$\bar{B}_{d,s} \to D^{*}_{d,s} V$ and $\bar{B}_{d,s}^* \to D_{d,s} V$ decays within QCD Factorization and Possible Puzzles

Motivated by the rapid development of heavy-flavor experiments, phenomenological studies of nonleptonic $\bar{B}_{d,s} \to D^{*}_{d,s} V$ and $\bar{B}_{d,s}^* \to D_{d,s} V$~($V=\rho\,,K^*$) decays are performed within the framework of QCD Factorization. Relative to previous works, the QCD corrections to the transverse amplitudes are evaluated at next-to-leading order. The theoretical predictions of the observables are updated. For the measured $\bar{B}_{d,s} \to D^{*}_{d,s} V$ decays, two tensions between theoretical results and experimental measurements, {\it i.e.}"$R_{ds}^{V}$ puzzle"and"$D^{*} V$~(or $R_{V/\ell\bar{\nu}_\ell}$) puzzle", are presented after detailed analyses. For the $\bar{B}_{d,s}^* \to D_{d,s} V$ decays, they have relatively large branching fractions of the order $\gtrsim{\cal O}(10^{-9})$ and are in the scope of Belle-II and LHCb experiments. Moreover, they also provide a way to crosscheck the above-mentioned possible puzzles through the similar ratios $R_{ds}^{\prime V}$ and $R_{V/\ell\bar{\nu}_\ell}^{\prime}$. More refined experimental measurements and theoretical efforts are required to confirm or refute such two anomalies.


Introduction
Thanks to the efforts of BABAR and Belle collaborations in the past years, most of the B u,d mesons decays with branching fractions O(10 −7 ) have been measured. With the particle physics entering the LHC era, more rare decays of B mesons, especially of B s meson, are expected to be well measured. In addition, most recently, the upgrading SuperKEKB/Belle-II experiment has started test operations and succeeded in circulating and storing beams in the electron and positron rings. So, in the near future, the measurements of B meson decays are expected to reach unprecedented precision, which will provide a much more fertile ground for testing the flavor picture of the Standard Model (SM) and exploring underlying mechanisms.
For the nonleptonic two-body B meson decays, the theoretical evaluation is generally complicated due to the nontrivial QCD dynamics related to the hadronic final states. In order to evaluate the strong interaction corrections to the amplitude, several attractive QCD-inspired approaches, including QCD factorization (QCDF) [1,2], the pQCD approach [3,4] and the soft-collinear effective theory (SCET) [5][6][7][8], have been presented. For the case of two light final states, the theoretical evaluations with QCDF approach have been fully developed in the heavy quark limit, for instance Refs. [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. For the case of heavy-light final states, such as B → Dπ and D * ρ et al., the calculation is generally much more complicated due to the un-negligible c quark mass. In Refs. [2,23], the factorization formula at two-loop order has been proven, and the explicit results of QCD corrections forB → DP , DV , D * P and longitudinally polarized D * V (P and V are light pseudoscalar and vector mesons, respectively) decays have been presented.
The b → c induced nonleptonic B decays, which are tree-dominated and CKM-favored, have relatively large branching fractions and have been widely studied in various theoretical frameworks, for instance Refs. [24][25][26][27][28][29][30][31][32][33]. In the previous works for theB 0 → D * + V − decays based on the QCDF, the QCD corrections to the longitudinal amplitude have been fully evaluated at next-to-leading order (NLO) [2] and next-to-next-to-leading order (NNLO) [23]. Even though theB 0 → D * + V − decays are dominated by the longitudinally polarized final states, to keep fairness and consistence, the power-suppressed transverse amplitudes should also be calculated to the same order as the longitudinal one, which is also essential for relatively accurate theoretical results, especially for the polarization fractions. That is what we would like to do in this paper. Moreover, due to the refined experimental measurements, it is also worth performing detailed phenomenological analyses and testing whether the data and the theoretical results are in agreement.
In addition to B mesons, their excited states, such as B * mesons with quantum number of n 2s+1 L J = 1 3 S 1 and J P = 1 − , also could decay through the same transitions as B mesons at quark level. Thanks to the rapid development of heavy-flavor experiments [34][35][36], even though B * decays are dominated by the electromagnetic processes B * → Bγ, the B * weak decays with branching fractions O(10 −9 ) are still hopeful to be observed by Belle-II as analyzed in Refs. [37,38]. Moreover, owing to the much larger beauty production cross section of pp collisions [39], the LHC experiments may also provide a lot of experimental information for B * decays, such as the leptonic B * s → l + l − decay with branching fraction ∼ O(10 −11 ) analyzed in Ref. [40]. Recently, a few interesting theoretical studies of B * weak decays have been made, for instance Refs. [40][41][42][43][44]. In our previous work [38], theB * 0 → D * + V − decays have been studied in the framework of naive factorization (NF). In this paper, the QCD corrections at NLO will be evaluated with QCDF approach for relatively accurate prediction, and the phenomenological studies will be updated simultaneously.
Our paper is organized as follows. In section 2, the theoretical framework and calculations forB 0 d,s → D * + d,s V − andB * 0 d,s → D + d,s V − decays are presented with QCDF approach. Section 3 is devoted to the numerical results and discussions. Finally, we give our summary in section 4.

Theoretical Framework and Calculation
The effective Hamiltonian responsible for where G F is the Fermi coupling constant, V cb V * uq is the product of CKM matrix elements, and Q 1,2 are local tree four-quark operators defined as The corresponding Wilson coefficients C 1,2 (µ) summarize the physical contributions above scale of µ and are calculable with the perturbation theory [45].
In order to obtain the decay amplitudes, the remaining work is to accurately calculate the hadronic matrix elements of every local operators in effective Hamiltonian. The simplest way to evaluate the hadronic matrix elements is the NF scheme [46,47]. However, in the framework of NF, the amplitudes are renormalization-scale-dependent, and the non-factorizable contributions dominated by the hard gluon exchange are lost. In order to remedy these deficiencies, the QCDF approach is proposed by BBNS [1,2]. In the framework of QCDF, up to power corrections of order Λ QCD /m b , the hadronic matrix elements M 1 M 2 |Q i |B ( * ) (M 1 is heavy and M 2 is light ) obey the factorization formula [1,2], where FB is the light-cone distribution amplitude (LCDA) for the quark-antiquark Fock state of meson M 2 ; and T ij (x) denotes the hardscattering function, which is calculable order by order from the first principle of perturbative QCD theory.
Applying the QCDF formula, the amplitude ofB → D * + V − decay could be written as where λ = 0 , ± denotes the helicity of V meson; H λ is the product of matrix elements of current operators, i.e., H λ ≡ V |qγ µ (1 − γ 5 )u|0 D * |cγ µ (1 − γ 5 )b|B ; α λ 1 is the effective flavor coefficient and includes the nonfactorizable contributions. Without the QCD corrections, the NF result, α λ 1 = C 1 + C 2 /N c , is recovered. The amplitude ofB * → D + V − decay is obtained from the formula above by replacingB →B * and D * → D. The explicit expressions of H λ and α λ 1 are given in the following. The decay constant of emitted vector meson is defined through the current matrix element, where m V and ε 2 denote the mass and the polarization vector, respectively. Meanwhile, with the same conventions as Ref. [48], the form factors are defined by forB → D * transition, and meson, and the sign convention 0123 = −1 is taken. Then, after contracting the current matrix elements, we finally obtain forB → D * + V − decays, and and p c is obtained from p c by replacing m B → m B * and m D * → m D .
The effective coefficient α λ 1 in the amplitude, Eq. (4), includes the nonfactorizable contributions from QCD radiative vertex corrections (the penguin diagrams do not contribute tō decays at the order of α s ), and could be written as After calculation, we get the explicit expressions of the vertex corrections V λ 1 written as where Φ V (u) is the leading-twist LCDA and conventionally expanded in Gegenbauer polynomials [49,50], It could be found that only the leading-twist LCDA of emitted vector meson contributes to V 0 1 and twist-3 ones contribute to V ∓ 1 . In addition, the loop functions g 0,∓ (u) in Eqs. (16) and (17) are written as in which, In the Eqs. (20) and (21), the functions ζ(r c ) and ξ ∓ (r c ) contain all of the anti-symmetrical contributions under the transformation m c → −m c (or r c → −r c ). They are written as in which, In the limit of m c → 0, both ζ(r c ) and ξ ∓ (r c ) vanish, i.e., lim mc→0 ζ(r c ) = lim mc→0 ξ ∓ (r c ) = 0. Moreover, after taking the limit m c → 0, one can find that the results of B → V V decays, which have been presented in the Eqs. (A.7) and (A.8) of Ref. [11], could be recovered from Eqs. (16), (17), (20) and (21).
For g 0 (u), the only difference between longitudinally polarized B → D * L and B → DL (L is a light meson) decays is the overall sign of ζ(r c ) (or the sign of r c ), which has been pointed out in Ref. [2] and confirmed in Ref. [23]. Such difference could be easily understood from that: (i) after computing the one-loop correction, the (qq ) pair (q ( ) are light quarks) always retains its (V − A) structure, but the (cb) pair has not only (V − A) but also (V + A) structure due to the un-negligible m c (In this paper, the contributions of the later are exactly collected into the functions ζ(r c ) and ξ ∓ (r c ), i.e., Eqs. (23) and (24)); (ii) only the V -current contributes to B → DL and only the A-current contributes to longitudinally polarized B → D * L. One can also refer to the section 4.4 in Ref. [2] for detailed explanation. Our result of V 0 1 is in consistence with the result in Ref. [2], while the transverse results V ∓ 1 and g ∓ (u) are first presented. For g ∓ (u), because both V -and A-currents contribute to the transversely polarized B → D * V decay, the overall factor k ∓ instead of overall sign exists in Eq. (24). In addition, it should be noted that the strong phase can be obtained by recalling that r 2 c is r 2 With the amplitudes given above, the branching fraction ofB → D * V decay is defined as where Γ tot (B) is the total decay width ofB meson. ForB * → DV decays, the definition is obtained from Eq. (25) by replacingB →B * , D * → D, p c → p c and multiplying by an additional factor 1/3, which is caused by averaging over the spin of initial stateB * . Besides of the branching fraction, the polarization fractions are also very important observables, which are defined as where A and A ⊥ are parallel and perpendicular amplitudes, and could be easily gotten through

Numerical Results and Discussions
Before presenting our numerical results, we would like to clarify the input parameters used in the evaluations. For the CKM matrix elements, we adopt the Wolfenstein parameterization [51] and choose the parameters A and λ as [52] For the well-known Fermi coupling constant G F , the masses of mesons and the total decay widths (or lifetimes) of B mesons, we take the central values given by PDG [53]. However, for have been given in various theoretical models [54][55][56][57][58][59][60]. In this paper, we take the central values of the latest results [59,60] which are in agreement with most of the other theoretical predictions.
Then, the residual inputs are the QCD form factors V (q 2 ) and The QCD form factors V (q 2 ) and A 1,2 (q 2 ) are obtained through the relation [63] where the ratio The ω dependence of h A 1 (ω) and R 1,2 (ω) reads [63] To be conservative, 10% uncertainties are assigned to the values above. With the assumption of the nearest pole dominance, the q 2 dependences of form factors read [65,66] where B c (J P ) is the state of B c with quantum number of J P (J and P are the quantum numbers of total angular momenta and parity, respectively).
With the theoretical formula and inputs given above, we then present our numerical results and discussions. Within the QCDF framework, the QCD corrections are contained in the effective coefficients α λ i , which are generally renormalization-scale-dependent and the dependence is expected to be reduced after the higher order QCD corrections are taken into account.
In Fig. 1, we plot the dependence of tree coefficient α λ 1 (D * ρ) on the renormalization scale µ. As Fig. 1 (b) shows, the imaginary part Im[α 1 ], which is zero at LO (NF result), arises after    Fig. 1 (a) shows, the reduction effect is not very significant, which is attributed to that the NLO QCD corrections to α 1 associated with small C LO 2 is color-suppressed. As found in Refs. [14,23], after taking the NNLO correction, which is no longer color-suppressed, into account, the scale dependence will be significantly improved further. Numerically, the LO and NLO results of coefficient α λ 1 (D * ρ) at µ = m b /2 , m b , 2m b are summarized in Table 1   In Table 2, we have summarized our numerical results for the observables ofB → D * V decays. Moreover, the experimental data and the results of some previous works are also listed in Table 2 for comparison. In Table 3, our theoretical predictions for the observables of Similarly, forB * → DV decays, Eqs. (12) and (13) could be simplified as From the simplified expression given above, one can obtain the relation |H 0 | : For the branching fractions, it could be found from Table 2 that our results are in consistence with the ones based on the instantaneous Bether-Salpeter method (BSm) [29,30], the Heavy quark symmetry (HQS) [25] and the QCDF with the NNLO corrections to the longitudinal polarization amplitude [23], as well as the other theoretical results in Refs. [31,32], but a bit larger than the result of the pQCD approach [26]. In addition, the most recent updated pQCD Compared with the experimental data, it could be found from Table 2 that: (i) our QCDF result for B(B 0 s → D * + s ρ − ) is in a good agreement with the data; (ii) however, the results (central values) for B(B 0 → D * + ρ − ) and B(B 0 → D * + K * − ) are much larger than the data. Moreover, such deviation would be further enlarged if the NNLO QCD corrections, which provide about 4% enhancement to the branching fractions, are included [23]. In addition, the results within the other theoretical frameworks also deviate from the data more or less, which can be seen exactly from Table 2. To clarify such possible mismatch, one can define the quantity Comparing with theB 0 s → D * + s V − (V = ρ , K * ) decay appeared in denominator, theB 0 d → D * + V − decay appeared in numerator receives additional weak-annihilation corrections, which is however power-suppressed and numerically trivial for the tree-dominated decays [2]. So, in the limit of U-spin flavor symmetry, the result R V ds 1 is expected. Using the experimental data listed in Table 2 and the error transfer formula, we get in which the number in the round brackets is the result gotten by using the direct measurement In order to check if the "D * V puzzle" is stable, an improved way is to perform an measurement on the ratio defined by in which = e , µ. Firstly, we would like to estimate its theoretical result. Neglecting the lepton mass, the differential decay rate can be written as where H 00,±± are the helicity amplitudes and are q 2 -dependent. One may refer to Refs. [68][69][70][71] et al. for the details. At q 2 = m 2 V , we get | p|| q 2 =m 2 V = p c and |H 00,−−,++ | 2 where H 0,−,+ have been given by Eqs. (10) and (11), and the explicit expression for H 00,−−,++ could be found in Ref. [71]. Then, further considering that |α 0 1 | 2 ≈ |α − 1 | 2 ≈ |α + 1 | 2 numerically, we finally obtain which is independent of the form factors, and principally could be precisely determined. Numerically, forB 0 → D * + ρ − andB 0 → D * + K * − decays, one can easily get the theoretical Using the most recent result |α 1 | = 1.07 at NNLO in QCDF [23], one can obtain the similar results R NNLO ρ/ ν = 3.01 and R NNLO K * / ν = 0.17. Then, using the distribution of dΓ(B → D * ν )/dω measured by Belle collaboration [72] and the data of B(B 0 → D * + ρ − ) and B(B 0 → D * + K * − ) listed in Table 2, we obtain the experimental results In the estimation, to get the differential decay rate at q 2 = m 2 V , we pick out the measurement at the bin ω ∈ [1.45, 1.50], which covers the point q 2 = m 2 ρ,K * , and take the approximation . Comparing Eqs. (53) and (54) with Eqs. (55) and (56), one may find that the SM expectations for R ρ/ ν and R K * / ν deviate from the experimental results by about 2.3σ and 3.0σ, respectively. In Ref. [23], the authors have also pointed out that the the deviation is at the level of 2 − 3σ to the NNLO accuracy. A specific experimental measurement or analysis on R V / ν at a very narrow bin covering q 2 = m 2 ρ,K * is required for a much more reliable result.
ForB * → D + V − decays, it could be found from Table 3 that all decays have the branching fractions O(10 −9 ), and therefore, are hopeful to be measured by Belle-II experiment, which have been pointed out in Ref. [38]. Moreover, they are also in the scope of LHCb experiment, which can be seen from the following analysis. To make an estimate, we takeB 0 → D + π − decay as a reference. Using the data corresponding to integrated luminosities of 1.0f b −1 of pp collisions at a center-of-mass energy of √ s = 7TeV, about 1.06 × 10 5B0 → D + π − decay events have been measured by LHCb collaboration [73]. After high-luminosity upgrade, a data sample of 50 fb −1 will be collected by LHCb collaboration at a much higher √ s = 14 TeV, which results in a further enhancement of bb production by a factor about 2 [35,74]. Moreover, the most of B mesons detected at LHC are mainly produced through B * → Bγ decays because B * mesons are often produced by about 3 times more than the B mesons [75]. Finally, further considering B(B * 0 →D + ρ − ) B(B 0 →D + π − ) = 15.1×10 −9 26.8×10 −4 0.56 × 10 −5 , we estimate that of the order of O(100) B * 0 → D + ρ − decay events could be collected by LHCb. Furthermore, takingB 0 s → D + s π − as reference instead ofB 0 → D + π − and revisiting the estimate above, one may find about O(10) B * 0 s → D + s ρ − decay events are expected to be observed in the high-luminosity LHC era. TheB * 0 → D + V − decays occur with the same transition asB 0 → D * + V − decays at quark level, and have similar amplitudes asB 0 → D * + V − decays. So, once measured in the future, they are expected to provide crosschecking of the possible R V ds and D * V (or R V / ν ) puzzles mentioned above. For this point, one can define the ratios which are similar to the ones for B decays, Eqs. (45) and (49) Comparing R V ds with R V ds , one can find that the main difference is induced by the lifetimes of B d,s and B * d,s , i.e., τ Bs /τ B d ≈ 1 but τ B * s /τ B * d ≈ 0.46 (see Eqs. (28) and (29)). All of the findings above for B * d,s decays are awaiting the future experimental test.

Summary
Motivated by the rapid development of heavy-flavor experiments, we perform phenomenological studies of nonleptonicB d,s → D * d,s V andB * d,s → D d,s V (V = ρ , K * ) decays in detail. The amplitudes are calculated carefully in the framework of QCD Factorization, in which, relative to the previous works, the NLO QCD corrections to the transverse amplitudes are evaluated.
Our theoretical results for the branching fractions and polarization fractions are summarized in Tables 2 and 3. After detailed analyses, two possible puzzles relevant toB d,s → D * d,s V decays are presented, which are: (i) the current theoretical results for the ratio R V ds , which reflects the effects of flavor-symmetry-breaking and is defined by Eq.