Form factors of $P\to T$ transition within the light-front quark models

In this paper, we calculate the vector, axial-vector and tensor form factors of $P\to T$ transition within the standard light-front~(SLF) and covariant light-front~(CLF) quark models~(QMs). The self-consistency and Lorentz covariance of CLF QM with two types of correspondence schemes are investigated. The zero-mode effects and the spurious $\omega$-dependent contributions to the form factors of $P\to T$ transition are analyzed. Employing a self-consistent CLF QM, we present our numerical predictions for the vector, axial-vector and tensor form factors of $c\to (q,s)$~($q=u,d$) induced $D \to (a_2,K^*_2)$, $D_s \to (K^*_2,f'_{2})$, $\eta_c(1S) \to (D^*_2,D^*_{s2})$, $ B_c \to (B^*_2,B^*_{s2})$ transitions and $b\to (q,s,c)$ induced $B \to (a_2,K^*_2,D^*_2)$, $B_s \to (K^*_2,f'_2,D^*_{s2})$, $B_c \to (D^*_2,D^*_{s2},\chi_{c2}(1P))$, $\eta_b(1S) \to (B^*_2,B^*_{s2})$ transitions. Finally, in order to test the obtained form factors, the semileptonic $B\to \bar{D}_2^*(2460)\ell^+\nu_\ell$~($\ell=e,\mu$) and $\bar{D}_2^*(2460)\tau^+\nu_{\tau}$ decays are studied. It is expected that our results for the form factors of $P\to T$ transition can be applied further to the relevant phenomenological studies of meson decays.

The B and D meson decays induced by heavy-to-light transition provide a fertile ground for testing the Standard Model (SM) and searching for new physics (NP). Some discrepancies between the experimental data and the SM predictions have been found, for instance, the SM predictions for R D ( * ) deviate from data by more than 3σ errors [2]. If these tensions are confirmed by the forthcoming experiments, the discrepancies should also be seen in B transitions to tensor mesons in addition to B decays to pseudoscalar or vector mesons. The decay modes involving tensor final states are of great interest because the tensor meson has additional polarization states compared with the (pseudo)scalar and (axial)vector mesons and thus the relevant decays may have more kinematical quantities related to the underlying helicity structure. Some theoretical studies on these decays have been made in, for instance, Refs. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
In the calculation of the amplitudes of semi-leptonic, non-leptonic and radiative B and D meson decays, the form factors serve as the basic and important input parameters.
Unfortunately, the covariance of matrix element in fact can not be fully recovered in such traditional CLF QM because there are still some residual ω-dependent spurious contributions associated with B functions [33,44]. In addition, the traditional CLF QM suffers from a selfconsistence problem for a long time, for instance, it has been found that the CLF results for f V obtained respectively via longitudinal (λ = 0) and transverse (λ = ±) polarization states are inconsistent with each other [45], [f V ] λ=0 CLF = [f V ] λ=± CLF , because the former receives an additional contribution characterized by the B (2) 1 function. Another self-consistence problem has also been found in Ref. [46].
In order to deal with the self-consistence problem, Choi and Ji present a modified correspondence scheme between the covariant BS approach and the LF approach (named as type-II scheme) [47], which requires an additional M → M 0 replacement relative to the traditional type-I correspondence scheme. By using such improved self-consistent CLF QM, one can obtain the self-consistent results [47], and moreover, the covariance of matrix element can be fully recovered [44,46,48,49]. In this paper, we would like to further test the self-consistence and covariance of the self-consistent CLF QM via the form factors of P → T transition.

Theoretical results in the SLF QM
In order to clarify the convention and notation used in this paper, we would like to review briefly the framework of SLF QM. One may refer to, for instance, Refs. [28][29][30][31] for details.
The main work of LF approach is to evaluate the current matrix element, which will be further used to extract the form factors. The meson bound-state (q 1q2 ) with total momentum p and spin J can be written as where k 1 and k 2 are the on-mass-shell light-front momenta and can be written in terms of the internal relative momentum variables (x, k ⊥ ) as The momentum-space wavefunction (WF) for a 2S+1 L J meson, Ψ JJz LS (k 1 , h 1 , k 2 , h 2 ), in Eq. (10) satisfies the normalization condition and can be expressed as where, ψ(x, k ⊥ ) is the radial WF and responsible for describing the momentum distribution of the constituent quarks in the bound-state; S h 1 ,h 2 (x, k ⊥ ) is the spin-orbital WF and responsible for constructing a state of definite spin (S, S z ) out of the LF helicity (h 1 , h 2 ) eigenstates. For the former, we adopt the Gaussian type WF where, k z is the relative momentum in z-direction, where, For P and T mesons, the vertices Γ are written as whereˆ µν = µν (M → M 0 ).
Using the formulae given above, the matrix element of M → M transition can be written as where S ( ) and ψ ( ) are the WFs of initial (final) state; with Γ given by Eq. (9) for P → T transition. In the further calculation, it is covariant to use the Drell-Yan-West frame, q + = 0, and take a Lorentz frame, p ⊥ = 0. In this frame, the momenta of constituent quarks in initial and final states are written as In the SLF QM, in order to extract the form factors of P → T transition defined by Eqs. (1)(2)(3), one has to take explicit values of µ and/or λ. In this work, we take the following strategy 1 : • For the vector form factor V , we take λ = +2 and multiply both sides of Eq. (1) by µ .
• For the axial-vector form factors, we take µ = +, and then use B + SLF with λ = +2 and +1 to extract A 2 and A 1 , respectively; we take λ = +2, and multiply both sides of Eq. (2) by q µ to extract A 0 .
After some derivations and simplifications, we finally obtain the SLF results for the form factors of P → T transition written as where, the integrands are

Theoretical results in the CLF QM
In order to treat the complete Lorentz covariance of matrix elements and investigate the effect of zero-mode contribution, a theoretical framework of CLF approach is developed by Jaus with the help of a manifestly covariant BS approach as a guide to the calculation. One may refer to Refs. [33,45,51] for detail. In the CLF QM, the matrix element is obtained by calculating the Feynman diagram shown by Fig. 1. For the P → T transition, the matrix element can be expressed as where 2 1 +iε and N 2 = k 2 2 −m 2 2 +iε come from the fermion propagators, H P,T are vertex functions, and is the polarization tensor of tensor meson. The trace term S B is associated with the fermion loop and is written as where Γ P,T are the vertex operators written as [45] iΓ P = −iγ 5 , As has been stressed in Ref. [33], the covariant calculation and the calculation of the lightfront formalism give identical results at the one-loop level if the vertex functions H P,T are analytic in the upper complex k − 1 plane. By closing the contour in the upper complex k − 1 plane and assuming that H P,T are analytic within the contour, the integration picks up a residue at k 2 2 =k 2 2 = m 2 2 corresponding to putting the spectator antiquark on its mass-shell. After integrating the minus component of loop momentum, the covariant calculation becomes the LF one. Such manipulation ask for the following replacements [33,45] and where the LF form of vertex function, h P (T ) , is given by The Eq. (32) shows the correspondence between manifestly covariant and LF approaches [33,45], the correspondence between χ and h can be clearly derived by matching the CLF expressions to the SLF ones via some zero-mode independent quantities [33,45]. However, the validity of the correspondence for the D factor appearing in the vertex operator cannot been clarified explicitly [47]. In fact, the traditional type-I correspondence may give self-contradictory results for some quantities. An obvious example is f V noted by the authors of Ref. [45]. In order to get self-consistent results for f V , a much more generalized correspondence scheme is proposed [47], Within this updated self-consistent scheme, the CLF QM can give a self-consistent result for f V,A and form factors of P → (V, A) and V → V transitions [44,[47][48][49], while the self-consistency of the form factors of P → T transition remains to be tested.
Using above formulas and integrating out k − 1 , the matrix element, Eq. (27), can be reduced to the LF formB However, the matrix element obtained in this way contains spurious ω µ -dependent contributions, which violate the covariance of matrix element. It should be noted that the contribution of the zero-mode from the k + 1 = 0 is not taken into account in the contour integration. It is interesting that the spurious ω µ -dependent terms inB can be eliminated by the zero-mode contributions [33]. The inclusion of zero-mode contributions in practice amount to the following where A and B functions are given by 1 ; The ω-dependent terms associated with the C functions are not shown because they can be eliminated exactly by the inclusion of the zero-mode contributions [33]. However, there are still some residual ω-dependences associated with the B functions, which are irrelevant zero-mode contribution [33], and possibly result in the inconsistence and covariance problems [44,[47][48][49].
Using the formulae given above, one can obtain CLF results for the matrix elements of to the definitions of form factors given by Eqs. (1-3), the CLF results for the form factors of P → T transition can be extracted directly. They can be written as where the integrands are 3 ) The CLF results given above are independent of µ and λ, which implies that the CLF contributions are irrelevant to the self-consistence and covariance problems. This is a significant advantage of CLF QM compared the SLF QM. However, it should be noted that the contributions associated with B functions are not included in above formulae. These contributions may result in the self-consistence and covariance problems of CLF QM except they are equal to zero numerically, and will be analyzed in the next section.
In our following discussions, we also need the valence contributions, which can be obtained by assuming k + 2 = 0 (or p + = 0). This assumption ensures the pole of N 2 is safely located in the contour of k − 1 integral ( the pole of N 2 is finite) and implies that the zero-mode contributions are not taken into account. At this moment, the replacements fork µ 1 given above have to be disregarded. Instead, one just need to directly use the on-mass-shell condition of spectator antiquark, k 2 2 = m 2 2 , and the conservation of four-momentum at each vertex. The valence contributions to the form factors can also be expressed as Eq. (50) with the integrands written

Numerical results and discussion
With the theoretical results given above and the values of input parameters collected in appendix A, we then present our numerical results and discussions in this section. As has been mentioned in the last section, the contributions associated with B functions are not included in the CLF results, Eqs. (51)(52)(53)(54)(55)(56)(57). These contributions to the matrix elements of P → T transition can be written as where, the integrands are 1 + 2B After extracting their contributions to the form factors, [F] B , one can obtain the full result of form factor in the CLF QM, which can be expressed as Based on these results, we have following discussions and findings: • Here, we take B B (Γ = σ µν γ 5 q ν ) given by Eq. (69) as an example. From this equation, x  it can be found that the third term is proportional to ω µ . This spurious ω µ -dependent contribution corresponds to an unphysical form factor and may violate the covariance of matrix element if it is non-zero. The other terms would present contributions to the tensor form factors, T 2 and T 3 . For convenience of discussion, we take T 3 as an example, which could receive the contribution from the second term written as which is obviously dependent on the choice of λ . For different values of λ , T B 3 can be explicitly written as Further considering the fact that [F] CLF is independent of λ , it is clearly seen that • In order to clearly show the possible self-consistence problem caused by B functions within type-I and type-II schemes, we define the contributions of B functions ∆ B (x) as which is equal to N c transitions as examples, the dependence of ∆ B (x) on x are shown in Fig. 2. It can be seen that the self-consistence is violated within the type-I scheme, but it can be satisfied within the type-II scheme due to In order to further confirm such finding, we list the numerical results of [T 3 ] full for B c → D * 2 transition at q 2 ⊥ = (0, 1, 4, 9) GeV 2 with λ = (0, ±1, ±2) in Table 1, in which the SLF, valence and CLF results are also given for comparison. From these numerical results, it within the traditional type-I scheme, while which confirms again that the contribution associated with B functions vanishes numerically within the type-II scheme, even though it exists formally in the expression of form factor given by Eq. (70). In addition, one can also find from Table 1  above, it is found that this spurious ω µ -dependent contribution is (non)zero within the type-II (I) scheme, which implies that the Lorentz covariance of B µ is violated within the type-I correspondence scheme, but such problem can be avoided by employing the type-II scheme.
• As has been mentioned above, the spurious ω µ -dependent contributions associated with C functions can be canceled by the zero-mode contributions [33].  Fig. 3. It can be found that zero-mode presents nonzero contributions within the traditional type-I correspondence scheme; while, these contributions, although existing formally, vanish numerically in the type-II correspondence scheme, i.e., [T 3 (q 2 )] z.m.= 0 (type-II), because the contribution with small x and the one with large x cancel each other out exactly at each q 2 ⊥ point. This can also be found from the numerical results given by Table 1.
The tensor form factors of P → T transition have also been calculated by Cheng and Chua (CC) [50] within the traditional CLF QM, their results are given in appendix B. The contributions associated with B functions are not considered in their calculation. Besides, comparing CC's results with ours, it is found that they are the same for T 1 , but are obviously different for T 2 and T 3 in form. In addition, it has been checked that their numerical results for T 2 and T 3 are different either within type-I scheme. After checking our and CC's calculations, we find another inconsistent problem caused by the different way for dealing with the trace term To clarify the origin of this inconsistent problem, we take the term 2ig νλ g αµ g βσ (P + q) β k σ 1 k α 1 k 1δ appeared in S P →T µνλδ as an example. As has been mentioned in the last section, some replacements are needed to take the zero-mode contribution into account after integrating out k − 1 . In the CC's calculation, the replacement fork σ 1k α 1k δ 1 is used directly though σ is a dummy indices, i.e., 2ig νλ g αµ g βσ (P + q) βk σ 1k α 1k 1δ =2ig νλ g αµ g βσ (P + q) β (g ασ P δ + g α δ P σ + g σ δ P α )A 1 + (P · q + q 2 )A 2 ] In our calculation, we employ the standard procedure of CLF calculation, and obtain 2ig νλ g αµ g βσ (P + q) βk σ 1k α 1k 1δ =2ig νλk 1µk 1δk 1 · (P + q) 3 )    In order to clearly show the divergence between CC's and our results, we define the difference where F = T 2,3 . Then, taking D → K * 2 and B c → D * 2 as examples, the dependences of ∆ CLF T 2,3 (x, q 2 ⊥ ) on x in type-I and -II schemes are shown in Fig. 4. It can be easily seen from Fig. 4 that our and CC's numerical results for T CLF 2,3 are inconsistent within the type-I scheme; however, such inconsistence problem vanishes in the type-II scheme due to 1 0 dx∆ CLF T 2,3 (x)=0. From above discussions, one can conclude that the CLF QM with type-II corresponding scheme can make sure the Lorentz covariance of matrix elements and give self-consistent results for the form factors. Using the values of input parameters collected in appendix A and employing the self-consistent type-II scheme, we then present our numerical predictions for transitions. It should be noted that the CLF calculation is made in the q + = 0 frame, which implies that the form factors are known only for space-like momentum transfer, q 2 = −q 2 ⊥ 0, and the results in the time-like region need an additional q 2 extrapolation. For the phenomenological applications, we adopt the BCL version of the z-series expansion [52] in the form adopted in Refs. [53,54], where, z(q 2 , t 0 ) =         the masses of resonances collected in Table 2, we take the values given by PDG [1] and lattice QCD [55,56]. In the practice, we will truncate the expansion at N = 1. The parameter b k will be obtained by fitting to the results computed directly by CLF QMs.
Using the parameterization scheme given by Eq. (78), we present our numerical results of Tables 3 and 4, respectively. The q 2 dependence of form factors are shown in Figs. 5 and 6. Some remarks on these results are given in order.
• Firstly, we would like to test the legality of the truncation-scheme N = 1 employed in this paper. In the expansion, Eq. (78), the values of Z k (q 2 ) ≡ z(q 2 , t 0 ) k − z(0, t 0 ) k in general  → (q, s, c) , which can be found from the values listed in Table 5 (for convenience of discussion, we mainly study the effect of k = 2 term, and take B → D * 2 and B → K * 2 transitions as examples). Therefore, the k = 2 term can be neglected ) transitions with the parameterization scheme given by Eq. (78).   Table 6, it can be found that the values of b 1 and b 2 are at the same level. Thus, the truncation N = 1 employed in this paper is acceptable. It can be also clearly seen from Fig. 7 that the effect of truncation-scheme N = 2 on the q 2dependences of form factors are not significant compared with truncation-scheme N = 1.
Such finding can be easily understood because the CLF result for the form factors can be well reproduced within the truncation-scheme N = 1, and thus the higher-order terms are trivial. Some discussions on the effects of higher-order terms have been made in, for instance, Refs. [54,64].
• Just like the η − η mixing in the pseudoscalar case, the physical isoscalar tensor states transitions with the parameterization scheme given by Eq. (78). f 2 (1270) and f 2 (1525) also have a mixing. In order to exhibit their flavor components, the mixing relation can be written as Table 5: Numerical results of Z k (q 2 ) at q 2 = (3, 5, 7) GeV 2 for B → D * 2 transition, and at q 2 = (6, 10, 14) GeV 2 for B → K * 2 transition.
where θ is the mixing angle. It is obvious that the mixing angle should be small because f 2 (1270) and f 2 (1525) decay predominantly into ππ and KK, respectively. Numerically, it is found that θ = 9 • ± 1 • [1]. Therefore, in our calculation, the possible mixing effect is neglected, i.e., f 2 (1270) and f 2 (1525) are assumed to be pure (uū + dd) and (ss) states, respectively.
• From Table 4 and Fig. 6, it can be clearly found that all of transitions respect the relation which is essential to assure that the hadronic matrix element of P → T is divergence free at q 2 = 0. However, their dependence on q 2 is different, which can be applied further in the relevant phenomenological studies of meson decays.
: N=2 : N=1 : N=2 : N=1  • Compared the numerical results of P → T transition with the ones of P → V transition obtained in our previous works [46,48] at q 2 = 0, it is found that: (i) For the c → q and b → q (q = u, d, s) induced transition with a light spectator quark, the former are smaller than the later, which is favored by the experimental data of radiative decays. For instance, our result T given by PDG. Theoretically, for the relevant strong decays, the relation Γ(D * 0 2 → D ( * )− π + ) = Γ(D * − 2  for Γ(D * 0 2 → D ( * )− π + ), respectively. Theoretically, the differential decay widths of semileptonic B →D * 2 lν l decays can be written as [67,68] dΓ where λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc is the Källén function. Using the CLF results for (V, A 0 , A 1 , A 2 ) in type-I and -II schemes, and the values of the other input parameters given by PDG, we summarize our results for Table 8, in which the results obtained in Refs. [36,67] are also listed. It can be found that our results are much larger (smaller) than the ones given in Ref. [67] (Ref. [36]) due to the different form factors. Comparing our results with experimental data given in Eq. (87), we find that the type-II results are in good consistence with data, while the type-I results can not be excluded due to the large theoretical and experimental errors.
More theoretical and experimental efforts are needed to improve the accuracies of results and further test the legality of such two schemes. The errors caused by form factors can be well controlled by evaluating the ratio R D * 2 ≡ Γ(B→D * 2 τ + ντ ) Γ(B→D * 2 + ν ) . Our prediction is which are consistent with the LCSR prediction 0.041 ± 0.002 [36], but are different from the result 0.16 ± 0.04 [67]. Such ratio is expected to be measured in the future, it will test whether the R D ( * ) anomalies in the pseudoscalar (vector) channels exist also in the tensor channel or not, and play a similar role as R D ( * ) in testing the lepton flavor universality.

Summary
In this paper, the matrix elements and relevant vector, axial-vector and tensor form factors of P → T transition are calculated within the CLF approach. The SLF results are also calculated for comparison. The self-consistency and Lorentz covariance of the CLF QM are analyzed in detail. It is found that the CLF QM with the traditional correspondence scheme (type-I) between the manifest covariant BS and the LF approaches has two kinds of self-consistence problems: one is caused by the non-vanishing ω-dependent spurious contributions associated with the B functions, which also violate the strict Lorentz covariance of CLF QM; another one is caused by the different strategies for dealing with the trace term in the calculation of matrix element. The self-consistence and Lorentz covariance problems can be resolved by employing the improved self-consistent type-II correspondence scheme which requires an additional replacement M → M 0 relative to type-I scheme. Within the self-consistent type-II scheme, the zero-mode contributions to the form factors exist only in form but vanish numerically, and the valence contributions are exactly the same as the SLF results. Theses findings confirm again the conclusion obtained via P → V , P → A and V → V transitions in our previous works.  Tables 3 and 4. Some form factors are first predicted in this work. Our predictions for the form factors of B → a 2 and B → K * 2 transitions are generally in consistent with the results obtained by employing LCSR and PQCD approaches, and show that the selfconsistent type-II scheme can significantly improve the CLF prediction. Compared with the form factors of P → V transition, it is also found that the form factors of P → T transition are smaller than the ones of P → V at q 2 = 0 point when T is a light tensor meson, which is in consistence with the experimental data. Using the obtained form factors, we also present the predictions for B →D * 2 (2460) + ν ( = e, µ) andD * 2 (2460)τ + ν τ decays. It is expected that our results for the form factors of P → T transition can be applied further to the relevant phenomenological studies of meson decays. Appendix A: Input parameters The constituent quark masses and Gaussian parameters β are essential inputs for computing the form factors. The quark masses are model dependent, and their values obtained in the previous works [45,[58][59][60][61][62][63] are different from each other more or less. In this work, we take m q = 230 ± 40 MeV , m s = 430 ± 60 MeV , m c = 1600 ± 300 MeV , m b = 4900 ± 400 MeV .
which suggested values given in the previous works [49], it covers properly the others values and therefore can reflect roughly the uncertainties induced by the model dependence of quark mass. Then, the parameters β listed in Table 9 [46], in which it have been assumed that β qq is same for V and T due to the lack of tensor meson decay constant data . In addition, the type-II correspondence scheme is employed in the fits, while the fitting results do not affect following comparison between type-I and -II schemes.