Charmless \(B_{(s)}\rightarrow VV\) decays in factorizationassisted topologicalamplitude approach
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Abstract
Within the factorizationassisted topologicalamplitude approach, we studied the 33 charmless \(B_{(s)} \rightarrow VV\) decays, where V stands for a light vector meson. According to the flavor flows, the amplitude of each process can be decomposed into eight different topologies. In contrast to the conventional flavor diagrammatic approach, we further factorize each topological amplitude into decay constant, form factors and unknown universal parameters. By \(\chi ^2\) fitting 46 experimental observables, we extracted 10 theoretical parameters with \(\chi ^2\) per degree of freedom around 2. Using the fitted parameters, we calculated the branching fractions, polarization fractions, CP asymmetries and relative phases between polarization amplitudes of each decay mode. The decay channels dominated by tree diagram have large branching fractions and large longitudinal polarization fraction. The branching fractions and longitudinal polarization fractions of colorsuppressed decays become smaller. Current experimental data of large transverse polarization fractions in the penguin dominant decay channels can be explained by only one transverse amplitude of penguin annihilation diagram. Our predictions of the not yet measured channels can be tested in the ongoing LHCb experiment and the BelleII experiment in the future.
1 Introduction
Charmless hadronic Bmeson decays have been of significant interest, as they can provide us an abundant source of information on the flavor physics, within and beyond the standard model (SM). After the very successful first generation of Bfactory experiments, BaBar and Belle [1], the interest in this field is reinforced by the LHCb experiment [2] and the upcoming start of BelleII experiment [3, 4]. In the long term run, the LHCb upgrade plan promises excellent future opportunity [5, 6]. The FCCee, as well as CEPC, the proposal for future electron–positron collider will give further chance for the flavor physics study [7]. In the theoretical side, beyond the naive factorization approach [8, 9, 10], three major QCDinspired approached had been proposed to deal with charmless nonleptonic B decays, based on the effective theories, namely, the QCD factorization (QCDF) [11, 12], perturbative QCD (PQCD) [13, 14], and softcollinear effective theory (SCET) [15, 16, 17, 18]. The difference between them is only on the treatment of dynamical degrees of freedom at different mass scales, namely the power counting. Within these approaches, most decay processes have been studied, including the branching fractions and the CP asymmetries. However, the factorization for hadronic matrix elements is only proved at the leading order in \(1/m_b\), with \(m_b\) denoting the b quark mass. The precision of these approaches is limited to the leading power calculations.
In contrast to the above approaches based on the perturbative QCD, another idea based on the topological diagrams and flavor SU(3) symmetry was also proposed [19, 20, 21], where the nonperturbative parameters are extracted directly from the experimental data. Therefore, the extracted parameters include the effects of strong interactions to all orders, as well as longdistance rescattering. This idea has been used to analyze hadronic B meson decays extensively [22, 23, 24, 25, 26, 27, 28], as well as D meson decays [29, 30]. Although no direct power expansion is needed in this approach, flavor SU(3) symmetry is required to reduce the number of free parameters to be fitted from experiments. As the experimental precision is better and better, the limitation of theoretical precision is retarded. Recently, the improved version, the socalled factorizationassisted topologicalamplitude approach (FAT) [31, 32] was proposed, in order to deal with the SU(3) breaking effects. By using some of the welldefined factorization formulas to include most of the SU(3) breaking effects, the theoretical results of the twobody nonleptonic D decays accommodate experimental data very well. Recently, the FAT approach has been utilized to study the twobody charmed nonleptonic B mesons decays [33]. Within four universal nonperturbative parameters fitted from 31 experimental observations, 120 charmed B decay modes were calculated. Both branching fractions and CP asymmetry parameters are in agreement with experimental data well. Very recently, the charmless \(B_{(s)} \rightarrow PP\) and \(B_{(s)}\rightarrow PV\) processes are also studied using this approach [34]. The longstanding \(B\rightarrow \pi ^0 \pi ^0\) and \(B\rightarrow K\pi \) CP puzzles can be explained simultaneously.
In contrast to \(B_{(s)} \rightarrow PP\) and PV decays, charmless \(B_{(s)} \rightarrow V V\) decays are much more complicated, because more helicity amplitudes will be considered. Due to angular momentum conservation, there are three independent configurations of the finalstate spin vectors: a longitudinal component where both resonances are polarized in their direction of motion, and two transverse components with perpendicular and transverse polarizations. For the \(VA\) coupling of the SM, a specific pattern of the three helicity amplitudes is naively expected [35], such that the longitudinal polarization fraction \(f_\mathrm{L}\) should be close to unity, while the transversal contributions are suppressed by \(\Lambda _\mathrm{QCD}/m_B\). In 2004, large transverse polarization fractions (around 50%) of \(B\rightarrow \phi K^*\) have been measured in the experiments. Later on, some other penguindominated strangenesschanging decays, such as \(B\rightarrow \rho K^*\) and \(B_s\rightarrow \phi \phi \), have also been found with large transverse polarization fractions. These large unexpected transverse polarization fractions have attracted much theoretical attention with several explanations based on the QCDF [36, 37, 38, 39, 40, 41, 42], PQCD [43, 44, 45], even on the new physics scenarios [46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. These decays have rich observables, some of which are regarded as good places for testing the SM and searching for possible effects of new physics beyond the SM.
To our knowledge, these decays with two vector meson final states have not been studied in the flavor diagram approach. In this work, we shall explore the charmless twobody nonleptonic \(B_{(s)} \rightarrow V V\) decays, in the newly established FAT approach. The branching fractions, CP asymmetries, as well as the angular distributions will be investigated. The organization of the paper is as follows: In Sect. 2, we give the definitions for helicity amplitudes, angular variables and polarization observables. The calculation of the \(B\rightarrow VV\) decay amplitudes in the FAT framework is briefly reviewed. Section 3 provides the numerical results and the phenomenological discussions. We will summarize our work in Sect. 4.
2 Framework

T, denoting the colorfavored tree diagram with external W emission;

C, denoting the colorsuppressed tree diagram with internal W emission;

E, denoting the Wexchange diagram;

A, denoting the annihilation diagram.

P, denoting the QCDpenguin diagram;

S, denoting the flavorsinglet QCDpenguin diagram;

\(P_E\), denoting the timelike penguin annihilation diagram;

\(P_A\), denoting the spacelike penguin annihilation diagram.
The mass and decay constant of meson (in units of GeV)
Meson  Mass  Decay constant 

\(B^{\pm /0}\)  5.28  0.190 
\(B_s^{0}\)  5.36  0.225 
\(\rho \)  0.77  0.213 
\(\omega \)  0.78  0.192 
\(\phi \)  1.01  0.225 
\(K^*\)  0.89  0.220 
3 \(\chi ^2\) fitting and numerical results
To characterize the flavor SU(3) breaking effects in our calculation, we need input parameters of various meson masses [58] and decay constants that are summarized in Table 1. Unlike the meson masses, the value of the decay constants is not known in experiment, but they can be given from theoretical calculations, such as QCD sum rules [59, 60], the Bethe–Salpeter equation [61, 62], and lattice QCD [63, 64, 65], and we have taken the value from [34] with \(5\%\) uncertainty.
For the calculation of the colorfavored tree diagram T and the QCD penguin diagram P, we also need the input of the form factors for \(B \rightarrow V\) transitions. There are many calculations for form factors, such as lightcone sum rules [66, 67, 68], perturbative QCD approach [69, 70, 71], and lattice QCD [72, 73, 74] etc. The central values we used in this work of the transition form factors at \(q^2=0\) are shown in Table 2. To estimate the theoretical uncertainty of the numerical results in our calculation, we include the uncertainties of all form factors as large as \(10\%\). In fact, they belong to the major sources of theoretical uncertainty in our numerical results, especially for processes dominated by the colorfavored tree (T) diagram. Since the finalstate meson mass is small compared with the large B meson mass, the \(q^2\) dependence of the form factors will be neglected. In fact, the effects of \(q^2\) dependence on the numerical results are negligible, which has been indicated in \(B\rightarrow PP,PV\) decays [34].
The transition form factor of \(B\rightarrow V\) at \(q^2=0\)
\(B \rightarrow \rho \)  \(B \rightarrow K^*\)  \(B\rightarrow \omega \)  \(B_s\rightarrow K^*\)  \(B_s\rightarrow \phi \)  

V(0)  0.33  0.41  0.29  0.31  0.42 
\(A_0(0)\)  0.32  0.38  0.28  0.36  0.44 
\(A_1(0)\)  0.25  0.29  0.22  0.23  0.31 
3.1 The \(\chi ^2\) fit for theoretical parameters
Unlike \(B\rightarrow D^{(*)}M\) and \(B\rightarrow PP(V)\) decay modes, the amplitudes of \(B\rightarrow VV\) modes are much complicated, because each decay process has three polarization contributions, which means that the number of parameters will increase threefold. From previous section, we do not introduce any new parameter in colorfavored tree diagram T and QCDpenguin diagram P. For the colorsuppressed tree diagram C, the flavorsinglet QCDpenguin diagram S, the Wexchange diagram and QCDpenguin annihilation diagram \(P_A\), we have altogether 24 parameters \(\chi _C^{0,\parallel ,\perp }\), \(\phi _C^{0,\parallel ,\perp }\), \(\chi _{S}^{0,\parallel ,\perp }\), \(\phi _{S}^{0,\parallel ,\perp }\), \(\chi _{E}^{0,\parallel ,\perp }\), \(\phi _{E}^{0,\parallel ,\perp }\), \(\chi _{P_A}^{0,\parallel ,\perp }\) and \(\phi _{P_A}^{0,\parallel ,\perp }\). So many free parameters are difficult to determine from the limited number of experimental measurements. It will also decrease the predictive power of this FAT approach. As indicated from the QCD factorization approach [41, 42] and the perturbative QCD approach [45] calculations, the colorsuppressed tree diagram C, W exchange diagram E and the flavorsinglet QCDpenguin diagram S are dominated by longitudinal polarization contributions. For simplicity, we here drop the negligible transverse contributions of these topological diagrams to set \(\chi _C^{\parallel }=\chi _C^{\perp }=0\), \(\chi _E^{\parallel }=\chi _E^{\perp }=0\), and \(\chi _{S}^{\parallel }=\chi _{S}^{\perp }=0\). Therefore, the only transverse polarization amplitudes to be fitted are from the penguin annihilation diagram. According to the power counting of this diagram [37], the negativehelicity amplitude is chirally enhanced, while the positivehelicity amplitude is still suppressed (to be neglected). This results in the relation \(\chi _{P_A}^{\parallel } \approx \chi _{P_A}^{\perp }\). Thus, there are only 10 universal parameters left, which will be fitted by experimental data.
In the experimental sides, after the first decay modes \(B\rightarrow \phi K^*\) were measured, more and more observables have been measured, involving the branching fractions, CP asymmetries, polarization fractions, and relative phases of helicity amplitudes. Since some observables are measured with very poor precision, the data with less than \(3\sigma \) significance will not be used in our fitting program. Then we have 46 experimental data, involving 18 branching fractions, 20 polarization fractions, six relative phases, and two direct CP asymmetries.
3.2 Numerical results and discussions
Branching fractions and the direct CP asymmetries of \(\overline{B} \rightarrow VV\) decay modes. The experimental data [58] are also given for comparison
Class  Decay mode  Branching fraction/\(10^{6}\)  \(A_\mathrm{CP}\)/percent  

Theory  Exp  Theory  Exp  
T  \(B^\rightarrow \rho ^\rho ^0\)  \(21.7\pm 1.8\pm 4.2\pm 2.2\)  \(24.0\pm 1.9\)  0  \(5\pm 5\) 
\(\overline{B}^0\rightarrow \rho ^\rho ^+\)  \(29.5\pm 1.9\pm 5.4\pm 3.0\)  \(28.3\pm 2.1\)  \(8.10\pm 2.94\)  
\(B^\rightarrow \rho ^\omega \)  \(18.2\pm 1.5\pm 2.8\pm 1.6\)  \(15.9\pm 2.1\)  \(3.45\pm 5.38\)  \(20\pm 9\)  
\(\overline{B}_s^0\rightarrow K^{*+}\rho ^\)  \(38.6\pm 0.1\pm 7.3\pm 3.9\)  \(10.9\pm 3.0\)  
C  \(\overline{B}^0\rightarrow \rho ^0\rho ^0\)  \(0.94\pm 0.46\pm 0.11\pm 0.14\)  \(0.97\pm 0.24\)  \(49.7\pm 13.4\)  
\(\overline{B}^0\rightarrow \rho ^0\omega \)  \(1.48\pm 0.71\pm 0.06\pm 0.20\)  \({<}1.6\)  \(38.5\pm 13.6\)  
\(\overline{B}^0\rightarrow \omega \omega \)  \(1.20\pm 0.49\pm 0.12\pm 0.18\)  \(1.2\pm 0.4\)  \(28.1\pm 13.8\)  
\(\overline{B}_s^0\rightarrow K^{*0}\rho ^0\)  \(1.18\pm 0.39\pm 0.21\pm 0.12\)  \({<}767\)  \(4.9\pm 18.3\)  
\(\overline{B}_s^0\rightarrow K^{*0}\omega \)  \(0.97\pm 0.33\pm 0.16\pm 0.10\)  \(32.2\pm 16.0\)  
P  \(B^\rightarrow \rho ^\overline{K}^{*0}\)  \(10.4\pm 1.6\pm 1.7\pm 1.1\)  \(9.2\pm 1.5\)  \(1.00\pm 0.17\)  \(1\pm 16\) 
\(B^\rightarrow \rho ^0 K^{*}\)  \(5.83\pm 0.66\pm 0.76\pm 0.65\)  \(4.6\pm 1.1\)  \(34.6\pm 8.3\)  \(31\pm 13\)  
\(\overline{B}^0\rightarrow \rho ^0\overline{K}^{*0}\)  \(5.09\pm 0.75\pm 0.82\pm 0.53\)  \(3.9\pm 1.3\)  \(0.6\pm 4.0\)  \(6\pm 9\)  
\(\overline{B}^0\rightarrow \rho ^+ K^{*}\)  \(10.5\pm 1.3\pm 1.4\pm 1.2\)  \(10.3\pm 2.6\)  \(34.3\pm 6.3\)  \(21\pm 15\)  
\(B^\rightarrow \omega K^{*}\)  \(4.24\pm 0.70\pm 0.32\pm 0.51\)  \({<}7.4\)  \(30.1\pm 13.8\)  \(29\pm 35\)  
\(\overline{B}^0\rightarrow \omega \overline{K}^{*0}\)  \(3.10\pm 0.75\pm 0.27\pm 0.38\)  \(2.0\pm 0.5\)  \(11.7\pm 4.0\)  \(45\pm 25\)  
\(B^\rightarrow \phi K^{*}\)  \(9.31\pm 1.90\pm 1.83\pm 0.97\)  \(10\pm 2\)  \(1.00\pm 0.27\)  \(1\pm 8\)  
\(\overline{B}^0\rightarrow \phi \overline{K}^{*0}\)  \(8.64\pm 1.76\pm 1.70\pm 0.90\)  \(10\pm 0.5\)  \(1.00\pm 0.27\)  \(0\pm 4\)  
\(\overline{B}_s^0\rightarrow \overline{K}^{*0} K^{*0}\)  \(14.9\pm 2.0\pm 1.9\pm 2.3\)  \(28\pm 7\)  \(0.78\pm 0.19\)  
\(\overline{B}_s^0\rightarrow K^{*} K^{*+}\)  \(15.9\pm 1.7\pm 1.7\pm 2.6\)  \(21.1\pm 7.1\)  
\(\overline{B}_s^0\rightarrow \phi \phi \)  \(26.4\pm 4.8\pm 4.5\pm 3.8\)  \(19.3\pm 3.1\)  \(0.83\pm 0.28\)  
P  \(B^\rightarrow K^{*0} K^{*}\)  \(0.66\pm 0.10\pm 0.13\pm 0.08\)  \(1.2\pm 0.5\)  \(24.8\pm 2.6\)  
\(\overline{B}^0\rightarrow K^{*0}\overline{K}^{*0}\)  \(0.61\pm 0.09\pm 0.12\pm 0.07 \)  \(24.8\pm 2.6\)  
\(\overline{B}_s^0\rightarrow \phi K^{*0}\)  \(0.70\pm 0.11\pm 0.13\pm 0.08\)  \(1.13\pm 0.30\)  \(17.3\pm 5.6\)  
S  \(B^\rightarrow \phi \rho ^{}\)  \(0.06\pm 0.02\pm 0.01\pm 0.01\)  \({<}3.0\)  0  
\(\overline{B}^0\rightarrow \phi \rho ^{0}\)  \(0.03\pm 0.01\pm 0.01\pm 0.00\)  \({<}0.33\)  0  
\(\overline{B}^0\rightarrow \phi \omega \)  \(0.02\pm 0.01\pm 0.00\pm 0.002\)  \({<}0.7\)  0  
\(\overline{B}_s^0\rightarrow \phi \rho ^0\)  \(0.07\pm 0.03\pm 0.01\pm 0.01\)  \({<}617\)  0  
\(\overline{B}_s^0\rightarrow \phi \omega \)  \(3.69\pm 1.19\pm 0.74\pm 0.37\)  \(15.0\pm 7.0\)  
E(\(P_E\))  \(\overline{B}^0\rightarrow K^{*+} K^{*}\)  \(1.43\pm 0.91\pm 0\pm 0.29\)  \({<}2.0\)  0  
\(\overline{B}_s^0\rightarrow \rho ^\rho ^+\)  \(0.10\pm 0.06\pm 0\pm 0.02\)  0  
\(\overline{B}_s^0\rightarrow \rho ^0\rho ^0\)  \(0.05\pm 0.03\pm 0\pm 0.01\)  \({<}320\)  0  
\(\overline{B}_s^0\rightarrow \rho ^0\omega \)  \(0.08\pm 0.05\pm 0\pm 0.01\)  0  
\(\overline{B}_s^0\rightarrow \omega \omega \)  \(0.03\pm 0.02\pm 0\pm 0.01\)  0 
In Table 3, we list the branching fractions, and the theoretical errors correspond to the uncertainties due to variation of: (1) the fitted universal \(\chi \) values, (2) the heavytolight form factors and (3) the uncertainty of decay constants. The error of the variation of the CKM matrix elements is negligible. We note that, for the other observable, we have combined these uncertainties by adding them in quadrature and show the resulting uncertainty, due to the space limitations in the tables. The decay \(B \rightarrow \phi \phi \) is absent in our tables, because it is a pure annihilation type process induced only by the timelike penguin diagram \(P_E\), which contribution is neglected in this work, though its branching fraction is estimated to be of the order of \(10^{8}\) based on PQCD approach [75]. Comparing our predictions with experimental data, one can find that our results can accommodate the data well, within uncertainties from both the theoretical and the experimental sides. For the decays that have not been measured, our prediction can be tested in the ongoing LHCb experiment and the forthcoming BelleII experiment.
The polarization fractions and relative phases of \(\overline{B} \rightarrow VV\) decay modes. The experimental data are taken from [58]
Decay mode  \(f_\mathrm{L}\)/percent  \(f_\perp \)/percent  \(\phi _\parallel \)/rad  \(\phi _\perp \)/rad  

Theory  Exp  Theory  Exp  Theory  Exp  Theory  Exp  
\(B^\rightarrow \rho ^\rho ^0\)  \(95.5\pm 1.1\)  \(95\pm 1.6\)  \(2.22\pm 0.64\)  \(0.09\pm 0.05\)  \(0.09\pm 0.05\)  
\(\overline{B}^0\rightarrow \rho ^\rho ^+\)  \(92.6\pm 1.6\)  \(98.8\pm 2.6\)  \(3.65\pm 0.91\)  \(0.27\pm 0.08\)  \(0.27\pm 0.08\)  
\(B^\rightarrow \rho ^\omega \)  \(92.7\pm 1.4\)  \(90\pm 6\)  \(3.60\pm 0.76\)  \(0.23\pm 0.07\)  \(0.23\pm 0.07\)  
\(\overline{B}_s^0\rightarrow K^{*+}\rho ^\)  \(94.4\pm 1.2\)  \(2.74\pm 0.64\)  \( 0.08\pm 0.03\)  \(0.08\pm 0.03\)  
\(\overline{B}^0\rightarrow \rho ^0\rho ^0\)  \(81.7\pm 10.8\)  \(60\pm 23\)  \(9.21\pm 5.50\)  \(0.04\pm 0.44\)  \(0.03\pm 0.44\)  
\(\overline{B}^0\rightarrow \rho ^0\omega \)  \(82.7\pm 9.5\)  \( 8.68\pm 4.82\)  \(0.98\pm 0.22\)  \(0.98\pm 0.22\)  
\(\overline{B}^0\rightarrow \omega \omega \)  \(92.2\pm 3.6\)  \(3.94\pm 1.85\)  \(1.46\pm 0.26\)  \(1.45\pm 0.26\)  
\(\overline{B}_s^0\rightarrow K^{*0}\rho ^0\)  \(79.8\pm 8.0\)  \(10.2\pm 4.1\)  \(0.94\pm 0.28\)  \(0.94\pm 0.28\)  
\(\overline{B}_s^0\rightarrow K^{*0}\omega \)  \(77.9\pm 9.2\)  \(11.2\pm 4.7\)  \(0.73\pm 0.31\)  \(0.73\pm 0.31\)  
\(B^\rightarrow \rho ^\overline{K}^{*0}\)  \(46.0\pm 12.9\)  \(48\pm 8\)  \(27.2\pm 7.0\)  \(2.07\pm 0.22\)  \(2.08\pm 0.22\)  
\(B^\rightarrow \rho ^0 K^{*}\)  \(40.7\pm 10.6\)  \(78\pm 12\)  \(29.8\pm 5.9\)  \(2.24\pm 0.20\)  \(2.24\pm 0.20\)  
\(\overline{B}^0\rightarrow \rho ^0\overline{K}^{*0}\)  \(48.7\pm 12.3\)  \(40\pm 14\)  \(25.8\pm 6.7\)  \(2.08\pm 0.21\)  \(2.09\pm 0.21\)  
\(\overline{B}^0\rightarrow \rho ^+ K^{*}\)  \(38.9\pm 11.3\)  \(38\pm 13\)  \(30.8\pm 6.3\)  \(2.18\pm 0.22\)  \(2.18\pm 0.22\)  
\(B^\rightarrow \omega K^{*}\)  \(29.9\pm 6.8\)  \(41\pm 19\)  \(35.3\pm 4.5\)  \(0.02\pm 0.85\)  \(0.03\pm 0.85\)  
\(\overline{B}^0\rightarrow \omega \overline{K}^{*0}\)  \(29.4\pm 17.5\)  \(69\pm 13\)  \(35.6\pm 9.4\)  \(2.62\pm 0.53\)  \(2.61\pm 0.53\)  
\(B^\rightarrow \phi K^{*}\)  \(48.0\pm 16.0\)  \(50\pm 5\)  \(25.9\pm 8.6\)  \(20\pm 5\)  \(2.47\pm 0.27\)  \(2.34\pm 0.18\)  \(2.47\pm 0.27\)  \(2.58\pm 0.17\) 
\(\overline{B}^0\rightarrow \phi \overline{K}^{*0}\)  \(48.0\pm 16.0\)  \(49.7\pm 1.7\)  \(26.0\pm 8.6\)  \(22.4\pm 1.5\)  \(2.47\pm 0.27\)  \(2.43\pm 0.11\)  \(2.47\pm 0.27 \)  \(2.53\pm 0.09\) 
\(\overline{B}_s^0\rightarrow \overline{K}^{*0} K^{*0}\)  \(34.3\pm 12.6\)  \(33.2\pm 6.9\)  \(38\pm 11\)  \(2.10\pm 0.23\)  \(2.10\pm 0.23\)  
\(\overline{B}_s^0\rightarrow K^{*} K^{*+}\)  \(30.9\pm 10.4\)  \(34.9\pm 5.8\)  \(2.19\pm 0.22\)  \(2.20\pm 0.22\)  
\(\overline{B}_s^0\rightarrow \phi \phi \)  \(39.7\pm 16.0\)  \(36.2\pm 1.4\)  \(31.2\pm 8.9\)  \(30.9\pm 1.5\)  \(2.53\pm 0.28\)  \(2.55\pm 0.11\)  \(2.56\pm 0.27\)  \(2.67\pm 0.23\) 
\(B^\rightarrow K^{*0} K^{*}\)  \(58.3\pm 11.1\)  \(75\pm 21\)  \(20.8\pm 6.0\)  \(2.10\pm 0.20\)  \(2.09\pm 0.20\)  
\(\overline{B}^0\rightarrow K^{*0}\overline{K}^{*0}\)  \(58.3\pm 11.1\)  \(80\pm 13\)  \(20.8\pm 6.0\)  \(2.10\pm 0.20\)  \(2.09\pm 0.20\)  
\(\overline{B}_s^0\rightarrow \phi K^{*0}\)  \(38.9\pm 14.7\)  \(51\pm 17\)  \(31.4\pm 8.1\)  \(2.52\pm 0.27\)  \(2.55\pm 0.27\) 
Amplitudes of each diagram of \(B\rightarrow \phi K^*\) (\(\times 10^{8} \mathrm {GeV}\))
P  S  \(P_A\)  

\(A^0\)  \(1.36+5.01 i\)  \(1.390.43 i\)  \(0.212.29 i\) 
\(A^\parallel \)  \(0.33+1.22 i\)  \(0+0 i\)  \(0.602.60 i\) 
\(A^\perp \)  \(0.33+1.23 i\)  \(0+0 i\)  \(0.60 2.60 i\) 
For \(B_s\) decays dominated by T and C diagrams, they have the same manner as B decays, for example, the colorallowed decay \(\overline{B}_s^0 \rightarrow \rho ^K^{*+}\) has large branching fraction and large longitudinal fraction, while the branching fraction of colorsuppressed \(\overline{B}_s^0 \rightarrow K^{*0}\rho ^0(\omega )\) decays is a bit smaller and the transverse polarizations are about \(20\%\). Comparing the branching fractions of \(\overline{B}_s^0 \rightarrow \rho ^K^{*+}\) with \(\overline{B}^0\rightarrow \rho ^\rho ^+\), we find that the former is larger than the latter one. The reason is that the form factor \(A_0^{B_s \rightarrow K^*}\) is larger than \(A_0^{B \rightarrow \rho }\) by \(13\%\). Considering the life time difference, the large gap between these two branching fractions can be well understood. Since the order of these three branching fractions is about \(10^{6}\), they should be measurable in the running LHCb experiment.
We now discuss the 14 decay modes dominated by QCDpenguin diagrams P. Among these decays, 11 decays induced by \(b\rightarrow s\) transition have branching fractions up to \(10^{5}\) due to large CKM matrix elements \(V_{tb}V_{ts}^*\). Some of them have been measured precisely in the experiments, including branching fractions, polarization fractions, and even CP asymmetries. The first measured one, and also the bestmeasured channel, is \(B\rightarrow \phi K^*\) modes that are induced by the \(b\rightarrow ss\bar{s}\) transition. In this decay, the magnitudes of QCDpenguin diagram P, the flavorsinglet penguin diagram S and the penguin annihilation diagram \(P_A\) are at the same order magnitude. For illustration, numerical results of each diagram are provided in Table 5. It is easy to see that the penguin annihilation diagram has a very large transverse polarization contribution. In QCDF [38, 40], in order to increase the effects of the annihilation diagrams, the free parameters \(\rho _A\) and \(\phi _A\) introduced for the power suppressed penguin annihilation diagram are required to be very large. In the socalled PQCD approach, the large effects of annihilation are arrived at, by including the transverse momenta of inner quarks. So, we then conclude that the larger transverse polarizations in \(B\rightarrow \phi K^*\) arise from the annihilation diagrams.
Another decay mode induced by \(b\rightarrow ss\bar{s}\) is \(B_s \rightarrow \phi \phi \). From Tables 3 and 4, one can see that although our estimation of branching fractions agree with data within uncertainties, our center value is a bit larger than the experimental data, and the predicted polarization fractions are in agreement with the data. The acceptable divergency in the branching fraction is due to the larger form factor \(A_0^{B_s \rightarrow \phi }\) we adopted, which in fact is also related to the branching ratio of \(B_s \rightarrow \phi K^*\). If we consider \(B_s \rightarrow \phi \phi \) alone, the present data favor a smaller \(A_0^{B_s \rightarrow \phi }\). On this point, the precise calculations of the form factor in lattice QCD and other effective approach are needed.
The predictions of the CP asymmetries of longitudinal and perpendicular polarizations, as well as relative phase differences of \(B\rightarrow VV\) decays
Decay mode  \(A_\mathrm{CP}^0\)/percent  \(A_\mathrm{CP}^\perp \)/percent  \(\Delta \phi _\parallel \)/rad  \(\Delta \phi _\perp \)/rad 

\(B^\rightarrow \rho ^\rho ^0\)  0  0  0  0 
\(\overline{B}^0\rightarrow \rho ^\rho ^+\)  \(1.30\pm 0.54\)  \(16.3\pm 8.2\)  \(0.41\pm 0.05\)  \(0.41\pm 0.05\) 
\(B^\rightarrow \rho ^\omega \)  \(2.38\pm 0.86\)  \(30.2\pm 11.6\)  \(0.70\pm 0.07\)  \(0.72\pm 0.07\) 
\(\overline{B}_s^0\rightarrow K^{*+}\rho ^\)  \(0.91\pm 0.45\)  \(15.4\pm 9.5\)  \(0.52\pm 0.06\)  \(0.54\pm 0.06\) 
\(\overline{B}^0\rightarrow \rho ^0\rho ^0\)  \(10.5\pm 9.6\)  \(46.9\pm 13.9\)  \(1.89\pm 0.19\)  \(1.89\pm 0.19\) 
\(\overline{B}^0\rightarrow \rho ^0\omega \)  \(8.68\pm 7.72\)  \(41.6\pm 13.3\)  \(1.47\pm 0.09\)  \(1.47\pm 0.09\) 
\(\overline{B}^0\rightarrow \omega \omega \)  \(2.10\pm 1.87\)  \(24.7\pm 14.1\)  \(1.43\pm 0.07\)  \(1.43\pm 0.07\) 
\(\overline{B}_s^0\rightarrow K^{*0}\rho ^0\)  \(0.47\pm 4.69\)  \(1.89\pm 18.3\)  \(2.03\pm 0.10\)  \(2.02\pm 0.10\) 
\(\overline{B}_s^0\rightarrow K^{*0}\omega \)  \(8.37\pm 7.28\)  \(29.5\pm 16.3\)  \(1.58\pm 0.13\)  \(1.58\pm 0.13\) 
\(B^\rightarrow \rho ^\overline{K}^{*0}\)  \(1.40\pm 0.56\)  \( 1.19\pm 0.19\)  \(0.01\pm 0.00\)  \(0.01\pm 0.00\) 
\(B^\rightarrow \rho ^0 K^{*}\)  \(35.0\pm 19.8\)  \(24.2\pm 9.0\)  \(0.82\pm 0.15\)  \(0.82\pm 0.15\) 
\(\overline{B}^0\rightarrow \rho ^0\overline{K}^{*0}\)  \(0.41\pm 4.3\)  \(0.39\pm 4.06\)  \(0.13\pm 0.04\)  \(0.13\pm 0.04\) 
\(\overline{B}^0\rightarrow \rho ^+ K^{*}\)  \(37.2\pm 18.9\)  \(23.8\pm 6.9\)  \(0.66\pm 0.12\)  \(0.65\pm 0.12\) 
\(B^\rightarrow \omega K^{*}\)  \(93.4\pm 25.9\)  \(39.6\pm 13.0\)  \(2.09\pm 0.90\)  \(2.09\pm 0.90\) 
\(\overline{B}^0\rightarrow \omega \overline{K}^{*0}\)  \(27.7\pm 19.4\)  \(11.5\pm 4.0\)  \(0.04\pm 0.12\)  \(0.04\pm 0.12\) 
\(B^\rightarrow \phi K^{*}\)  \(1.26\pm 0.71\)  \(1.16\pm 0.30\)  \(0.02\pm 0.00\)  \(0.02\pm 0.00\) 
\(\overline{B}^0\rightarrow \phi \overline{K}^{*0}\)  \(1.26\pm 0.71\)  \(1.16\pm 0.30\)  \(0.02\pm 0.00\)  \(0.02\pm 0.00\) 
\(\overline{B}_s^0\rightarrow \overline{K}^{*0} K^{*0}\)  \(1.81\pm 0.69\)  \(0.94\pm 0.20\)  \(0.005\pm 0.003\)  \(0.004\pm 0.003\) 
\(\overline{B}_s^0\rightarrow K^{*} K^{*+}\)  \(32.6\pm 20.2\)  \(14.8\pm 7.3\)  \(0.70\pm 0.14\)  \(0.70\pm 0.14\) 
\(\overline{B}_s^0\rightarrow \phi \phi \)  \(1.55\pm 0.85\)  \(1.02\pm 0.29\)  \(0.01\pm 0.00\)  \(0.01\pm 0.00\) 
\(B^\rightarrow K^{*0} K^{*}\)  \(20.2\pm 8.0\)  \(28.3\pm 3.3\)  \(0.16\pm 0.04\)  \(0.16\pm 0.04\) 
\(\overline{B}^0\rightarrow K^{*0}\overline{K}^{*0}\)  \(20.2\pm 8.0\)  \(28.3\pm 3.3\)  \(0.16\pm 0.04\)  \(0.16\pm 0.04\) 
\(\overline{B}_s^0\rightarrow \phi K^{*0}\)  \(32.9\pm 15.0\)  \(21.0\pm 5.7\)  \(0.27\pm 0.08\)  \(0.27\pm 0.08\) 
Prediction of the timedependent CP violation (\(\%\))
Decay mode  \(S_f\)  \(C_f\)  

Theory  Exp  Theory  Exp  
\( \bar{B}^{0} \rightarrow (\rho ^{} \rho ^{+})_L\)  \(3.67\pm 3.02\)  \(6\pm 17\)  \(6.80\pm 3.12\)  \(5\pm 13\) 
\( \bar{B}^{0} \rightarrow (\rho ^{0} \rho ^{0})_L\)  \(41.7\pm 27.8\)  \(30\pm 70\)  \(57.2\pm 17.4\)  \(20\pm 90\) 
\( \bar{B}^{0} \rightarrow (\omega \omega )_L\)  \(15.7\pm 13.3\)  \(30.0\pm 15.1\)  
\( \bar{B}^{0}_s \rightarrow (K^{*} K^{*+})_L\)  \(80.9\pm 14.4\)  \(50.2\pm 20.7\)  
\( \bar{B}^{0}_s \rightarrow (\phi \phi )_L\)  \(2.16\pm 0.76\)  \(2.39\pm 0.82\) 
In Table 4, as we expected, the longitudinal polarization fractions of the tree diagram dominant decays are predicted to be near unity with errors in the (5–\(10)\%\) range. The CP asymmetries in the longitudinal polarizations of these decays are less than \(5\%\), as shown in Table 6. Although the CP asymmetries of the perpendicular polarizations of these decays shown in Table 6 are large, they are difficult to measure, since their fractions are too small. For the decays controlled by the C diagram, although the longitudinal polarization factions become smaller, they still play primary roles with large uncertainties. Furthermore, the uncertainties of \(A^0_\mathrm{CP}\) are large, some of them can reach \(10\%\). Our theoretical result of \(f_\mathrm{L}\) for \(B^\rightarrow \rho ^\rho ^0\) is a bit larger than the data, but the situation of \(\overline{B} ^0 \rightarrow \rho ^+\rho ^\) is in the opposite direction.
Because each \(B \rightarrow VV\) decay has three polarizations, the possible timedependent CP violation is complicated, it is very hard to measure them precisely in the experiments. For the treedominant decays, the transverse parts can be neglected, and the measurement of the timedependence CP violation of these decays becomes plausible. In this work, we calculated \(C_{f}\) and \(S_{f}\) of the longitudinal parts of \(\overline{B}^0 \rightarrow \rho ^+\rho ^\) and \(\overline{B}^0 \rightarrow \rho ^0\rho ^0\), as shown in Table 7. Obviously, our results agree with experiment, though there are large uncertainties on both the theoretical and the experimental sides. Also, we present the numerical results of \( \bar{B}^{0}_s \rightarrow (K^{*} K^{*+})_L\) and \( \bar{B}^{0}_s \rightarrow (\phi \phi )_L\), which may be measured in the future, as both modes have large branching fractions and the final states are easy to identify. Note that the precise measurement of \(C_{\rho \rho }^L\) and \(S_{\rho \rho }^L\) will help us to determine the CKM angles \(\alpha \) and \(\gamma \) [79].
We then come to the decay modes \(B \rightarrow \rho K^*\) and \(B_s \rightarrow K^* K^*\). For the purepenguin process \(B^ \rightarrow \rho ^ \overline{K}^{*0}\), which is similar to the decays \(B \rightarrow \phi K^*\), the penguin annihilation diagram will give a large transverse polarization fraction. For \(B^ \rightarrow \rho ^0 K^{*}\), to which the tree operators also contribute, the destructive interference between tree and penguin operators reduce the longitudinal amplitude. So, the smaller longitudinal polarization fraction of \(B^ \rightarrow \rho ^0 K^{*}\) is obtained. The large longitudinal polarization fraction \(f_\mathrm{L}\) of this decay is only measured by the BABAR experiment. We hope the Belle or Belle II experiment can help to resolve this puzzle. Due to the factor \(1/\sqrt{2}\), the branching fraction of \(B^ \rightarrow \rho ^0 K^{*}\) is about half of that of \(B^ \rightarrow \rho ^ \overline{K}^{*0}\). The analysis and the result of the modes \(B \rightarrow \omega K^*\) and \(B_s \rightarrow K^* K^*\) should be similar to those of \(B \rightarrow \rho K^*\). From Tables 4 and 6, we find that for all penguin dominant decays \(f_\parallel \approx f_\perp \), \(\phi _\parallel \approx \phi _\perp \) and \(\Delta \phi _\parallel \approx \Delta \phi _\perp \), which indicates that the positivehelicity amplitudes are about zero. In fact, due to the suppression of leading QCDpenguins, the \(\rho K^*\) final states have also been used to probe the electroweak penguin effect [38]; however, this kind of contributions has been neglected in the present work due to insufficient experimental data.
As for the last five pure annihilation type decays, they all have two kinds of contributions: the W exchange diagram (E) and the timelike penguin annihilation diagram (\(P_E\)). As discussed in previous section, there are not enough experimental data to determine the amplitude of timelike penguin annihilation diagram (\(P_E\)). In our fitting, we have to set it to 0. Since all \(B_s\) decays in this category are dominated by this \(P_E\) contribution except \(\overline{B}_s^0 \rightarrow \rho ^0\omega \), due to the small CKM matrix elements in W exchange diagram (E), the branching fractions of these decay modes are not stable in Table 3 with large uncertainties. Only \(\overline{B}^0\rightarrow K^{*+} K^{*}\) decay has a relatively larger CKM matrix elements in the W exchange diagram (E), which makes a relatively larger branching fractions, but still with large uncertainty. And the pure W exchange diagram (E) channel \(\overline{B}_s^0 \rightarrow \rho ^0\omega \) also has a large uncertainty because of the large error of \(\chi _E^0\). Therefore, we conclude that the current experimental data cannot help us to make predictions on this kind of decays, but we need to wait for the running LHCb experiment, BelleII or other future colliders.
4 Summary
In the work, we preformed analysis of 33 charmless twobody \(B_{(s)} \rightarrow VV\) decays within the factorizationassisted topologicalamplitude approach. In contrast to the charmless \(B\rightarrow PP\) and \(B\rightarrow PV\) decays, more parameters (triple number in principle) are needed to describe the three polarization amplitudes of \(B_{(s)} \rightarrow VV\) decays. However, with the current 46 experimental data, we can only fit 10 universal parameters of them. For the decays with large transverse polarization fractions, such as the penguin diagram contribution dominated decays, we need only one transverse polarization amplitude in penguin annihilation diagram to explain all the polarization data. We calculated many decay modes not yet measured, involving the branching fractions, the polarization fractions, CP violation parameters, as well as relative strong phases. These results will be tested in the LHCb experiment and future BelleII experiment.
Notes
Acknowledgements
We are grateful to W. Wang, F.S. Yu, SH Zhou, Y.B. Wei, J.B. Liu, X.D. Gao and Q. Qin for helpful discussions. The work is partly supported by National Natural Science Foundation of China (11575151, 11375208, 11521505, 11621131001 and 11235005) and the Program for New Century Excellent Talents in University (NCET) by Ministry of Education of P. R. China (NCET130991). Y. Li and C.D. Lü are also supported by the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No.Y5KF111CJ1). Y.Li is also support by the Natural Science Foundation of Shandong Province (Grant No. ZR2016JL001).
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