence problems.
It has been noted for a long time that the traditional CLF approach [9] suffers from a self-consistence problem in the vector meson system. For instance, the CLF results for the decay constant of vector meson, f V , obtained via longitudinal (λ = 0) and transverse (λ = ±) polarization states are inconsistent with each other, i.e. [f V ] λ=0 = [f V ] λ=± [60], because the former receives an additional contribution characterized by the B (2) 1 function. Some analyses has been made in Ref. [84], and the authors present a possible solution to the self-consistence problem by introducing a modified correspondence between the covariant BS approach and the LF approach (named as type-II scheme [84]), which requires an additional M → M 0 replacement relative to the traditional correspondence scheme (named as type-I scheme [84]).
In our previous works [85][86][87], the self-consistence problem has also been studied in detail via f P,V,A and form factors of P → (P, V ) and V → V transitions associated with the (axial-)vector current, and the modified type-II correspondence scheme as a solution to the selfconsistence problem [84] is carefully tested. Besides, we have also found that: the covariance of the traditional CLF QM in fact can not be maintained strictly due to the residual ω-dependent contributions; the self-consistence and covariance problems have the same origin and can be resolved simultaneously by employing the modified type-II scheme. In this paper, we would like to extend our previous works on above issues to the tensor form factors of P → P, S, V and A transitions, and update the theoretical results within a self-consistence scheme. In addition, we will also show another "new" self-consistence problem of the CLF QM, which has not been noted before.
Our paper is organized as follows. In section 2, we review briefly the SLF and the CLF QMs for convenience of discussion, and then present our theoretical results for the tensor form factors of P → P, S, V and A transitions. In section 3, the self-consistency and covariance of CLF QM are discussed in detail, and our numerical results for the tensor form factors of some c → q, s and b → q, s , c (q = u, d) induced P → P, S, V and A transitions are presented.
Finally, our summary is given in section 5. Some previous theoretical results are collected in appendix A for convenience of discussion and comparison, and the values of input parameters used in the computation are collected in appendix B.
The main work of LF approaches is to evaluate the current matrix element of M → M transition, B ≡ M (p )|q 1 (k 1 )Γq 1 (k 1 )|M (p ) , Γ = σ µν , σ µν γ 5 , ... (9) which will be further used to extract the form factors by matching to the definitions given above.
In the framework of the SLF QM, the matrix element, Eq. (9), can be written as [85][86][87] where corresponds to the operator in Eq. (9), x and k ⊥ are the internal LF relative momentum variables. The momenta of quark q 1 and spectator anti-quarkq 2 in the initial state have been written in terms of (x, k ⊥ ) as where,x = 1 − x. For convenience of calculation, it is usually assumed that the initial state moves along with z-direction, which implies that p ⊥ = 0. Taking the convenient Drell-Yan-West frame, q + = 0, where q ≡ p − p = k 1 − k 1 is the momentum transfer, the momentum of quark q 1 in the final state can be written as where k ⊥ = k ⊥ −xq ⊥ .
For the former, we adopt commonly used Gaussian-type WFs, which are written as for s-wave and p-wave mesons, respectively. The Gaussian parameters β can be determined by fitting to data, and k z is the relative momentum in the z-direction and can be written as with the invariant mass defined by For the later, S h 1 ,h 2 (x, k ⊥ ), it can be obtained by the interaction-independent Melosh transformation, and finally written as a covariant form [13,60], For the P , S, V and A states, Γ M has the form where Using the formulas given above, one can obtain the explicit expression of B SLF , which is further used to extract the form factors. The form factor in the SLF QM can be written as For the P → P and P → S transitions, taking µ = + and ν = ⊥, we finally obtain where, " F SLF transitions, we take λ = + and multiply both sides of Eqs. (3) and (4) by ( µ q ν , µ P ν , µ * ν ) for convenience of extracting the form factors T (1,2,3) . The final results are written as It should be noted that only the D -terms are kept in T should not be applied to the m 1 in D factor.

Theoretical results in the CLF QM
In order to maintain manifest covariance and explore the zero-mode effects, a CLF approach is presented in Refs. [9,59,60] with the help of a manifestly covariant BS approach as a guide to the calculation. In the CLF QM, the matrix element for M → M transition is obtained by calculating the Feynman diagram shown in Fig. 1, and can be written as a manifest covariant form, where d 4 k 1 = 1 2 dk − 1 dk + 1 d 2 k ⊥ , E P,S = 1 and E V,A = µ , the denominators N where Γ M ( , ) is the vertex operator and can be written as [60,84] iΓ P = −iγ 5 , Integrating out the minus component of loop momentum, one goes from the covariant calculation to the LF one. By closing the contour in the upper complex k − 1 plane and assuming that H M ,M are analytic within the contour, the integration picks up a residue at k 2 2 =k 2 2 = m 2 2 corresponding to put the spectator antiquark on its mass-shell. Consequently, integrating out the minus component, one has the following replacements [9,60] and where the LF forms of vertex functions, h M , for P , S, V and A mesons are given by Eq. (40) shows the correspondence between the manifestly covariant and the LF approaches.
In Eq. (40), the correspondence between χ and ψ can be clearly derived by matching the CLF expressions to the SLF ones via some zero-mode independent quantities, such as f P and [9,60], however, the validity of the correspondence for the D factor appearing in the vertex operator, D V,con → D V,LF , has not yet been clarified explicitly [84]. Instead of the traditional type-I correspondence, a much more generalized correspondence, It should be noted that B receives additional spurious contributions proportional to the light-like vector ω µ = (0, 2, 0 ⊥ ), and these undesired spurious contributions are expected to be cancelled out by the zero-mode contributions [9,60]. The inclusion of the zero-mode contribution in practice amounts to some proper replacements fork 1 andN 2 inŜ under integration [9]. In this work, we needk where A and B functions are written as 1 , B 1 ; In above formulas, the ω-dependent terms associated with the C functions are not given since they are eliminated exactly by the inclusion of the zero-mode contributions [9].
In the CLF QM, the tensor form factors can be obtained directly by matchingB CLF to their definitions given by Eqs. (1), (2) and (5-8) 1 . Our final CLF results for the tensor form factors can be written as where, the integrands are Similar to the case of SLF results, only the D -terms are kept in T , Eq. (56), is exactly the same as the one in Refs. [81,82], Eq. (91), however, the results for T CLF 2,3 are different. This inconsistence will be analyzed in detail in the next section.
In the CLF QM, for a given quantity (Q), the CLF result (Q CLF ) can be expressed as a sum of valence (Q val. ) and zero-mode (Q z.m. ) contributions [84], Q CLF = Q val. + Q z.m. , in which the CLF results for the tensor form factors has been given above. It has been found in Ref. [84] and our previous works [85,86] that Q CLF= Q val. = Q SLF within type-II correspondence scheme, where "=" denotes that two quantities are equal to each other only in numerical value, while "=" means that two quantities are exactly the same not only in numerical value but also in form. In order to check the universality of such relation and clearly show the effects of zero-mode contributions, we have also calculated the valence contributions, which are written as F val.
It can be easily found that the tensor form factors of P → (P, S) transitions are free from the zero-mode effects, while the ones of P → (V, A) transitions are zero-mode dependent.

Numerical results and discussions
Using the theoretical results given in the last section and input parameters collected in appendix B, we then present our numerical results and discussions in this section. It has been mentioned above that most of the spurious ω-dependent contributions are neutralized by zero-mode contributions, but there are still some residuals associated with B functions, which possibly violate the self-consistence and covariance of CLF QM, but are not taken into account in Eqs. (54)(55)(56)(57)(58)(59)(60) and are not considered in the previous works [80][81][82] either. These residual ω-dependent contributions to the tensor matrix elements of P → V transition ( l.h.s. of Eqs. (5) and (6)) can be written as where, B µ B = 0 for the P → (P, S) transitions, and B µ B for the P → A transitions can be obtained from above results by the replacements similar to Eqs. (59) and (60). Taking the contributions associated with B functions into account, the full results for the tensor form factors in the CLF QM can be expressed as Based on these formulas, we have following discussions and findings: • In Eq. (69), the first term would introduce a spurious unphysical form factor, and thus is expected to vanish. Unfortunately, it is equal to zero for λ = 0 but is nonzero for λ = ± within type-I scheme. The last three terms give additional contributions to T 1 , which are however λ-dependent. Explicitly, these contributions to T 1 can be written as    Table 1; moreover, the dependence of ∆ B (x) defined as where F = T 1 , on x are shown in Fig. 2. From these results, it can be easily find that the self-consistence is violated in the traditional type-I scheme (i.e., [ , but can be recovered by using the type-II scheme due to [T 1 ] B λ=0,±= 0, i.e.,  70), the first and the second terms give additional contributions to T 2 and T 3 , the last term is proportional to ω µ and corresponds to a unphysical form factor. We take T 3 as an example for convenience of discussion. The correction of B function to T 3 is which can be explicitly rewritten as λ-dependent form, Comparing with the B function contribution to

given by
Eqs. (4.5) and (4.6) in Ref. [87], it can be found that [ [87] in detail, and have obtained the same conclusion as we have obtained in the last item via T 1 .
• The covariance of the matrix element of tensor operators in the type-I scheme is violated due to the non-zero ω-dependent contributions associated with B function (for instance, the last term proportional to ω µ in Eq. (70)); while, the Lorentz covariance can be naturally recovered in the type-II scheme because all of the contributions associated with B function exist only in form but vanish numerically.
• Taking This relation is also valid for the form factors of P → A and P → (P, S) transitions, while, for the later, the notation "=" should be replaced by "=" because F T and U T are zero-mode independent. The analyses and findings mentioned above confirm again the main conclusion obtained in our previous works [85,86] and Ref. [84]. In addition to above-mentioned self-consistency problem of CLF QM caused by the contributions associated with B function, we note a new inconsistence problem, which will be discussed in the following.
The tensor form factors T 1,2,3 have also been obtained by Cheng and Chua (CC) in Ref. [82] within the CLF QM, these results are collected in the appendix A ( Eqs. (91-93) ) for convenience of discussion. Comparing our results given by Eqs. (56-58) with CC's results, one can easily find that the results for T 1 are consistent with each other, but the ones for T 2 and T 3 are obviously different. In addition, for T 2 and T 3 , it is found that our and CC's numerical results are also inconsistent with each other in the traditional type-I scheme. After carefully checking our and CC's calculations, we find that such new inconsistence problem is caused by the different ways to deal with the trace term S µνλ related to the fermion-loop in B P →V CLF [Γ = σ µν γ 5 ], where B CLF and S have been given by Eq. (32) and Eq. (33), respectively. Explicitly, B P →V CLF [Γ = σ µν γ 5 ] is written as The trace term, S µνλ , can be related to S ρσλ by using the identity 2σ µν γ 5 = iε µνρσ σ ρσ , where S ρσλ is the trace term in B P →V CLF [Γ = σ ρσ ] corresponding to T 1 . Explicitly, it is written as For convenience of discussion, we take the last term, [S µν λ ] last term = −2iε µνρσ ε σλαβ k 1ρ k α 1 (P +q) β , as example.
In the CC's calculation [82], the obtained results forŜ ρσλ is used directly to calculateŜ µν λ by usingŜ µν λ = i 2 ε µνρσŜ ρσλ , which is formally similar to Eq. (79). It implies that, after integrating out k − 1 , the replacement fork ρ 1k α 1 is made directly by using Eq. (48) even though ρ and α are dummy indices; then, in the CC's way, using the identity it is obtained that where only the terms proportional to g ν λ g µ α g ρ β are shown for convenience of comparison with our corresponding result given in the following.
In our calculation, we employ the standard procedure of CLF calculation instead of directly using the obtained result forŜ ρσλ . Firstly, we write [S µν λ ] last term as where only the terms proportional to g ν λ g µ α g ρ β corresponding to the CC' result, Eq. (82), are shown, by using Eq. (81) and Then, after integrating out k − 1 , we further make replacements fork µ 1 andk µ 1N 2 (note that µ is free index) by using Eqs. (47) and (49). Finally, we arrive at    Table 3: Same as Table 2 except for D q,s → S and B q,s,c → S transitions.
in Fig. 4. In can be easily found from Fig. 4  CLF (x) = 0; however, it is interesting that the consistence can be achieved numerically within the type-II scheme because CLF (x) = 0. The case of P → A transition is similar to the one of P → V transition.
From above analyses and discussions, it can be concluded that the type-II scheme provides a feasible solution to the covariance and self-consistency problems of the CLF QM. Therefore, we would like to update the CLF predictions for the tensor form factors of some b → c, s , q and c → s, q (q = u, d) induced P → P, S, V and A transitions by employing self-consistent type-II Table 4: Same as Table 2 except for D q,s → V and B q,s,c → V transitions.   Table 2 except for D q,s → 1 A and B q,s,c → 1 A transitions.    Table 2 except for D q,s → 3 A and B q,s,c → 3 A transitions.
which is suitable for most of form factors considered in this paper. However, for T Using the values of input parameters collected in appendix B, we then present our numerical predictions for the tensor form factors in Tables 2-6; and the q 2 -dependences are shown in Figs. 5-9. From these results, it can be found that the CLF results obtained in the spacelike region can be well reproduced by Eqs. (89) and (90), and are further extrapolated to the time-like space. In addition, our results for P → V and A transitions respect the relation that

Summary
In this paper, motivated by the problems of LFQMs, we have investigated the tensor matrix elements and relevant form factors of P → P, S, V and A transitions within the SLF and the CLF approaches. The self-consistency and Lorentz covariance of the CLF predictions for the tensor matrix elements and form factors are analyzed in detail, and moreover, the zeromode effects and the relation between valence contribution and SLF result are studied. As has been pointed out in our previous works, the covariance is in fact violated in the CLF QM with the traditional correspondence scheme (type-I) between the manifest covariant BS and the LF approach; moreover, for P → V and A transitions, the tensor form factors extracted via          found that such two problems have the same origin (the non-vanishing ω-dependent spurious contributions associated with B functions), and can be resolved simultaneously by employing the improved type-II correspondence scheme which requires an additional replacement M → M 0 relative to the traditional type-I scheme. Within the type-II scheme, the zero-mode corrections are only responsible for neutralizing spurious ω-dependent contributions associated with C functions, but do not contribute numerically to the form factors; and the valence contributions in the CLF QM are exactly the same as the SLF results. The findings mentioned above confirm again the main conclusions obtained in Ref. [84] and our previous works [85][86][87] .
Besides, we find a "new" self-consistence problem of CLF approach with traditional type-I scheme. It is found that different strategies for dealing deal with the trace term, S, in the CLF matrix element would result in different formulas for the tensor form factors T 2(3) of P → V and A transitions, and the numerical results are also inconsistent with each other within type-I scheme; but interestingly, this new inconsistence problem can also be overcome numerically by employing type-II scheme. Finally, using the CLF approach with the covariant and self-consistent type-II scheme, the theoretical predictions for the tensor form factors of c → q , s (q = u , d) induced D q,s → P, S, V, A and b → q , s , c induced B q,s,c → P, S, V, A transitions are updated. Appendix B: Input parameters The masses of valence quark and Gaussian parameters β are essential inputs for computing the form factors. For the former, we take [87] m q = 230 ± 40 MeV , m s = 430 ± 60 MeV , m c = 1600 ± 300 MeV , m b = 4900 ± 400 MeV , which can cover properly the fitting results and suggested values given in the previous works, for instance, the result obtained via variational analyses of meson mass spectra for the Hamiltonian with a smeared-out hyperfine interaction [88], the values obtained by the variational principle for the linear and harmonic oscillator (HO) confining potentials, respectively [89], the fitting results obtained via decay constants and mean square radii of mesons [29], some commonly used values in the LFQMs [60,61] and so on. For the later, its value for a given meson can be obtained by fitting to the data of decay constant. Using the data of decay constant, f P,V , collected in Ref. [85] and the default values of quark masses given by Eq. (94), we obtained the values of β collected in Table 7, in which it have been assumed that β q 1q2 is universal for P (V ) and S(A) mesons due to the lack of data for f S,A . In addition, the self-consistent type-II scheme is employed in computing decay constants.