Abstract
In this paper, we studied systematically the semileptonic decays with by using the perturbative QCD (PQCD) and the “PQCD+Lattice” factorization approach, respectively. We first evaluated all relevant form factors in the low- region using the PQCD approach, and we also took the available lattice QCD results at the high- region as additional input to improve the extrapolation of from the low- region to the endpoint . We then calculated the branching ratios and many other physical observables , , , and and the clean angular observables and . From our studies, we find the following points: (a) the PQCD and “PQCD+Lattice” predictions of are about , which agree well with the LHCb measured values and the QCD sum rule prediction within still large errors; (b) we defined and calculated the ratios of the branching ratios and ; (c) the PQCD and “PQCD+Lattice” predictions of the longitudinal polarization , the CP-averaged angular coefficients , and the CP asymmetry angular coefficients agree with the LHCb measurements in all considered bins within the still large experimental errors; and (d) for those currently still unknown observables , and , we suggest LHCb and Belle-II Collaboration to measure them in their experiments.
1. Introduction
In the standard model (SM) of particle physics, one treats these three generations of the charged leptons as exact copies of each other. These charged leptons behave in the same way but differ only in the masses determined by their Yukawa coupling to the Higgs boson. The lepton flavor universality (LFU), i.e., the equality of the coupling to the all electroweak gauge bosons among three families of leptons, has been regarded as an exact symmetry for quite a long time [1]. In recent years, however, some physics observables associated with the flavor-changing neutral current (FCNC) transitions have exhibited deviations from the SM expectations. These include the LFU-violating(LFUV) ratios and [2, 3], whose measurements deviate from universality [4–6] by around . More notably, the measurements of the angular observable of decay in the large recoil region [7–11] as reported by the LHCb [12, 13] and Belle Collaboration [14] point to a deviation of about with respect to the SM prediction [15].
As is well known, the FCNC transition is forbidden at tree-level, but proceeds by way of loop diagrams with a very low rate. Due to the strong suppression within SM, such kinds of FCNC decays may be sensitive to the possible new physics (NP) effects. Therefore, the semileptonic decay has received striking attentions by means of measurements of the inclusive and/or the exclusive decays and their comparison with the SM predictions. Besides the decay rates, many angular observables of the semileptonic decays have also been measured previously [12–14]. The precision of the experimental measurements will also be expected to upgrade remarkably in the forthcoming year.
The semileptonic decay , which is closely relevant to the decay , offers an alternative scene to check out the same fundamental quark process, in a different hadronic background. On the theoretical side, various studies on the quark level transition and the exclusive decays by using rather different theories or models have been performed within the SM, such as the constituent quark model or covariant quark model [16, 17], the light front quark model [18], the QCD factorization (QCDF) [19], and the light-cone sum rule (LCSR) [20–25], and beyond the SM, such as the universal extra dimension [26, 27] and the supersymmetric theory [28]. On the experimental side, the decay mode was first observed and studied by the CDF Collaboration [29] and subsequently by the LHCb Collaboration [30–34]. Beyond the measurement of the branching ratio, a rich phenomenology of various kinematical distributions can be presented. While the angular distributions were found to be consistent with the SM expectations obtained in References [22, 23], however, LHCb also observed a deficit with respect to the SM prediction for the branching ratio in the low- region: the tension between the theory and experiment is about in the region , where the form factors are evaluated by using the combined fit of lattice and the LCSR results [22, 23].
In a previous paper [35], the semileptonic decays have been studied by us using the perturbative QCD (PQCD) factorization approach [36–47]. In this paper, we will make systematic studies for the semileptonic and present the theoretical predictions for many physical observables: (1)For decays, we treat them as a four-body decay described by four kinematic variables: the lepton invariant mass squared and three angles . We defined and calculated the full angular decay distribution, the transverse amplitudes, the partially integrated decay amplitudes over the angles , the CP-averaged differential branching, the ratios and of the branching ratios, the forward-backward asymmetry , the polarization fraction , the CP-averaged (asymmetry) angular coefficients (), and the optimized observables and . Following Reference [48], where the authors approved that the possible S-wave correction to the branching fractions of decays is small and may modify the differential decay widths by about only, we therefore will take the S-wave correction to the branching fractions as an additional uncertainty of 5% in magnitude(2)We used both the PQCD factorization approach and the “PQCD+Lattice” approach to determine the values and their -dependence of the transition form factors. We used the -series parametrization to make the extrapolation for all form factors from the low- region to the endpoint . We will calculate the branching ratios and all other physical observables by using the PQCD approach itself and the “PQCD+Lattice” approach, respectively, and compare their predictions with those currently available experimental measurements
The paper is organized as follows: In Section 2, we give a short review for the kinematics of the decays including distribution amplitudes of and mesons and the effective Hamiltonian for the quark level . In Section 3, we define explicitly all physical observables for decays. In Section 4, we present our theoretical predictions of all relevant physical observables of the considered decay modes, compare these predictions with those currently available experimental measurements, and make some phenomenological analysis. A short summary is given in the last section.
2. Kinematics and Theoretical Framework
2.1. Kinematics and Wave Functions
We treat the meson at rest as a heavy-light system. The kinematics of the semileptonic decays in the large-recoil (low-) region will be discussed below, where the PQCD factorization approach is applicable to the considered decays. In the rest frame of meson, we define the meson momentum and the momentum in the light-cone coordinates as Reference [41]. We also use to denote the momentum fraction of light antiquark in each meson and set the momentum and (the momenta carried by the spectator quark in and meson) in the following forms: where the mass ratio and the factor is defined in the following form: where is the lepton-pair four-momentum. For the final state meson, its longitudinal and transverse polarization vector can be written in the form of and .
For the meson wave function, we use the same kind of parameterizations as in References [42–44].
Here, only the contribution of the Lorentz structure is taken into account, since the contribution of the second Lorentz structure is numerically small [49, 50] and has been neglected. We adopted the distribution amplitude of the meson in the similar form as that of -meson in the limit being widely used in the PQCD approach [42–44]
In order to estimate the theoretical uncertainties induced by the variations of , one usually take for meson [38, 42]. The normalization factor depends on the values of the shape parameter and the decay constant and defined through the normalization relation: [42, 44].
For the vector meson , the longitudinal and transverse polarization components can both provide the contribution. Here, we adopt the wave functions of the vector as in Reference [42]: where and are the momentum and the mass of the meson, and correspond to the longitudinal and transverse polarization vectors of the vector meson , respectively. The twist-2 DAs and in Equations (5) and (6) can be reconstructed as a Gegenbauer expansion [42]: where and are the Gegenbauer moments, while are the Gegenbauer polynomials as given in Reference [42]. and are the longitudinal and transverse components of the decay constants of the vector meson with and as given in Reference [42]. For the relevant Gegenbauer moments, we use the same ones as those in References [42, 46, 47, 51].
The twist-3 DAs and in Equations (5) and (6) are the same ones as those defined in Reference [42]: where .
2.2. Effective Hamiltonian for Decays
The effective Hamiltonian for the considered semileptonic decay is defined by the same one as in References [52–56]: where is the Fermi constant and are the CKM matrix elements. For the operators , we adopt those as defined in the so-called -free basis [57, 58]. Following Reference [59], the operators can be written in the following form: where are the current-current operators, are the QCD penguin operators, are the electromagnetic and chromomagnetic penguin operators, respectively, and finally, are the semileptonic operators. The inclusion of the factors in the definition of the operators serves to allow a more transparent organisation of the expansion of the relevant Wilson coefficients as defined in References [59, 60] up to next-to-next-to-leading order (NNLO). They are then evolved from the scale down to the scale using the renormalization group equations.
Since the contributions from the subleading chromomagnetic penguin, quark loop, and annihilation diagrams are highly suppressed for the considered decays [55], we will neglect them in our calculations. Using the effective Hamiltonian in Equation (11), the decay amplitude for loop transition can be decomposed as a product of a short-distance contributions through Wilson coefficients and long-distance contribution which is further expressed in terms of form factors: where and are the effective Wilson coefficients, defined as in References [44, 61].
The term in Equation (14) is the absorptive part of transition and was given in Reference [61]. where and . The explicit expressions of the term and in Equation (15) are of the following form [56, 62–65]: where , , , and the CKM ratio . In Equation (17), the function is the soft-gluon correction to the matrix element of operator . The function in Equations (17) and (18) is related to the basic fermion loop. The contributions from four quark operators are usually combined with coefficient into an “effective” one. One can find the explicit expressions of the function and easily, for example, in Reference [35] and references therein.
The term in Equations (15) and (17) defines the short-distance perturbative part that involves the indirect contributions from the matrix element of the four quark operators [62–65] and lies at the place far away from resonance regions.
The term in Equations (15) and (18) describes the long-distance resonant contributions related with the transitions in the resonance regions, where are the light vector mesons and charmonium states. Up to now, the term cannot be calculated from the first principle of QCD and may also introduce the double-counting problem with the term . For more details about such kinds of double-counting problem, one can see the discussions as given in References [66, 67]. In this paper, we checked the possible effects on the theoretical predictions for the branching ratios and other considered physical observables by including the term or not in our numerical calculations, and we found that the resulted variations of the theoretical predictions are less than . It is much smaller than the total theoretical errors, say around . According to the argument in Reference [68], the term is also generally small. Because of its smallness and the possible double-counting problem, we here simply drop the term out in our numerical evaluations for all physical observables considered in this paper.
2.3. Transition Form Factors
For the vector meson with polarization vector , as usual, the relevant form factors for transitions are and of the vector and axial vector currents and of the tensor currents. Between the form factors at the point , there is an exact relation in order to avoid the kinematical singularity. Between the form factor , there also exists a relation in an algebraic manner which is implied by the identity with the convention for the Levi-Civita tensor.
Using the well-studied wave functions as given in previous subsection, the PQCD factorization formulas for the relevant form factors of decays can be calculated and written in the following form: where , the twist-2 DAs , and the twist-3 DAs have been defined in Equations (7) and (10). The function in above equations is of the following form:
The hard functions come form the Fourier transform of virtual quark and gluon propagators and they can be defined by the following: where and are the modified Bessel functions and where and . The hard scales in Equation (20) are chosen as the largest scale of the virtuality of the internal particles in the hard -quark decay diagram, including :
The threshold resummation factor in Equation (20) is adopted from [49]: with a fitted parameter [39] and [69]. The function is normalized to unity. The function in Equation (20) contains the Sudakov logarithmic corrections and the renormalization group evolution effects of both the wave functions and the hard scattering amplitude; for more details of function , one can see References [37, 49].
3. Observables for Mode Decays
In experimental analysis, the decay is treated as the four body differential decay distribution and has been described in terms of the four kinematic variables [10, 11, 13, 19]: the lepton invariant mass squared and the three decay angles . The angle is the angle between the direction of flight of and meson in the rest frame of , is the angle made by with respect to the meson in the dilepton rest frame, and is the azimuthal angle between the two planes formed by dilepton and .
With the hadronic and leptonic amplitudes defined in Equation (13), we write down the fourfold differential distribution of four-body decay [13, 55, 70, 71]: where the functions can be written in terms of a set of angular coefficients and trigonometric functions [70]:
For the CP-conjugated mode , the corresponding expression of the angular decay distribution is as follows: where the function is obtained from in Equation (26) by making the complex conjugation for all weak phases in [70] and numerically by the following substitution:
The minus sign in Equation (28) is a result of the convention that, under the previous definitions of three angles , a CP transformation interchanges the lepton and antilepton, i.e., leading to the transformation and .
The angular coefficients , which are functions of only, are usually expressed in terms of the transverse amplitudes [8, 13]. In the limit of massless leptons, there are six such complex amplitudes: , , and , where and refer to the chirality of the leptonic current. For the massive case, an additional complex amplitude is required, where the timelike component of the virtual gauge boson (which can later decay into dilepton) couples to an axial vector current.
In Table 1, we summarize the treatment of the angular distribution by decomposition of the angular coefficients into seven transverse amplitude and as well as the corresponding trigonometric factor . Here, we will not consider scalar contribution to facilitate the comparison with Reference [35]. Notice that the distribution including lepton masses (but neglecting scalar ) contains eleven where only 10 of them are independent [8, 72]. In the limit of massless leptons, it is easy to obtain the relations and [70].
The seven transverse amplitudes , , , and of decay, in turn, can be parameterized by means of the relevant form factors [70, 73]: where , , and the normalization constants are given as follows:
In numerical calculations, we take from PDG 2018 [74]. It is easy to see that the narrow width approximation works well in the case of meson since .
Analogous to Reference [70], to separate CP-conserving and CP-violating effects, one can define the CP-averaged angular coefficients and CP asymmetry angular coefficients normalized by the differential (CP-averaged) decay rate to reduce the theoretical uncertainties: where and have been defined in Equations (26), (27), and (28) and Table 1, and the differential decay rate reads (analogously for )
Based on the definition of , one can find the relation . Consequently, all established observables can be expressed in terms of and : (1)The CP asymmetry:(2)The lepton forward-backward (CP) asymmetry:(3)The polarization fractions:
In the massless limit, since the CP-averaged observable obey the relations and , the definitions of the polarization fractions can be simplified directly as follows: (4)The clean (no S-wave pollution) observables and in the natural basis can be defined in terms of the coefficients through the following relations [71, 75, 76]:where . (5)In the massless limit of leptons, the optimized observables [8] can be transformed as the following form:
One should know that our definitions of the CP-averaged angular coefficients , the CP asymmetry angular coefficients , and the clean observable and differ from those adopted by the LHCb Collaboration. To be specific, the reasons are the following: (1)Our conventions for the angles to define the kinematics are identical to the Reference [70] but different from the LHCb choices [13, 31]. The corresponding relations are the following:
Some angular coefficients , , and , consequently, will have different signs:
Other remaining coefficients ( and ), however, have the same sign in both conventions. (2)Our definitions of the clean observables and in Equation (38) in terms of may be different from those defined and used by the LHCb Collaboration, for example, in Reference [13]. The resultant differences of the sign and normalization are of the following:
For more details about the angular conventions of the angular observables of the semileptonic decays , one can see Reference [77].
4. Numerical Results and Discussions
In the numerical calculations, we use the following input parameters (here, masses and decay constants are in units of GeV) [42, 74]:
For the CKM matrix elements and angles, we adopt the following values as given in Reference [74]:
4.1. The Form Factors
For the considered semileptonic decays, the differential decay rates and other physical observables strongly rely on the value and the shape of the relevant form factors and for decays. These form factors have been calculated in rather different theories or models [23, 36–38, 42]. Since the PQCD predictions for the considered form factors are valid only at the large hadronic recoil (low-) region, we usually calculate explicitly the values of the relevant form factors at the low- region, say , and then make an extrapolation for all relevant form factors from the low- towards the high- region by using the pole model parametrization [78, 79] or other different methods.
In References [80–82], we developed a new method: the so-called “PQCD+Lattice” approach. Here, we still use the PQCD approach to evaluate the form factors at the low- region, but take those currently available lattice QCD results for the relevant form factors at the high- region as the lattice QCD input to improve the extrapolation of the form factors up to . In References [81, 82], we used the Bourrely-Caprini-Lellouch (BCL) parametrization method [83, 84] instead of the traditional pole model parametrization.
In Table 2, we list the PQCD predictions for all seven relevant form factors , , and for transition at the point . The dominant theoretical errors come from the uncertainties of the parameter [42], the Gegenbauer moments and [42, 46, 47, 51], and the decay constants and [42].
In Table 3, as a comparison, we also list the central values of the theoretical predictions for the form factors at evaluated in the PQCD approach [42, 85] and in other different theories or models [17, 20, 21, 23, 86–91]. One can see that there exist always some differences between different authors, even among the authors using the same approach. Taking the calculations based on the LCSR method as an example, the authors of Reference [23] introduced the hadronic input parameters, Ball and Zwicky considered the one-loop radiative corrections [20], and Yilmaz included the radiative and higher twist corrections and SU(3) breaking effects [86].
In Table 4, for the transition form factors , we quote directly the values of the lattice QCD results at two or three reference points of the high- region, say and , as listed in Table XXXI of Reference [92]. In Reference [92], the authors defined the helicity form factors and from the ordinary form factors and : where the kinematic variable with . From above two equations and the numerical values of as given in Reference [92], we can find the corresponding lattice QCD results of and at the two points by direct numerical calculations. When , however, the parameter in Equation (45) is also approaching zero simultaneously; one therefore cannot determine and reliably from the values of and as given in Reference [92]. Consequently, and are absent in Table 4.
In this work, we will use both the PQCD factorization approach and the “PQCD+Lattice” approach to evaluate all relevant form factors over the whole range of . (1)In the PQCD approach, we use the definitions and formulae to calculate the values of all relevant form factors , , and at some points in the region of . We then make the extrapolation for these form factors to the large region up to by using the selected parametrization method directly(2)In the “PQCD+Lattice” approach, we take the lattice QCD results of the form factors at the points of as the new input in the high- region and then make a combined fit of the PQCD results in low- region and the lattice QCD results in high- region to determine the relevant parameters in the -series expansion and then complete the extrapolation(3)For both approaches, we always use the model-independent -series parametrization, which is based on a rapidly converging series in the parameter , as in References [23, 55] to make the extrapolation. The entire cut -plane will be mapped onto the unit disc under the conformal transformation as [93]where and is an auxiliary parameter which can be optimized to reduce the maximum value of in the physical range of the form factors and will be taken in the same way as in Reference [94]: . The form factors are finally parameterized in the BCL version of the -series expansion [83].
Since the term in the whole considered region, the high-order terms in Equation (48) should be very small in magnitude and therefore can be neglected. After the truncation at , the coefficient for the corresponding form factor can be determined by fitting to the PQCD predictions at low- region and the lattice QCD results in the high- region. Taking the form factor as an example, we calculate first by employing the PQCD approach in the sixteen points in the low region, take the lattice QCD results and as additional input, and finally make the fitting for the parameter and find that with the goodness-of-fit . For other form factors , we find the results by following the same kind of procedure. The input values of the various -resonance mass in Equation (48) can be found from Reference [74] and are collected in Table 5. For further discussions on the systematic uncertainties due to the dependence of truncation schemes and on the implementation of the strong unitary constraints, one can see References [94, 95]. (4)In Figure 1, we show the theoretical predictions of the form factors , , and for transition based on the PQCD approach (red curves) and the “PQCD+Lattice” approach (blue curves) with the extrapolation from to by applying the -series parameterizations. The shaded bands represent the total theoretical error obtained by adding in quadrature of the separate errors from the uncertainty of the parameter , , , and . The black error bars in the low- region correspond to the PQCD predictions of the corresponding form factors, while the error bars in the high- region are the currently known lattice QCD results as collected in Table 3
4.2. Numerical Results
We now proceed to explore the phenomenological aspects of the cascade decays , which allow us to define and compute a number of physical observables and compare them with those measured by experiments. We first compare our results for the branching ratios and angular observables with the experimental data reported by the LHCb Collaboration [31, 33]. As studied systematically in the last section, the physical observables accessible in the semileptonic decays are the CP-averaged differential branching fraction [31, 33], the CP-averaged meson longitudinal polarization fraction , the forward-backward asymmetry , the angular coefficients and , and the optimized observables and [25]. The CP asymmetry angular coefficients in the SM are induced by the weak phase from the CKM matrix. For the transition, the CP asymmetries proportional to , which is of order [19] as measured by the LHCb Collaboration (see Table 3 in Reference [31]), but the statistical uncertainties are still large. For these reasons, we will focus on the CP-averaged quantities when taking the binned observables into consideration.
We begin with the branching ratios of the decays . From the differential decay rates as defined in Equation (32), it is straightforward to make the integration over the range of . In order to be consistent with the choices made by LHCb Collaboration in their data analysis, we here also cut off the regions of dilepton-mass squared around the charmonium resonances and ; i.e., and for cases. We display the PQCD and “PQCD+Lattice” predictions for the differential branching ratios in Figure 2 for the cases of , including currently available LHCb results in six or eight bins of [31, 33] indicated by the crosses for decay. From Figure 2, one can see that both PQCD and “PQCD+Lattice” predictions for the differential branching ratios do agree well with the LHCb results within the still large errors. Since the theoretical prediction for the differential branching ratio of the electron mode is almost identical with the one of the muon, we do not draw the figure of in Figure 2.
In Table 6, we present the theoretical predictions of the total branching fractions for with obtained by the integration over the six bins using the PQCD (the first row) and “PQCD+Lattice” approach (the second row), respectively. The major theoretical errors from different sources, such as the form factors (FFs) as listed in Table 2, the scale , and the CKM matrix element and , are also listed. As in Reference [31], a correction factor is applied to account for the contribution in the veto bins for cases. As a comparison, we also show the LHCb measured value [31] and [33] and the QCDSR predictions for all three decay modes [87]. For decay, for instance, the theoretical predictions and the LHCb measurement [31, 33] (in unit of ) are the following:
From the numerical results in above equation and Table 6, one can see that (1)The PQCD and “PQCD+Lattice” predictions for the branching ratio with do agree well with each other within the errors, while the “PQCD+ Lattice” predictions of have smaller errors than those of the PQCD predictions(2)Both PQCD and “PQCD+Lattice” predictions of do agree well with currently available LHCb measured values [31, 33] within errors. For the electron and tau mode, however, we have to wait for the future experimental measurements(3)For all three decay modes, our theoretical predictions of the branching ratios do agree well with the theoretical predictions obtained from the QCD sum rule [87]
Since the large theoretical uncertainties of the branching ratios could be largely canceled in the ratio of the branching ratios of decays, one can define and check the physical observables and [5]. In the region , where only the modes are allowed, there is a large enhancement due to the scaling of the photon penguin contribution [96]. In order to remove the phase space effects in the ratio and keep consistent with other analysis [5], we here also use the lower cut of for both the electron and muon modes in the definition of the ratio as in Reference [5]:
For the case of the ratio , we have where the total error is the combination of the individual errors in quadrature. We suggest the LHCb and Belle-II to measure these two ratios.
For decay, we show our theoretical predictions for the -dependence of the longitudinal polarization , the CP-averaged angular coefficients , and the CP asymmetry angular coefficients in Figure 3. As a comparison, the currently available LHCb measurements for these observables of decay in the six bins [31] are also shown by those crosses explicitly. One can see from Figure 3 that (1)For the longitudinal polarization , although both PQCD and “PQCD+Lattice” predictions all agree well with the LHCb measurements in the six bins, our theoretical predictions in the region of the fourth and fifth bin are little larger than the measured ones(2)For the CP-averaged angular coefficients , the PQCD and “PQCD+Lattice” predictions agree very well with each other and are consistent with the LHCb results within the still large experimental errors. For the last two high- bins, the LHCb results of () are a little larger (smaller) than our theoretical predictions(3)For the CP asymmetry angular coefficients , the PQCD and “PQCD+Lattice” predictions are very small: in the range of to . For the LHCb measurements in the six bins, they are clearly consistent with our theoretical predictions due to still large experimental errors
In Figure 4, we show our theoretical predictions for the -dependence of the forward-backward asymmetry , the optimized observables , and for decay. Unfortunately, there exist no any experimental measurements for these observables. We have to wait for future LHCb and Belle-II measurements. Analogous to Figure 3, the vertical grey blocks in Figure 4 also denote the two experimental veto regions of : and .
In Table 7, we list the theoretical predictions for the values of the observables , , , , , and , obtained after the integrations over the whole kinematic region of for the semileptonic decays with in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches, respectively. Of course, the regions corresponding to resonance and , say and numerically, are also cut off here. The total errors are the combinations of the individual errors from the form factors, the renormalization scales, and the relevant CKM matrix elements. The above theoretical predictions should be tested in the near future LHCb and Belle-II experiments. For the considered meson decays, one should consider the effects from the - mixing [6]. The theoretical framework for examining the time-dependent decays with the inclusion of such mixing effects can be found in Reference [97]. The authors of Reference [97] proved that the mixing effects on the values of decay rates and CP-averaged observables are generally within a few percent and could be neglected.
4.3. The -Binned Predictions
For decay mode, the LHCb Collaboration has reported their experimental measurements for many physical observables in several bins [31, 33]. In order to compare our theoretical predictions with the LHCb results bin by bin, we make the same choices of the bins as LHCb did, calculate, and show our theoretical predictions for the branching ratio and the asymmetry with in Tables 8 and 9 and the observables with in Table 10. For observables and , in fact, our theoretical predictions for their values are very small, say in the range of in magnitude, but still agree with the LHCb measurements in different bins [31] due to still large experimental errors. For the observables and , they are also very small in size: in the range of and there exist no corresponding data at present. For observables and , on the other hand, although there exist no experimental measurements for them at present, they are relatively large in size and may be measured in the near future LHCb and Belle-II experiments, so we calculate and list the theoretical predictions of these observables bin by bin for the cases of in Tables 11 and 12. Very recently, LHCb reported some new measurements for the angular observables of decay [34] in the bins different from those in their previous work [31], which will be studied in our next work.
The definitions of the -binned observables are the following:
From the numerical values as shown in Figure 3 and in Tables 8–10, we find the following points about the relevant physical observables of the considered decays in bins: (1)For decay, besides the good consistency between the theory and the LHCb data for the integrated total branching ratio as listed in Equation (50), the PQCD and “PQCD+Lattice” predictions for in most bins do agree well with the measured ones within errors. For the first low- bin , however, the central value of the LHCb result is larger than the theoretical ones by roughly a factor of three. The LHCb results of in different bins of as listed in the third column of Table 9 are obtained from the results as given in References [31, 33] by multiplying the LHCb measured values of differential decay rate with the width of the corresponding bin . The theoretical errors of our theoretical predictions of the branching ratios in bins are still relatively large, while the differences between the PQCD and “PQCD+Lattice” predictions for with are small(2)In the first low- bin , both the PQCD and “PQCD+Lattice” predictions for are larger than the LHCb measured results [31]. For other bins, both PQCD and “PQCD+Lattice” predictions of for muon mode do agree very well with currently available LHCb measured values [31] within errors. It is worth of remaining that our theoretical predictions of have a little error of due to the strong cancellation of the theoretical errors in the ratios. The theoretical predictions for and in different bins of as listed in Table 9 will be tested by future experimental measurements(3)For the observables , as listed in Table 10, the PQCD and “PQCD+Lattice” predictions for their values in all bins are in the range of and show a good agreement with the LHCb measured values [31]. The errors of the theoretical predictions are also very small, in magnitude, because of their nature of the ratios. In all bins, the LHCb measured values of are still consistent with zero due to their still large errors, which is a clear feature as can be seen easily from the numerical values in Table 10 and the crosses in Figure 3
In Tables 11 and 12, we show the PQCD and “PQCD+Lattice” predictions for the physical observables and in six bins. These physical observables could be tested in the near future LHCb and Belle-II experiments.
5. Summary
In this paper, we made a systematic study of the semileptonic decays with using the PQCD and the “PQCD+Lattice” factorization approach, respectively. We first evaluated all relevant form factors in the low- region using the PQCD approach, and we also took currently available lattice QCD results at the high- points as additional input to improve the extrapolation of the form factors from the low to the high- region. We calculated the branching ratios , the CP-averaged longitudinal polarization fraction , the forward-backward asymmetry , the CP-averaged angular coefficients , the CP asymmetry angular coefficients , the optimized observables , and . For decay mode, we calculated the binned values of the branching ratio and the observables and in the same bins as defined by LHCb Collaboration [31] in order to compare our theoretical predictions with those currently available LHCb measurements bin by bin directly.
Based on the analytical evaluations, the numerical results, and the phenomenological analysis, we found the following main points: (1)For the branching ratio , the PQCD and “PQCD+Lattice” predictions are and , respectively, which agree well with the LHCb measured values [31, 33] and the QCDSR prediction within still large errors. For the electron and tau mode, our theoretical predictions for their decay rates are also well consistent with the corresponding QCDSR predictions and to be tested by future experimental measurements(2)For the ratios of the branching ratios and , the PQCD and “PQCD+Lattice” predictions agree with each other and with small theoretical errors because of the strong cancellation of the theoretical errors in such ratios. We suggest the LHCb and Belle-II Collaboration to measure these ratios(3)For the longitudinal polarization , both PQCD and “PQCD+Lattice” predictions agree with the LHCb measurements in the considered bins within the errors. For the CP-averaged angular coefficients , the PQCD and “PQCD+Lattice” predictions in all bins are small in magnitude, in the range of , and agree well with the LHCb results within the still large experimental errors. For the CP asymmetry angular coefficients , the PQCD and “PQCD+Lattice” predictions are very small, in the range of , and clearly consistent with the LHCb measurements in the six bins(4)For the physical observables , , and , the experimental measurements are still absent now; we think that the PQCD and “PQCD+Lattice” predictions for these physical observables will be tested in the near future LHCb and Belle-II experiments
Data Availability
All relevant data are from LHCb Collaboration as given in References [31, 33].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work (The preprint number of this paper is arXiv: 2011.11409 with the web link: https://arxiv.org/abs/2011.11409) was supported by the National Natural Science Foundation of China under (Grant Nos. 11775117 and 11235005).