1 Introduction

According to the traditional quark model, the P-wave triplet states of the quark-antiquark system have the quantum number \(J^{P}\) \(=\) \(0^{+}\), and are called the scalar mesons. The scalar mesons mostly appear as the hadronic resonances, and have large decay widths. There will exist several resonances and decay channels within a short mass interval. The overlaps between resonances and background make it considerably difficult to resolve the scalar mesons. In addition, the di-boson combinations can also have the quantum number \(J^{P}\) \(=\) \(0^{+}\). In contrast to the ground pseudoscalar and vector mesons, the identification of the scalar is long-standing puzzle. To understand the internal structure of the scalar mesons is one of the most interesting topics in hadron physics. Generally, the scalar mesons have been identified as the ordinary quark-antiquark \(q\bar{q}\) states, tetraquark \(q\bar{q}q\bar{q}\) states, meson-meson molecular states or even those supplemented with a scalar glueball. There are many candidates with \(J^{PC}\) \(=\) \(0^{++}\) below 2 GeV, which cannot be accommodated in one SU(3) flavor nonet satisfactorily. From the mass spectrum of those scalar mesons and their chromatic as well as electromagnetic decays, a prospective picture (scenario 2, hereafter this text will be abbreviated as S2) suggests that the isovector \(a_{0}(1450)\), isodoublet \(K^{*}_{0}(1430)\), isoscalar \(f_{0}(1710)\) and \(f_{0}(1370)\) above 1 GeV can be assigned to be a conventional SU(3) \(q\bar{q}\) scalar nonet with the spectroscopy symbol of \(1^{3}P_{0}\) [1], while the scalar mesons \(a_{0}(980)\), \(K_{0}^{*}(700)\) (or \({\kappa }\)), \(f_{0}(980)\) and \(f_{0}(500)\) (or \({\sigma }\)) below 1 GeV form the unconventional \(q\bar{q}q\bar{q}\) exotic nonet [1,2,3]. Of course, the above assignments is tentative. In alternative schemes, the scalar mesons with mass below 1 GeV are interpreted as the lowest lying \(q\bar{q}\) states, while the scalars \(a_{0}(1450)\), \(K^{*}_{0}(1430)\), \(f_{0}(1710)\) and \(f_{0}(1370)\) are regarded as the radial excited states with the spectroscopy symbol of \(2^{3}P_{0}\) (scenario 1, namely S1).

It is widely known that the B mesons have rich decay modes. The light scalar mesons can be produced in the B meson decays. The B meson hadronic decays involving the final scalar mesons provides another efficient way to investigate the features and the possible inner structures of the scalar mesons. Experimentally, some of the B \({\rightarrow }\) SP, SV, SS, SX decays (where the symbols of S, P, V and X denote the light scalar mesons, pseudoscalar mesons, vector mesons and other particles, respectively), such as the B \({\rightarrow }\) \(K^{*}_{0}(1430)^{+}{\pi }^{-}\), \(K^{*}_{0}(1430)^{+}{\omega }\), \(K^{*}_{0}(1430)^{0}\overline{K}^{*}_{0}(1430)^{0}\), \(K^{*}_{0}(1430)^{0}{\pi }^{+}{\gamma }\) decays, have been measured by Belle, BaBar and LHCb groups [1]. With the running of high-luminosity Belle-II and LHCb experiments and the coming CEPC, FCC-ee and HL-LHC experiments, more and more data on the B meson decays will be available in the future, more and more B \({\rightarrow }\) SP, SV, SS, SX decays can be discovered and investigated, and the measurement precision will be higher and higher, which lays a solid experimental foundation to carefully study the scalar mesons and distinguish theoretical models. Phenomenologically, many of the B \({\rightarrow }\) SP, SV, SS, SX decays have been studied extensively with various theoretical models. For example, the study of the B \({\rightarrow }\) SP, SV decays with the QCD factorization (QCDF) approach [4,5,6,7,8,9,10,11,12,13,14,15], the B \({\rightarrow }\) SP, SV, SS decays with the perturbative QCD (PQCD) approach [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], the semileptonic B \({\rightarrow }\) SX decays with the sum rules and other approaches [39,40,41,42,43,44,45,46,47], and so on. It is easy to imagine that various scenarios, such as S1 and S2, will inevitably give different theoretical predictions on the B meson decays. It is noteworthy that some studies have shown that branching ratios for the charmless B \({\rightarrow }\) SS decays with the PQCD approach can be very large, for example, \({\mathcal {B}}(B_{s}{\rightarrow }K^{*}_{0}(1430)K^{*}_{0}(1430))\) \({\sim }\) \({\mathcal {O}}(10^{-4})\) [33], \({\mathcal {B}}(B{\rightarrow }K^{*}_{0}(1430){\sigma })\) \({\sim }\) \({\mathcal {O}}(10^{-4})\) [35], \({\mathcal {B}}(B_{s}{\rightarrow }{\sigma }{\sigma })\) \({\sim }\) \({\mathcal {O}}(10^{-4})\), \({\mathcal {B}}(B_{s}{\rightarrow }{\sigma }f_{0}(980))\) \({\sim }\) \({\mathcal {O}}(10^{-4})\), \({\mathcal {B}}(B_{s}{\rightarrow }f_{0}(980)f_{0}(980))\) \({\sim }\) \({\mathcal {O}}(10^{-4})\), \({\mathcal {B}}(B{\rightarrow }{\sigma }{\sigma })\) \({\sim }\) \({\mathcal {O}}(10^{-5})\) [38], \({\mathcal {B}}(B{\rightarrow }a_{0}(980)a_{0}(980))\) \({\sim }\) \({\mathcal {O}}(10^{-5})\) [33]. And the more striking phenomena are that branching ratios for the pure annihilation B \({\rightarrow }\) SS decays, which might be very small by intuition, are very large with the PQCD approach, for example, \({\mathcal {B}}(B_{s}{\rightarrow }a_{0}(980)a_{0}(980))\) \({\sim }\) \({\mathcal {O}}(10^{-5})\), \({\mathcal {B}}(B_{s}{\rightarrow }a_{0}(1450)a_{0}(1450))\) \({\sim }\) \({\mathcal {O}}(10^{-5})\), \({\mathcal {B}}(B_{d}{\rightarrow }{\kappa }^{+}{\kappa }^{-})\) \({\sim }\) \({\mathcal {O}}(10^{-6})\), \({\mathcal {B}}(B_{d}{\rightarrow }K^{*}_{0}(1430)^{+}K^{*}_{0}(1430)^{-})\) \({\sim }\) \({\mathcal {O}}(10^{-6})\) [37]. So the study of the B \({\rightarrow }\) SS decays is very promising and tempting both theoretically and experimentally. In order to deepen our understanding on the properties of the light scalar mesons and provide the ongoing and coming experimental analysis with additional theoretical references, in this paper, we will study the nonleptonic charmless B \({\rightarrow }\) SS decays with the QCDF approach, by considering scenarios S1 and S2 for the scalar mesons, where S \(=\) \(K_{0}^{*}(1430)\) and \(a_{0}(1450)\).

This paper is organized as follows. In Sect. 2, the theoretical framework are briefly reviewed, the next-to-leading order effective coefficients for the B \({\rightarrow }\) SS decays and the weak annihilation amplitudes are given with the QCDF approach. In Sect. 3, the values of the nonperturbative input parameters are fixed. The numerical results and our comments are presented in Sect. 4. Finally, we conclude with a summary in Sect. 5. The decay amplitudes are displayed in the Appendix.

2 Theoretical framework

2.1 The effective Hamiltonian

The low-energy effective Hamiltonian for the charmless nonleptonic B \({\rightarrow }\) SS decays is written as [48],

$$\begin{aligned} {\mathcal {H}}_{\textrm{eff}}= & {} \frac{G_{F}}{\sqrt{2}} \sum \limits _{q=d,s} \left\{ V_{ub} V_{uq}^{*} \Big [ C_{1}({\mu })O_{1}({\mu })+ C_{2}({\mu })O_{2}({\mu }) \Big ] \right. \nonumber \\{} & {} - V_{tb} V_{tq}^{*} \left[ \sum \limits _{i=3}^{10} C_{i}({\mu })O_{i}({\mu }) + C_{7{\gamma }} ({\mu })O_{7{\gamma }}({\mu })\right. \nonumber \\{} & {} \left. \left. + C_{8g}({\mu }) O_{8g}({\mu }) \right] \right\} + \mathrm{h.c.}, \end{aligned}$$
(1)

where the Fermi constant \(G_{F}\) and the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements \(V_{ij}\) have been well determined experimentally [1]. The Wilson coefficients \(C_{i}\), which summarize the short-distance physical contributions, are in principle computable with the perturbative theory order by order at the scale of \({\mu }\) \(=\) \(m_{W}\), and can be evaluated to the energy scale of the B meson decays \({\mu }\) \({\sim }\) \(m_{b}\) with the renormalization group equation (RGE) [48], where \(m_{W}\) and \(m_{b}\) are the mass of the gauge boson W of the weak interactions and the heavy b quark, respectively. The remaining theoretical work is to calculate the hadronic matrix elements (HMEs), \({\langle }S_{1}S_{2}{\vert }O_{i}{\vert }B{\rangle }\), where the local four-quark effective operators \(O_{i}\) are sandwiched between the initial B meson and the final scalar mesons.

In order to generate the essential strong phases for CP violations in the hadronic B meson decays, and cancel the unphysical \({\mu }\)-dependence of decay amplitude \({\mathcal {A}}\) \(=\) \({\langle }S_{1}S_{2}{\vert }{\mathcal {H}}_{\textrm{eff}}{\vert }B{\rangle }\) originating from the Wilson coefficients, the high order radiative corrections to HMEs are necessary and should be taken into consideration. However, the perturbative contributions embedded in HMEs became entangled with the nonperturbative contributions, which makes the theoretical calculations extremely complicated. How to properly and reasonably evaluate HMEs of the hadronic B meson decays has been an academic focus.

2.2 The QCDF decay amplitudes

The QCDF approach [49,50,51,52,53,54] is one of many QCD-inspired phenomenological remedies to deal with HMEs. Based on the power counting rules in the heavy quark limit and an expansion series in the strong coupling \({\alpha }_{s}\) assisted by the collinear approximation, the long- and short-distance contributions are factorized. The nonperturbative contributions in HMEs are either power suppressed by \(1/m_{b}\) or incorporated into the hadronic transition form factors and mesonic distribution amplitudes (DAs). Up to the leading power corrections of order \(1/m_{b}\), the QCDF factorization formula for HMEs concerned is written as [50],

$$\begin{aligned}&{\langle }S_{1}S_{2}{\vert }O_{i}({\mu }){\vert }B{\rangle }\nonumber \\&\quad = \sum \limits _{j} F_{j}^{B{\rightarrow }S_{1}}\, f_{S_{2}} {\int }dy\,{\mathcal {T}}_{ij}^{I}(x)\,\phi _{S_2}(x) + (S_{1}{\leftrightarrow }S_{2}) + f_{B}\,f_{S_{1}}\,f_{S_{2}}\nonumber \\&\qquad \times {\int } dx \,dy \,dz\, {\mathcal {T}}_{i}^{II}(x,y,z)\,{\phi }_{S_{1}}(y)\, {\phi }_{S_{2}}(x)\, {\phi }_{B}(z), \end{aligned}$$
(2)

where x, y and z are the longitudinal momentum fractions of the valence quarks. The form factors \(F_{j}^{B{\rightarrow }S}\), the decay constants \(f_{B}\) and \(f_{S}\), the mesonic light cone DAs \({\phi }_{B}\) and \({\phi }_{S}\), all of them are the nonperturbative parameters. These parameters are regarded to be universal and process-independent, and can be obtained from the experimental data, lattice QCD simulation, QCD sum rules, or by comparison with other exclusive processes. \({\mathcal {T}}^{I}\) and \({\mathcal {T}}^{II}\) are the hard-scattering functions describing the local interactions among quarks and gluons at the B meson decay scale. They are, in principle, perturbatively calculable to all orders in \({\alpha }_{s}\) at the leading power order of \(1/m_{b}\). At the leading order (LO) \({\alpha }_{s}^{0}\), \({\mathcal {T}}^{I}\) \(=\) 1 and \({\mathcal {T}}^{II}\) \(=\) 0. The convolution integrals of \({\mathcal {T}}^{I}\) and \({\phi }_{S}\) result in the decay constant of the emission scalar mesons. One can return from the QCDF formula Eq. (2) to the naive factorization (NF) approximations [55, 56], i.e., the four-quark HMEs can be written as the product of two diquark HMEs, and the diquark HMEs can be replaced by HMEs of the corresponding hadronic currents and then further parameterized by hadronic transition form factors and decay constants. Beyond the order \({\alpha }_{s}^{0}\), the radiative corrections to HMEs make \({\mathcal {T}}^{I,II}\) no longer trivial, and some information about the CP-violating strong phases and \({\mu }\)-dependence of HMEs can be retrieved naturally.

With the QCDF approach, the amplitudes for the concerned B \({\rightarrow }\) SS decays can be generally written as,

$$\begin{aligned} {\mathcal {A}} = {\langle }S_{1}S_{2}{\vert }{\mathcal {H}}_{\textrm{eff}} {\vert }B{\rangle } =\frac{G_{F}}{\sqrt{2}} \sum \limits _{i} {\lambda }_{i}\sum \limits _{j=1}^{10} a_{j} {\langle }S_{1}S_{2}{\vert }O_{j}{\vert }B{\rangle }_{\textrm{NF}},\nonumber \\ \end{aligned}$$
(3)

where the parameter \({\lambda }_{i}\) is the product of the CKM elements; the coefficient \(a_{j}\) including the nonfactorizable contributions beyond the leading order of \({\alpha }_{s}\) is the combinations of the Wilson coefficients; HMEs \({\langle }S_{1}S_{2}{\vert }O_{j}{\vert }B{\rangle }_{\textrm{NF}}\) are defined and evaluated with the NF approximation.

2.3 The QCDF coefficients

To simplify the decay amplitude expressions, we will use the notations in Ref. [57] and write the QCDF coefficients as follows.

$$\begin{aligned} {\alpha }_{1}(S_{1}\,S_{2})= & {} a_{1}(S_{1}\,S_{2}), \end{aligned}$$
(4)
$$\begin{aligned} {\alpha }_{2}(S_{1}\,S_{2})= & {} a_{2}(S_{1}\,S_{2}), \end{aligned}$$
(5)
$$\begin{aligned} {\alpha }_{3}^{p}(S_{1}\,S_{2})= & {} a_{3}^{p}(S_{1}\,S_{2})+ a_{5}^{p}(S_{1}\,S_{2}), \end{aligned}$$
(6)
$$\begin{aligned} {\alpha }_{4}^{p}(S_{1}\,S_{2})= & {} a_{4}^{p}(S_{1}\,S_{2}) + \bar{\gamma }_{\chi }^{S_{2}}\, a_{6}^{p}(S_{1}\,S_{2}), \end{aligned}$$
(7)
$$\begin{aligned} {\alpha }_{3,EW}^{p}(S_{1}\,S_{2})= & {} a_{9}^{p}(S_{1}\,S_{2}) + a_{7}^{p}(S_{1}\,S_{2}), \end{aligned}$$
(8)
$$\begin{aligned} {\alpha }_{4,EW}^{p}(S_{1}\,S_{2})= & {} a_{10}^{p}(S_{1}\,S_{2}) + \bar{\gamma }_{\chi }^{S_{2}}\, a_{8}^{p}(S_{1}\,S_{2}), \end{aligned}$$
(9)

where \(S_{1}\) denotes the recoiled scalar meson which absorbs the light spectator quark of the initial B mesons, and \(S_{2}\) denotes the emitted scalar meson. The ratio \(\bar{\gamma }_{\chi }^{S}\) is defined as

$$\begin{aligned} \bar{\gamma }_{\chi }^{S}({\mu })&= {\gamma }_{\chi }^{S}({\mu })\, \bar{\mu }_{S}^{-1}({\mu }) = \frac{2\,m_{S}}{\overline{m}_{b}({\mu })}, \end{aligned}$$
(10)
$$\begin{aligned} {\gamma }_{\chi }^{S}({\mu })&= \frac{2\,m_{S}^{2}}{\overline{m}_{b}({\mu }) \,[\overline{m}_{1}({\mu })-\overline{m}_{2}({\mu })]}, \end{aligned}$$
(11)
$$\begin{aligned} \bar{\mu }_{S}({\mu })&=\frac{m_{S}}{\overline{m}_{1}({\mu }) -\overline{m}_{2}({\mu })}, \end{aligned}$$
(12)

where \(m_{S}\) is the mass of the emission scalar meson, and the \({\mu }\)-dependent \(\overline{m}_{i}\) is the \(\overline{\textrm{MS}}\) running quark mass and can be evaluated with RGE. \(\overline{m}_{1}\) and \(\overline{m}_{2}\) correspond to the two valence quarks in a scalar meson.

Up to the next-to-leading order (NLO) in the coupling \({\alpha }_{s}\), the general form of the QCDF coefficients \(a_{i}^{p}\) is expressed as,

$$\begin{aligned} a_{i}^{p}(S_{1}\,S_{2})= & {} \Big ( C_{i} + \frac{ C_{i{\pm }1} }{N_{c}} \Big )\, N_{i}(S_{2}) + P_{i}^{p}(S_{2}) \nonumber \\{} & {} + \frac{ C_{i{\pm }1} }{N_{c}}\, \frac{ C_{F}\,{\alpha }_{s} }{4{\pi }} \Big [ V_{i}(S_{2}) +\frac{ 4{\pi }^{2} }{N_{c}} \, H_{i}(S_{1}\,S_{2}) \Big ],\nonumber \\ \end{aligned}$$
(13)

where the superscript p is to be omitted for i \(=\) 1 and 2, and the upper (lower) signs apply when i is odd (even). \(C_{i}\) is the Wilson coefficients, the color factor \(C_{F}\) \(=\) \((N_{c}^{2}-1)/(2\,N_{c})\) and the color number \(N_{c}\) \(=\) 3. Due to the relations between the scalar and vector decay constants for the scalar meson (see Eq. (51)), the factor \(N_{i}(S_{2})\) is

$$\begin{aligned} N_{i}(S_{2}) = \bigg \{\begin{array}{lcl} 1 &{} ~~ &{} \text {for}~~ i\, =\,6,8; \\ \bar{\mu }_{S}^{-1} &{} &{} \text {others}. \end{array} \end{aligned}$$
(14)

In Eq. (13), the terms proportional to \(N_{i}(S_{2})\) are the LO contributions. It is obvious that except for the coefficients of \(a_{6,8}\), the LO contributions are proportional to the mass difference \({\Delta }\overline{m}=\overline{m}_{1}- \overline{m}_{2}\). For the scalar mesons consisting of the light quarks, a common sense is that the mass difference \({\Delta }\overline{m}\) is usually very small. So, it is easy to picture that the LO contributions are suppressed by the chiral factors, and that the NLO contributions would be necessary and important for the \(B {\rightarrow } SS\) decays. The terms proportional to \({\alpha }_{s}\) are the NLO contributions, including the vertex corrections \(V_{i}(S_{2})\), penguin contributions \(P_{i}^{p}(S_{2})\), and hard spectator scattering amplitudes \(H_{i}(S_{1}\,S_{2})\). When the emission \(S_{2}\) meson can be decoupled from the \(B-S_{1}\) system, corresponding to the first line in Eq. (2), \(V_{i}(S_{2})\) and \(P_{i}^{p}(S_{2})\) are written as the convolution integrals of hard scattering kernels \(T^{I}(x)\) and mesonic DAs \({\phi }_{S_{2}}(x)\). When the initial B meson is entangled with the final states by the hard spectator scattering interactions, \(H_{i}(S_{1}\,S_{2})\) are written as the convolution integrals of hard scattering kernels \(T^{II}\) and all participating mesonic DAs, corresponding to the second line in Eq. (2). For the \(B {\rightarrow } SS\) decays, the explicit expressions of \(V_{i}(S_{2})\), \(P_{i}^{p}(S_{2})\) and \(H_{i}(S_{1}\,S_{2})\) have been shown in our previous paper [15] by using the replacements of the Gegenbauer moments \(a_{i}^{M_{j}} \rightarrow b_{i}^{S_{j}}\), the chiral factor \({\gamma }_{\chi }^{M_{i}} \rightarrow \bar{\gamma }_{\chi }^{S_{i}}\), and DAs \({\phi }_{M_{i}} \rightarrow {\phi }_{S_{i}}\). For example, by integrating out the momentum fraction, \(H_{i}(S_{1}\,S_{2})\) can be expressed as the functions of the Gegenbauer moments embedded in the mesonic DAs.

$$\begin{aligned}&H_{i}(S_{1}\,S_{2}) = \left\{ \begin{array}{lcl} 0, &{} ~ &{} \text {for}~ i\, =\, 6,8; \\ \displaystyle -\frac{ B_{S_{1}\,S_{2}} }{ A_{S_{1}\,S_{2}} } \frac{ m_{B} }{ {\lambda }_{B} } \Bigg [ 9 \sum \limits _{m=0}^{3} b_{m}^{S_{1}} \sum \limits _{j=0}^{3}(-1)^{j} b_{j}^{S_{2}} -3\,\bar{\gamma }_{\chi }^{S_{1}} X_{H} \sum \limits _{k=0}^{3} b_{k}^{S_{2}} \Bigg ], &{} ~ &{} \text {for}~ i\, =\, 5,7; \\ \displaystyle ~\frac{ B_{S_{1}\,S_{2}} }{ A_{S_{1}\,S_{2}} } \frac{ m_{B} }{ {\lambda }_{B} } \Bigg [ 9 \sum \limits _{m=0}^{3} b_{m}^{S_{1}} \sum \limits _{j=0}^{3} b_{j}^{S_{2}} -3\,\bar{\gamma }_{\chi }^{S_{1}} X_{H} \sum \limits _{k=0}^{3}(-1)^{k} b_{k}^{S_{2}} \Bigg ], &{} ~ &{} \text {others} \end{array} \right. \end{aligned}$$
(15)

with the common factors are

$$\begin{aligned} A_{S_{1}\,S_{2}}&= i\, \frac{ G_{F} }{\sqrt{2}} \, U_{0}^{B\,S_{1}}(m_{S_{2}}^2)\, \bar{f}_{S_{2}}\, ( m_{B}^{2}-m_{S_{1}}^{2}), \end{aligned}$$
(16)
$$\begin{aligned} B_{S_{1}\,S_{2}}&= i\, \frac{ G_{F} }{\sqrt{2}}\, f_{B} \, \bar{f}_{S_{1}}\, \bar{f}_{S_{2}}, \end{aligned}$$
(17)
$$\begin{aligned} \frac{ m_{B} }{ {\lambda }_{B} }&= {\int }_{0}^{1}\,dz\, \frac{ {\Phi }_{B}(z) }{ z }, \end{aligned}$$
(18)
$$\begin{aligned} X_{H}&= {\int }_{0}^{1}\, \frac{ dxy}{ 1-y } \rightarrow {\ln }\Big ( \frac{ m_{B} }{ {\Lambda }_{h} } \Big ) \, (1+{\rho }_{H}\, e^{i\,{\phi }_{H}}), \end{aligned}$$
(19)

where \(U_{0}^{B\,S_{1}}\) is the form factors, \(f_{B}\) is the decay constant for the B meson, \(\bar{f}_{S_{i}}\) is the scalar decay constant for the scalar mesons, the quantity \({\lambda }_{B}\) is used to parameterize our ignorance about the B mesonic DAs, and the phenomenological parameter \(X_{H}\) is introduced to regularize the end point singularities.

In addition, according to many practical application of the QCDF approach in the two-body hadronic B decays, such as Refs. [4,5,6,7,8,9,10,11,12,13,14,15, 50, 51, 57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72], it was shown that the weak annihilation (WA) contributions are important and worth of consideration, although they are formally power suppressed relative to the LO contributions based on the QCDF power counting rules in the heavy quark limits. The weak annihilation diagrams are shown by Fig. 1. The QCDF coefficients of the WA amplitudes for the B \({\rightarrow }\) SS decays have the same expression as those in Eq. (55) of Ref. [57], i.e.,

$$\begin{aligned} {\beta }_{i}^{p}&= -\frac{ B_{S_{1}\,S_{2}} }{ A_{S_{1}\,S_{2}} } \, b_{i}^{p}, \end{aligned}$$
(20)
$$\begin{aligned} b_{1}&= \frac{ C_{F} }{ N_{c}^{2} }\,C_{1}\,A_{1}^{i}, \quad \, b_{2}\, = \, \frac{ C_{F} }{ N_{c}^{2} }\,C_{2} \,A_{1}^{i}, \end{aligned}$$
(21)
$$\begin{aligned} b_{3}^{p}&= \frac{ C_{F} }{ N_{c}^{2} } \, \big [ C_{3}\, A_{1}^{i} + C_{5}\, ( A_{3}^{i} + A_{3}^{f} ) + N_{c}\, C_{6}\, A_{3}^{f} \big ], \end{aligned}$$
(22)
$$\begin{aligned} b_{4}^{p}&= \frac{ C_{F} }{ N_{c}^{2} } \, \big [ C_{4}\, A_{1}^{i} + C_{6}\, A_{2}^{i} \big ], \end{aligned}$$
(23)
$$\begin{aligned} b_{3,EW}^{p}&= \frac{ C_{F} }{ N_{c}^{2} } \, \big [ C_{9}\, A_{1}^{i} + C_{7}\, ( A_{3}^{i} + A_{3}^{f} ) + N_{c}\, C_{8}\, A_{3}^{f} \big ], \end{aligned}$$
(24)
$$\begin{aligned} b_{4,EW}^{p}&= \frac{ C_{F} }{ N_{c}^{2} } \, \big [ C_{10}\, A_{1}^{i} + C_{8}\, A_{2}^{i} \big ], \end{aligned}$$
(25)

where the subscripts 1, 2, 3 of \(A_n^{i,f}\) denote the annihilation amplitudes induced by \((V-A)(V-A)\), \((V-A)(V+A)\) and \((S-P)(S+P)\) operators, respectively; and the superscripts i and f refer to gluon emission from the initial and final-state quarks, respectively. Their explicit expressions are given by

$$\begin{aligned} A_1^{i}&=\pi \alpha _s\int d x dy\nonumber \\&\quad \times \left\{ \Phi _{S_2}(x)\Phi _{S_1}(y)\left[ {1\over y(1-x\bar{y})} +{1\over \bar{x}^2y}\right] \right. \nonumber \\&\quad \left. -\bar{r}_\chi ^{S_1}\bar{r}_\chi ^{S_2} \Phi _{S_2}^{s}(x)\Phi _{S_1}^{s}(y) \, {2\over \bar{x}y}\right\} \,, \end{aligned}$$
(26)
$$\begin{aligned} A_2^{i}&= \pi \alpha _s\int dxdy\nonumber \\&\quad \times \left\{ \Phi _{S_2}(x)\Phi _{S_1}(y)\left[ {1\over \bar{x}(1-x\bar{y})} +{1\over \bar{x}y^2}\right] \right. \nonumber \\&\quad \left. -\bar{r}_\chi ^{S_1}\bar{r}_\chi ^{S_2} \Phi _{S_2}^{s}(x)\Phi _{S_1}^{s}(y)\, {2\over \bar{x}y}\right\} \,, \end{aligned}$$
(27)
$$\begin{aligned} A_3^{i}&=-\pi \alpha _s\int dxdy\nonumber \\&\quad \times \left\{ \bar{r}_\chi ^{S_1}\Phi _{S_2}(x)\Phi _{S_1}^{s}(y) \,{2\bar{y}\over \bar{x}y(1-x\bar{y})}\right. \nonumber \\&\quad \left. +\bar{r}_\chi ^{S_2}\Phi _{S_1}(y)\Phi _{S_2}^{s}(x) \,{2x\over \bar{x}y(1-x\bar{y})}\right\} \,, \end{aligned}$$
(28)
$$\begin{aligned} A_3^{f}&=-\pi \alpha _s\int dxdy\nonumber \\&\quad \times \left\{ \bar{r}_\chi ^{S_1}\Phi _{S_2}(x)\Phi _{S_1}^{s}(y) \,{2(1+\bar{x})\over \bar{x}^2y}\right. \nonumber \\&\quad \left. -\bar{r}_\chi ^{S_2}\Phi _{S_1}(y)\Phi _{S_2}^{s}(x) \,{2(1+y)\over \bar{x}y^2}\right\} \,, \end{aligned}$$
(29)
$$\begin{aligned} A_1^f&= A_2^f=0, \end{aligned}$$
(30)

where \(\bar{x}=1-x\) and \(\bar{y}=1-y\). Here, we have adopted the convention that \(M_1\) contains an antiquark from the weak vertex with longitudinal fraction \(\bar{y}\), while \(M_2\) contains a quark from the weak vertex with momentum fraction x.

Fig. 1
figure 1

The weak annihilation contributions

Full size image

With the traditional parametrization scheme to regularize the end point singularities in the weak-annihilation amplitudes [57], the building blocks given above are respectively written as the functions of the Gegenbauer moments.

$$\begin{aligned} A_{1}^{i}&\approx 2\,{\pi }\,{\alpha }_{s} \left\{ 9\, \Bigg [ b_{0}^{S_{1}} \Bigg (b_{0}^{S_{2}} \,\left( X_{A} - 4 + \frac{ {\pi }^{2} }{ 3 } \right) \right. \nonumber \\&\quad + b_{2}^{S_{2}} \,\left( 6\, X_{A} - \frac{107}{3} + 2\,{\pi }^{2} \right) \nonumber \\&\quad + b_{1}^{S_{2}}\, ( 3\, X_{A} + 4 - {\pi }^{2} )\nonumber \\&\quad + b_{3}^{S_{2}}\, \left( 10\,X_{A} + \frac{23}{18} - \frac{10}{3}\,{\pi }^{2} \right) \Bigg )\nonumber \\&\quad - b_{1}^{S_{1}}\, \Bigg (b_{0}^{S_{2}} \, ( X_{A} + 29 - 3\,{\pi }^{2} )\nonumber \\&\quad + b_{2}^{S_{2}}\, ( 6\, X_{A} +754 -78\,{\pi }^{2} )\nonumber \\&\quad + b_{1}^{S_{2}}\, ( 3\,X_{A} - 213 + 21\, {\pi }^{2} )\nonumber \\&\quad + b_{3}^{S_{2}}\, \left( 10\,X_{A} - \frac{12625}{6} +210 \,{\pi }^{2}\right) \Bigg )\nonumber \\&\quad + b_{2}^{S_{1}}\, \Big (b_{0}^{S_{2}} \, ( X_{A} - 119 + 12\,{\pi }^{2} )\nonumber \\&\quad + b_{2}^{S_{2}}\, ( 6\, X_{A}-9609+972\,{\pi }^{2} )\nonumber \\&\quad + b_{1}^{S_{2}}\, ( 3\,X_{A} +1534-156\,{\pi }^{2} )\nonumber \\&\quad + b_{3}^{S_{2}}\, \left( 10\,X_{A}+\frac{118933}{3} -4020\,{\pi }^{2}\right) \Big )\nonumber \\&\quad - b_{3}^{S_{1}}\, \Bigg (b_{0}^{S_{2}} \, \left( X_{A}+\frac{2956}{9}-\frac{100}{3}\,{\pi }^{2}\right) \nonumber \\&\quad + b_{2}^{S_{2}}\, \left( 6\, X_{A} +\frac{198332}{3}-6700\,{\pi }^{2}\right) \nonumber \\&\quad + b_{1}^{S_{2}}\, \left( 3\,X_{A} -\frac{20743}{3}+700\,{\pi }^{2}\right) \nonumber \\&\quad + b_{3}^{S_{2}}\, \left( 10\,X_{A} -\frac{3585910}{9}+\frac{121100}{3}\,{\pi }^{2}\right) \Bigg )\Bigg ]\nonumber \\&\left. \quad - \bar{\gamma }_{\chi }^{S_{1}}\, \bar{\gamma }_{\chi }^{S_{2}}\, X_{A}^{2} \right\} , \end{aligned}$$
(31)
$$\begin{aligned} A_{2}^{i}&\approx 2\,{\pi }\,{\alpha }_{s} \left\{ 9\, \Bigg [ b_{0}^{S_{2}} \Bigg (b_{0}^{S_{1}} \, \left( X_{A} - 4 + \frac{ {\pi }^{2} }{ 3 }\right) \right. \nonumber \\&\quad + b_{2}^{S_{1}}\, \left( 6\, X_{A} - \frac{107}{3} + 2\,{\pi }^{2} \right) - b_{1}^{S_{1}}\, ( 3\, X_{A} + 4 - {\pi }^{2} )\nonumber \\&\quad - b_{3}^{S_{1}}\, \left( 10\,X_{A} + \frac{23}{18} - \frac{10}{3}\,{\pi }^{2} \right) \Bigg )\nonumber \\&\quad + b_{1}^{S_{2}}\, \Bigg (b_{0}^{S_{1}} \, ( X_{A} + 29 - 3\,{\pi }^{2} ) \nonumber \\&\quad + b_{2}^{S_{1}} \, ( 6\, X_{A} +754 -78\,{\pi }^{2} ) \nonumber \\&\quad - b_{1}^{S_{1}}\, ( 3\,X_{A} - 213 + 21\, {\pi }^{2} )\nonumber \\&\quad - b_{3}^{S_{1}}\, \left( 10\,X_{A} - \frac{12625}{6} +210 \,{\pi }^{2}\right) \Bigg )\nonumber \\&\quad + b_{2}^{S_{2}}\, \Bigg (b_{0}^{S_{1}}\, ( X_{A} - 119 + 12\,{\pi }^{2} ) \nonumber \\&\quad + b_{2}^{S_{1}}\, ( 6\, X_{A} -9609+972\,{\pi }^{2} ) \nonumber \\&\quad - b_{1}^{S_{1}}\, ( 3\,X_{A} +1534-156\,{\pi }^{2} )\nonumber \\&\quad - b_{3}^{S_{1}}\, \left( 10\,X_{A} +\frac{118933}{3}-4020\,{\pi }^{2}\right) \Bigg )\nonumber \\&\quad + b_{3}^{S_{2}}\, \Bigg (b_{0}^{S_{1}} \, \left( X_{A}+\frac{2956}{9}-\frac{100}{3}\,{\pi }^{2}\right) \nonumber \\&\quad + b_{2}^{S_{1}}\, \left( 6\, X_{A} +\frac{198332}{3}-6700\,{\pi }^{2}\right) \nonumber \\&\quad - b_{1}^{S_{1}}\, \left( 3\,X_{A} -\frac{20743}{3}+700\,{\pi }^{2}\right) \nonumber \\&\quad - b_{3}^{S_{1}} \, \left( 10\,X_{A}-\frac{3585910}{9} +\frac{121100}{3}\,{\pi }^{2}\right) \Bigg ) \Bigg ]\nonumber \\&\left. \quad - \bar{\gamma }_{\chi }^{S_{1}}\, \bar{\gamma }_{\chi }^{S_{2}}\, X_{A}^{2} \right\} , \end{aligned}$$
(32)
$$\begin{aligned} A_{3}^{i}&\approx -6\, {\pi }\, {\alpha }_{s} \left\{ \bar{\gamma }_{\chi }^{S_{1}} \, \Bigg [b_{0}^{S_{2}}\, \left( X_{A}^{2}-2\,X_{A} +\frac{{\pi }^{2}}{3} \right) \right. \nonumber \\&\quad +6\, b_{2}^{S_{2}}\, \left( X_{A}^{2} -\frac{16}{3}\,X_{A}+\frac{15}{2} +\frac{{\pi }^{2}}{3} \right) \nonumber \\&\quad +3\, b_{1}^{S_{2}}\, \left( X_{A}^{2}-4\,X_{A}+4 +\frac{{\pi }^{2}}{3}\right) \nonumber \\&\quad +10\,b_{3}^{S_{2}} \, \left( X_{A}^{2}-\frac{13}{9}\,X_{A}+\frac{191}{18} +\frac{{\pi }^{2}}{3} \right) \Bigg ]\nonumber \\&\quad +\bar{\gamma }_{\chi }^{S_{2}}\, \Bigg [b_{0}^{S_{1}} \, \left( X_{A}^{2}-2\,X_{A}+\frac{{\pi }^{2}}{3} \right) \nonumber \\&\quad +6\, b_{2}^{S_{1}}\, \left( X_{A}^{2}-\frac{16}{3}\,X_{A} +\frac{15}{2}+\frac{{\pi }^{2}}{3}\right) \nonumber \\&\quad -3\, b_{1}^{S_{1}}\, \left( X_{A}^{2}-4\,X_{A} +4 +\frac{{\pi }^{2}}{3} \right) \nonumber \\&\quad \left. -10\,b_{3}^{S_{1}} \, \left( X_{A}^{2}-\frac{13}{9}\,X_{A} +\frac{191}{18}+\frac{{\pi }^{2}}{3}\right) \Bigg ]\right\} , \end{aligned}$$
(33)
$$\begin{aligned} A_{1}^{f}&= A_{2}^{f}\, =\, 0, \end{aligned}$$
(34)
$$\begin{aligned} A_{3}^{f}&\approx -6\, {\pi }\, {\alpha }_{s}\, X_{A} \, \left\{ \bar{\gamma }_{\chi }^{S_{1}} \, \Bigg [ b_{0}^{S_{2}}\, ( 2\,X_{A}{-}1 ) {+} b_{2}^{S_{2}}\, ( 12\,X_{A}{-}31 ) \right. \nonumber \\&\quad + b_{1}^{S_{2}}\, ( 6\,X_{A}+11 ) + b_{3}^{S_{2}}\, \left( 20\,X_{A}-\frac{187}{3}\right) \Bigg ]\nonumber \\&\quad -\bar{\gamma }_{\chi }^{S_{2}} \, \Bigg [ b_{0}^{S_{1}}\, ( 2\,X_{A}-1 ) + b_{2}^{S_{1}}\, ( 12\,X_{A}-31 ) \nonumber \\&\left. \quad - b_{1}^{S_{1}}\, ( 6\,X_{A}+11) - b_{3}^{S_{1}}\, \left( 20\,X_{A}-\frac{187}{3}\right) \Bigg ] \right\} , \end{aligned}$$
(35)

where \(X_{A}\) has the similar definition as the the parameter \(X_{H}\) in Eq. (19), and is written as

$$\begin{aligned} X_{A} \, =\, {\ln }\Big ( \frac{ m_{B} }{ {\Lambda }_{h} } \Big )\, (1+{\rho }_{A}\, e^{i\,{\phi }_{A}}), \end{aligned}$$
(36)

with \({\Lambda }_{h}\) \(=\) 0.5 GeV [57], and \({\rho }_{A}\) and \({\phi }_{A}\) are the undetermined parameters. Theoretically, \(X_{H}\) and \(X_{A}\) are respectively related to the contributions from hard spectator scattering and weak annihilations, and their physical implication and significations are in nature different. What’s more, these parameters should depend on the specific process and hadrons, because they actually originate from the convolution integrals of hard scattering functions and hadronic DAs. In the practical application of the QCDF approach, \(X_{H}\) and \(X_{A}\) are usually and approximately regarded as the universal quantities to reduce the number of phenomenological model parameters. Here, we will consider two special cases. One case (C1) is to use the minimal parameters as possible, for example, \({\rho }_{H}\) \(=\) \({\rho }_{A}\) \(=\) 1 and \({\phi }_{H}\) \(=\) \({\phi }_{A}\) \(=\) \(-55^{\circ }\) [57]. The other case (C2) is that the factorizable and nonfactorizable WA contributions are treated independently, and two quantities \(X_{A}^{f}\) and \(X_{A}^{i}\) are introduced to replace \(X_{A}\). A global fit on the B \({\rightarrow }\) PP decays with an approximation \(X_{H}\) \({\approx }\) \(X_{A}^{i}\) gives \(({\rho }_{A}^{i},\,{\phi }_{A}^{i})\) \(=\) \((2.98,\,-105^{\circ })\) and \(({\rho }_{A}^{f},\,{\phi }_{A}^{f})\) \(=\) \((1.18,\,-40^{\circ })\) [68].

Instead of the traditional parametrization scheme for dealing with the end-point divergence in the weak-annihilation amplitudes, the authors of Ref. [73] have proposed a new way to treat the end-point contributions in the case of hard-collinear gluon exchange at the leading-twist order, and the amplitude of Fig. 1b for \((V-A)(V-A)\) current has been analyzed in detail.

The hard-collinear gluon exchange effects appear in the second term of \(A_1^i\) and \(A_2^i\) given by Eqs. (26) and (27).

Using the strategy of Ref. [73], the amplitude \(A_{1}^{i}\) given by Eq. (26) can be rewritten as

$$\begin{aligned} A_{1}^{i}= \pi \alpha _s({\mathcal {G}}_1 +{\mathcal {G}}_2 +{\mathcal {G}}_3), \end{aligned}$$
(37)

where

$$\begin{aligned} {\mathcal {G}}_1&=\int _0^{\infty } d \omega \int _0^1 d x \int _0^1 d y \nonumber \\&\quad \times \Phi _B^+ (\omega )\, \Phi _{S_2}(x) \, \Phi _{S_1}(y) \, {1\over y(1-x\bar{y})} \,, \end{aligned}$$
(38)
$$\begin{aligned} {\mathcal {G}}_2&=\int _0^{\infty } d \omega \int _0^1 d x \int _0^1 d y \nonumber \\&\quad \times \Phi _B^+ (\omega ) \, \Phi _{S_2}(x)\, \Phi _{S_1}(y) \, \frac{1}{\bar{x} \, y \, ( \bar{x} - {\omega / m_B} + i \epsilon )} \,, \end{aligned}$$
(39)
$$\begin{aligned} {\mathcal {G}}_3&=\int _0^{\infty } d \omega \int _0^1 d x \int _0^1 d y\nonumber \\&\quad \times \Phi _B^+ (\omega ) \,\Phi _{S_2}^s(x)\, \Phi _{S_1}^s(y) \, \bar{r}_\chi ^{S_1}\bar{r}_\chi ^{S_2}{-2\over \bar{x}y} \,, \end{aligned}$$
(40)

where \(\omega \) is the longitudinal momentum component for the light partonic constituent of the B meson state and is defined by the kinematic relation \(l_1 \cdot p_q = \bar{n} \cdot l_1 \, n \cdot p_q / 2 \equiv \bar{n} \cdot l_1 \, \omega / 2\) with the collinear-\(l_1\) approximation. The moment \(l_1\) and \( p_q\) carried by the quark in final \(M_1\) state and the light quark in initial B meson, respectively. The two light-cone vectors \(n_{\mu }\) and \(\bar{n}_{\mu }\) fulfill the constraints \(n^2=\bar{n}^2=0\) and \(n \cdot \bar{n}=2\).

The first term, \({\mathcal {G}}_1\), given by Eq. (38) is the twist-2 contribution due to the gluon emission from the bottom quark shown by Fig. 1a, and has been demonstrated to be calculable self-consistently [50]. Using the twist-2 DA of scalar meson given by Eq. (56), \({\mathcal {G}}_1\) can be further written as an analytic form

$$\begin{aligned} {\mathcal {G}}_{1}&= 18 \Bigg [b_0^{S1}\Bigg ( b_0^{S2} \left( -3 + \frac{\pi ^2}{3}\right) + b_1^{S2}(10-\pi ^2) \nonumber \\&\quad +b_2^{S2} \left( -\frac{59}{3}+2\pi ^2\right) + b_3^{S2} \left( \frac{593}{18}-\frac{10}{3}\pi ^2\right) \Bigg )\nonumber \\&\quad +b_1^{S1}\Bigg ( b_0^{S2}(-30+3\pi ^2) + b_1^{S2}(207 -21 \pi ^2 \nonumber \\&\quad +b_2^{S2}( -770 +78\pi ^2)+ b_3^{S2} \left( \frac{4145}{2}-210 \pi ^2\right) \Bigg )\nonumber \\&\quad +b_2^{S1}\Bigg ( b_0^{S2}(-118+12\pi ^2 )+ b_1^{S2} (1540-156\pi ^2) \nonumber \\&\quad +b_2^{S2}(-9593+972\pi ^2)+b_3^{S2} (39676-4020\pi ^2) \Bigg ) \nonumber \\&\quad +b_3^{S1}\Bigg (b_0^{S2}\left( -\frac{2965}{9} +\frac{100}{3}\pi ^2\right) \nonumber \\&\quad +b_1^{S2} \left( \frac{20725}{3}-700\pi ^2\right) \nonumber \\&\quad +b_2^{S2}\left( -\frac{198380}{3}+6700\pi ^2\right) \nonumber \\&\quad + b_3^{S2}\left( \frac{3585625}{9} -\frac{121100}{3}\pi ^2\right) \Bigg )\Bigg ]\,. \end{aligned}$$
(41)

The second term, \({\mathcal {G}}_2\), given by Eq. (39) is the twist-2 contribution due to the gluon emission from the light quark in initial state shown by Fig. 1b; it is obvious that the endpoint singularity can be avoided by employing the strategy of Ref. [73]. Employing the Grozin–Neubert model [74] for \(\Phi _B ^+(\omega )\), \({\mathcal {G}}_2\) can be simplified as

$$\begin{aligned} {\mathcal {G}}_{2}&\approx 18 (b_0^{S1} -b_1^{S1}+b_2^{S1}- b_3^{S1})\nonumber \\&\quad \times \Bigg [ b_0^{S2}\Bigg (T_0 \left( \ln {m_B\over \lambda _B} +\gamma _E-i\pi \right) -2\Bigg )\nonumber \\&\quad + b_1^{S2}\Bigg (T_1 \left( \ln {m_B\over \lambda _B} +\gamma _E-i\pi \right) -9+{36\lambda _B\over m_B}\Bigg )\nonumber \\&\quad + b_2^{S2}\Bigg (T_2 \left( \ln {m_B\over \lambda _B} +\gamma _E-i\pi \right) \nonumber \\&\quad -22+{180\lambda _B\over m_B} -{720\lambda _B^2\over m_B^2}\Bigg )\nonumber \\&\quad + b_3^{S2}\Bigg (T_3 \left( \ln {m_B\over \lambda _B} +\gamma _E-i\pi \right) -{125\over 3}\nonumber \\&\quad +{1900\lambda _B\over 3m_B} -{4200\lambda _B^2\over m_B^2}+{16800\lambda _B^3 \over m_B^3}\Bigg )\Bigg ] , \end{aligned}$$
(42)

where,

$$\begin{aligned} T_0&= 1-{2\lambda _B\over m_B},\quad T_1 = 3-{18\lambda _B\over m_B}+{36\lambda _B^2\over m_B^2}, \end{aligned}$$
(43)
$$\begin{aligned} T_2&= 6-{72\lambda _B\over m_B}+{360\lambda _B^2\over m_B^2} -{720\lambda _B^3\over m_B^3}, \end{aligned}$$
(44)
$$\begin{aligned} T_3&= 10-{200\lambda _B\over m_B}+{1800\lambda _B^2\over m_B^2} -{8400\lambda _B^3\over m_B^3}+{16800\lambda _B^4\over m_B^4}. \end{aligned}$$
(45)

After dropping out the subleading term \(\lambda _B/ m_B\) and using the asymptotic twist-2 DAs, the Eq. (12) in Ref. [73] can be recovered from \({\mathcal {G}}_{2}\). The corresponding endpoint singularity in \(A_2^i\) can be also regulated in the same way, and the explicit forms of \({\mathcal {G}}_{1}\) and \({\mathcal {G}}_{2}\) for \(A_2^i\) can be obtained by the replacements that \(b_{0,2}^{S1}\) \(\leftrightarrow \) \(b_{0,2}^{S2}\) and \(b_{1,3}^{S1}\) \(\leftrightarrow \) \(-b_{1,3}^{S2}\).

However, unfortunately, for the twist-3 contributions in \(A_{1,2}^i\) (for instance, \({\mathcal {G}}_3\) given by Eq. (40)) and \(A_3^{i,f}\) which involve logarithmic square divergence, \(X_A^2\), the endpoint singularities are still unavoidable, thus the traditional parameterization scheme (C1) has to be used. In our following discussion, above-mentioned way for dealing with the end-point singularities is named as case 3 (C3).

3 Input parameters

There are many input parameters in the numerical calculations. These parameters generally fall into two categories. One has been well determined experimentally or theoretically and listed explicitly in Ref. [1], such as the Fermi coupling constant \(G_{F}\), Wilson coefficients, the CKM elements, and hadron mass as well. Their central values in Ref. [1] will be regarded as the default inputs unless otherwise specified. The other is the nonperturbative parameters, such as the decay constants, mesonic transition form factors, and hadronic DAs, which lead to produce the main theoretical errors. The choice of these parameters requires certain caution.

Table 1 The scalar decay constants \(\bar{f}_{S}\) (in units of MeV) at the scale of \({\mu }\) \(=\) 1 GeV. Here the theoretical errors come mainly from the Gaussian parameter \({\beta }\) responsible for mesonic wave functions
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3.1 The CKM elements

The Wolfenstein parameterization is traditionally and commonly used for the unitary CKM matrix, due to the obvious power series in the Wolfenstein parameter \({\lambda }\) among the CKM elements.

The values of the four Wolfenstein parameters are [1],

$$\begin{aligned} A&= 0.790^{+0.017}_{-0.012}, \quad {\lambda } = 0.22650 {\pm } 0.00048,\nonumber \\ \bar{\rho }&= 0.141^{+0.016}_{-0.017}, \quad \bar{\eta } = 0.357 {\pm } 0.011. \end{aligned}$$
(46)

3.2 The decay constants

The lattice QCD results of the isospin averages of the B meson decay constants are [1],

$$\begin{aligned} f_{B_{u,d}}= & {} 190.0{\pm }1.3\,\text {MeV}, \end{aligned}$$
(47)
$$\begin{aligned} f_{B_{s}}= & {} 230.3{\pm }1.3\,\text {MeV}. \end{aligned}$$
(48)

There are two kinds of definitions of the decay constants for the scalar mesons, i.e.,

$$\begin{aligned}&{\langle } S(p)\,{\vert } \bar{q}_{1}\,{\gamma }^{\mu }\, q_{2} \,{\vert }\, 0 {\rangle }\, =\, f_{S}\, p^{\mu }, \end{aligned}$$
(49)
$$\begin{aligned}&{\langle } S(p)\,{\vert } \bar{q}_{1}\, q_{2}\, {\vert }\, 0 {\rangle }\, =\, m_{S}\, \bar{f}_{S}({\mu }). \end{aligned}$$
(50)

The scale-dependent scalar decay constant \(\bar{f}_{S}({\mu })\) and the vector decay constant \(f_{S}\) are related by the equation of motion,

$$\begin{aligned} f_{S} = \bar{f}_{S}({\mu })\, \bar{\mu }_{S}^{-1}({\mu }). \end{aligned}$$
(51)

Clearly, the vector decay constant \(f_{S}\) is proportional to the running mass difference, \({\Delta }\overline{m}\), between the two valence quarks resided in the scalar mesons. \(f_{S}\) for the light scalars should be seriously suppressed by the small \({\Delta }\overline{m}\), especially for the electrically neutral scalar mesons owing to charge conjugation invariance or conservation of vector current. For example, \(f_{S}\) will vanish for the \(a_{0}^{0}\) meson. At the same time the scalar decay constants \(\bar{f}_{S}\) remain finite. Here a preferable solution scheme is to use the scalar decay constants \(\bar{f}_{S}\). This is one of the main reasons for the factors in Eq. (14). In addition, the scalar mesons and its antiparticles have the same scalar decay constants, \(\bar{f}_{S}\) \(=\) \(\bar{f}_{\bar{S}}\). It means that the vector decay constants \(f_{S}\) \(=\) \(-f_{\bar{S}}\) from Eq. (12) and Eq. (51), which results in the \(f_{S}\) \(=\) 0 for the \(a_{0}^{0}\) meson.

Experimentally, these decay constants can be extracted from the purely leptonic decays of the scalar mesons. It is widely known that the scalar mesons usually appear as resonances and decay dominantly through the strong interactions, and the occurrence probability of the leptonic decays of the scalar mesons should in principle be very small. The leptonic decays of the scalar mesons have not been discovered by now. The experimental data on the decay constants of the scalar mesons is still unavailable. The theoretical values of the scalar decay constants \(\bar{f}_{S}\) corresponding to the S1 and S2 scenarios are listed in Table 1. It is clearly seen that for the S2 scenario, the central values of the decay constants \(\bar{f}_{S}\) obtained with the covariant light-front quark model (CLFQM) in this paper are generally in agreement with those from the QCD sum rules [5, 75] and light-cone sum rules [76] within an error range. Of course, the errors arising from the Gaussian parameter \({\beta }\) responsible for mesonic wave functions are still very large due to the inadequate data and our insufficient understanding on the scalar mesons for the moment, especially for \(\bar{f}_{K_{0}^{*}(1430)}\) with the S2 scenario. What’s more, the values of \(\bar{f}_{S}\) with the S2 scenario are about twice as larger as those with the S1 scenario, which will inevitably bring the obviously hierarchical relations with the branching ratios with these two different scenarios, because the decay amplitudes are directly proportional to the decay constants. A significant difference between branching ratios might be used to distinguish whether these scalar mesons are the 1P or 2P states.

3.3 Hadronic transition form factors

The form factors of B \({\rightarrow }\) S transitions are defined as [4,5,6],

$$\begin{aligned}{} & {} {\langle } S(k)\,{\vert } \bar{q}\,{\gamma }_{\mu } \, {\gamma }_{5} \,b\,{\vert }B(p){\rangle } \nonumber \\{} & {} \quad = -i\, \Bigg [ \Bigg ( P_{\mu } -\frac{ m_{B}^{2}-m_{S}^{2} }{ q^{2} }\,q_{\mu } \Bigg )\, U_{1}(q^{2}) +\frac{ m_{B}^{2}-m_{S}^{2} }{ q^{2} }\,q_{\mu }\, U_{0}(q^{2}) \Bigg ],\nonumber \\ \end{aligned}$$
(52)

where \(P_{\mu }\) \(=\) \(p_{\mu }\) \(+\) \(k_{\mu }\) and \(q_{\mu }\) \(=\) \(p_{\mu }\) − \(k_{\mu }\). \(U_{0}(q^{2})\) and \(U_{1}(q^{2})\) respectively denote longitudinal and transverse form factors. To regulate the singularities at the pole \(q^{2}\) \(=\) 0, the relations \(U_{0}(0)\) \(=\) \(U_{1}(0)\) is required. The values of \(U_{0,1}(0)\) can be obtained by fit the dependence of the form factors on \(q^{2}\) with the 3-parameter formula [5, 6],

$$\begin{aligned} U_{i}(q^{2}) = \frac{ U_{i}(0) }{ 1-a\,( q^{2}/m_{B}^{2}) +b\,( q^{2}/m_{B}^{2} )^{2}}. \end{aligned}$$
(53)
Table 2 Form factors for the B \({\rightarrow }\) S transitions obtained from CLFQM, considering the S1 and S2 scenarios for the scalar mesons
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Table 3 The values of the Gegenbauer moments at the scale of \({\mu }\) \(=\) 1 GeV, considering the S1 and S2 scenarios for the scalar mesons. The results in this work are obtained from CLFQM, and those of Ref. [6] from QCD sum rules
Full size table

The form factors obtained from CLFQM are listed in Table 2. It is clearly seen that (1) the central values of \(U_{0,1}(0)\) in this work are very close to those of Ref. [5]. They are slightly larger (smaller) than those given in Ref. [5] for the S2 (S1) scenario. The differences come mainly from the quark running mass and the Gaussian parameter \({\beta }\) as well. (2) For the S2 scenario, the SU(3) flavor symmetry among the central values of \(U_{0,1}(0)\) seems to be held well. (3) For the B \({\rightarrow }\) \(K_{0}^{*}(1430)\), \(a_{0}(1450)\) transition form factors, the differences between the S1 and S2 scenarios given in Ref. [5] are less obvious than those obtained in this work. Here the ratio of \(U_{0,1}^{B{\rightarrow }K_{0}^{*},a_{0}}(0)\) between the S1 and S2 scenarios is approximately 2/3, which will result in the ratio of branching ratio proportional to the square of the form factors is approximately 1/2. The bigger difference of the ratio, the easier the measurement becomes, and the more helpful it is to distinguish whether these scalar mesons are the 1P or 2P states from the semileptonic B \({\rightarrow }\) \(S{\ell }{\nu }\) decays in the future experiments, and to check the different theoretical predictions.

3.4 Mesonic light cone DAs

The definition of mesonic light cone DAs is [4, 5],

$$\begin{aligned}{} & {} {\langle } S(p)\,{\vert }\, q_{2{\beta }}(z_{2})\, q_{1{\alpha }}(z_{1})\,{\vert }0 {\rangle }\nonumber \\{} & {} \quad = \frac{1}{4}\,\bar{f}_{S}\,{\int }_{0}^{1}\,dx\, e^{ i\,(xp{\cdot }z_{2}+\bar{x}{\cdot }z_{1}) }\nonumber \\{} & {} \qquad \times \Bigg \{ \not {p}\,{\Phi }_{S}(x)+m_{S}\, \Bigg [{\Phi }_{S}^{s}(x)-{\sigma }_{{\mu }{\nu }}\,p^{\mu }\,z^{\nu } \frac{{\Phi }_{S}^{\sigma }(x)}{6} \Bigg ] \Bigg \}_{{\alpha }{\beta }},\nonumber \\ \end{aligned}$$
(54)

where the arguments \(\bar{x} =1 - x\) and \(z = z_{2} -z_{1}\). \({\Phi }_{S}\) is the twist-2 light cone DAs. The twist-3 light cone DAs \({\Phi }_{S}^{s,{\sigma }}\) are related by the equations of motion [5],

$$\begin{aligned} {\xi }\,{\Phi }_{S}^{s}(x) +\frac{1}{6}\, \frac{d\,{\Phi }_{S}^{\sigma }(x) }{d\,x}\, =\, 0, \end{aligned}$$
(55)

where \({\xi }\) \(=\) x − \(\bar{x}\) \(=\) \(2\,x\) − 1. The twist-2 DAs are written as [4, 5]

$$\begin{aligned} {\Phi }_{S}(x,\,{\mu }) =6\,x\,\bar{x}\, \Bigg \{ b_{0}^{S} +\sum \limits _{n=1}^{\infty } b_{n}^{S}({\mu }) \,C_{n}^{3/2}({\xi }) \Bigg \}, \end{aligned}$$
(56)

where the Gegenbauer moments \(b_{i}^{S}\), corresponding to the expansion coefficients of Gegenbauer polynomials \(C_{i}^{3/2}({\xi })\), are hadronic parameters. The asymptotic forms of the twist-3 DAs are respectively written as [5],

$$\begin{aligned} {\Phi }_{S}^{s}(x,\,{\mu })&= 1, \end{aligned}$$
(57)
$$\begin{aligned} {\Phi }_{S}^{\sigma }(x,\,{\mu })&= 6\,x\,\bar{x}. \end{aligned}$$
(58)
Fig. 2
figure 2

The twist-2 DAs for the \(a_{0}(1450)\) and \(K_{0}^{*}(1430)\). The dashed and solid lines correspond to the truncations up to n \(=\) 1 and 3, respectively

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The Gegenbauer moments \(b_{n}^{S}\) in the twist-2 DAs \({\Phi }_{S}\) are listed in Table 3. Our comments are (1) for either the S1 or S2 scenarios, the orbital angular momentum L \(=\) 1 between the two components of the scalar mesons. Using the parity of Gegenbauer polynomials and the isospin symmetry, the wave function should in principle be antisymmetric, \((-1)^{L}\), under the exchange of the longitudinal momentum fractions of the two valence quarks x \({\leftrightarrow }\) \(\bar{x}\), i.e., the Gegenbauer moments \(b_{n}^{S}\) with even n should be zero. This feature is clearly demonstrated for the \(a_{0}(1450)\) mesons in Table 3 and Fig. 2a. (2) For the \(K_{0}^{*}(1430)\) mesons, the flavor SU(3) symmetry breaking effects should be given due consideration. DAs for the \(K_{0}^{*}(1430)\) mesons should be asymmetric under x \({\leftrightarrow }\) \(\bar{x}\), i.e., the Gegenbauer moments \(b_{n}^{S}\) with both even and odd n are nonzero. This property is properly illustrated by our results in Table 3 and Fig. 2b. (3) According to the definition of hadronic matrix elements given by Eqs. (3.18) and (3.20) in Ref. [5], the Gegenbauer moments \(b_{1}^{S}\) in DAs and the decay constant \(\bar{f}_{S}\) are directly interrelated. The positive values of \(b_{1}^{S}\) correspond to the the negative value of \(\bar{f}_{S}\) listed in Table 1 for the S1 scenario, and vice versa for the S2 scenario. In this sense, the positive and negative sign for \(b_{1}^{S}\) from CLFQM in this work and QCD sum rules in Ref. [6] are self-consistent.

4 Numerical results and discussions

In the rest frame of the B meson, the CP-averaged branching ratio is defined as,

$$\begin{aligned} {\mathcal {B}} = \frac{{\tau }_{B}}{16{\pi }}\, \frac{p_{\textrm{cm}}}{m_{B}^{2}}\, \big \{{\vert }{\mathcal {A}} (B{\rightarrow }f){\vert }^{2}+{\vert }{\mathcal {A}}(\overline{B}{\rightarrow } \overline{f}){\vert }^{2} \big \}, \end{aligned}$$
(59)

where \({\tau }_{B}\) is the B meson lifetime, \(p_{\textrm{cm}}\) is the common momentum of final states.

The direct CP asymmetry is defined as,

$$\begin{aligned} A_{CP} = \frac{{\Gamma }(\overline{B}{\rightarrow }f)-{\Gamma }(B{\rightarrow }\overline{f})}{{\Gamma }(\overline{B}{\rightarrow }f)+{\Gamma }(B{\rightarrow }\overline{f})}. \end{aligned}$$
(60)

When the final states are common to the neutral \(B_{d,s}^{0}\) and \(\overline{B}_{d,s}^{0}\) decays, the CP violating asymmetry is defined as,

$$\begin{aligned} A_{CP}&= A_{CP}^{\textrm{mix}}\,{\sin }(x\,{\Delta }m\,t) -A_{CP}^{\textrm{dir}}\,{\cos }(x\,{\Delta }m\,t), \end{aligned}$$
(61)
$$\begin{aligned} A_{CP}^{\textrm{mix}}&= \frac{ 2\,{\mathcal {I}}m({\lambda }_{f}) }{ 1+{\vert }{\lambda }_{f}{\vert }^{2}}, \end{aligned}$$
(62)
$$\begin{aligned} A_{CP}^{\textrm{dir}}&= \frac{ 1-{\vert }{\lambda }_{f}{\vert }^{2} }{ 1+{\vert }{\lambda }_{f}{\vert }^{2} }, \end{aligned}$$
(63)
$$\begin{aligned} {\lambda }_{f}&= \left\{ \begin{array}{ll} \displaystyle \frac{ V_{tb}^{*}\,V_{td} }{ V_{tb}\,V_{td}^{*}} \,\frac{ {\mathcal {A}}(\overline{B}_{d}^{0}{\rightarrow }f)}{{\mathcal {A}}(B_{d}^{0}{\rightarrow }f) }, &{} ~~\text {for the}\ B_{d}^{0}\text {-}\overline{B}_{d}^{0} \ \text {system}, \\ \\ \displaystyle \frac{ V_{tb}^{*}\,V_{ts} }{ V_{tb}\,V_{ts}^{*}} \,\frac{ {\mathcal {A}}(\overline{B}_{s}^{0}{\rightarrow }f) }{{\mathcal {A}}(B_{s}^{0}{\rightarrow }f) }, &{} ~~\text {for the}\ B_{s}^{0}\text {-}\overline{B}_{s}^{0} \ \text {system}. \end{array} \right. \end{aligned}$$
(64)
Table 4 The CP-averaged branching ratios (in units of \({10}^{-6}\)) for the B \({\rightarrow }\) SS decays, where \(a_{0}\) \({\equiv }\) \(a_{0}(1450)\) and \(K_{0}^{*}\) \({\equiv }\) \(K_{0}^{*}(1430)\). The numbers in the “NF” columns corresponds to the LO results. The numbers in the “C1” and “C2” columns are results including the NLO contributions. “C1” corresponds to the case \(({\rho }_{A},\, {\phi }_{A})\) \(=\) \((1,\,-55^{\circ })\) and \(X_{H}\) \(=\) \(X_{A}\),, “C2” corresponds to \(({\rho }_{A}^{i},\,{\phi }_{A}^{i})\) \(=\) \((2.98,\,-105^{\circ })\), \(({\rho }_{A}^{f},\,{\phi }_{A}^{f})\) \(=\) \((1.18,\,-40^{\circ })\) and \(X_{H}\) \({\approx }\) \(X_{A}^{i}\), and “C3” corresponds to the case of using the strategy in Ref. [73] for treating the end-point contributions. The first uncertainties come mainly from the CKM elements. The second uncertainties come from the hadronic parameters including the decay constant, form factors, and the Gegenbauer moments in DAs
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Table 5 CP asymmetries (in units of \(\%\)) for the B \({\rightarrow }\) SS decays. Other legends are the same as those of Table 4
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The numerical results on the CP-averaged branching ratios and CP asymmetries for the B \({\rightarrow }\) SS decays are listed in Tables 4 and 5. Here we use the symbols T for the color favored tree processes, C for the color-suppressed tree processes, P for the penguin dominated processes, and A for the pure annihilation processes. Our comments are as follows.

  1. (1)

    As we have discussed earlier, the LO contributions are suppressed by the factor \(N_{i}(S_{2})\) in Eq. (13), then the NLO contributions will be very important for the B \({\rightarrow }\) SS decays. It is clearly shown in Table 4 that for both the S1 and S2 scenarios, the NLO contributions to branching ratios are generally significant, even a few fold changes to the LO contributions corresponding to the numbers in the “NF” columns for some processes, such as C-class B decays where the NLO contributions are proportional to the large Wilson coefficient \(C_{1}\).

  2. (2)

    The hard spectator scattering amplitudes \(H_{i}(S_{1}\, S_{2})\) belong to the nonfactorizable NLO contributions in Eq. (13) with the QCDF approach. So, the NLO contributions should be sensitive to the parameter \(X_{H}\). And the parameter \(X_{H}\) is closely related to the WA parameter \(X_{A}\) in this work. In Table 4, the differences of branching ratios between the C1 and C2 cases are still obvious, for example, for the B \({\rightarrow }\) \(a_{0}a_{0}\) and \(B_{s}\) \({\rightarrow }\) \(K_{0}^{*}\overline{K}_{0}^{*}\) decays, and the A-class decays as well. Additionally, the parameters \(X_{H,A}\) are always accompanied by the Gegenbauer moments in Eqs. (15) and (31—35). The smaller uncertainties of the Gegenbauer moments bring branching ratios with the smaller theoretical uncertainties, compared with those in Refs. [5, 6]. The branching ratios in C1 and C3 cases are close to each other for most of decay modes, but are significantly different for the pure annihilation decays, which is due to the fact that these decays are only related to \(A_{1,2}^i\) and the different strategies for treating the endpoint contributions in \(A_{1,2}^i\) are used in C1 and C3 cases.

  3. (3)

    As it is well known that the T-class B decays are induced by the external W boson emission interactions with the factorization approach, and their amplitudes are proportional to the large Wilson coefficient \(C_{1}\) or \({\alpha }_{1}\). These processes should theoretically have a relatively large branching ratio. It might be a little curious in Table 4 that branching ratios for the T-class B \({\rightarrow }\) SS decays are very small, \({\sim }\) \({\mathcal {O}}(10^{-7})\). Some even are less than the branching ratios of the purely WA decays. One of the main reasons is that the LO contributions of the T-class decay amplitudes are seriously suppressed by the factor \(N_{i}(S_{2})\) in Eq. (13), with \(N_{i}(a_{0})\) \({\sim }\) 0.002, and their NLO contributions are suppressed by both the factor \({\alpha }_{s}/N_{c}\) and the small coefficient \(C_{2}\).

  4. (4)

    It is obvious in Table 4 that for the all C1, C2 and C3 cases, branching ratios of the S2 scenario are larger than the corresponding ones of the S1 scenario, because the decay amplitudes are proportional to the product of the decay constants of the scalar mesons and form factors, and the numerical values of both the decay constants of the scalar mesons (see Table 1) and form factors (see Table 2) of the S2 scenario are larger than the corresponding ones of the S1 scenario. Specifically, the P-class B \({\rightarrow }\) \(a_{0} \overline{K}_{0}^{*}\) and \(B_{s}\) \({\rightarrow }\) \(K_{0}^{*} \overline{K}_{0}^{*}\) decays where the penguin contributions are largely enhanced by the CKM elements \(V_{tb}V_{ts}^{*}\) \({\sim }\) \({\mathcal {O}}({\lambda }^{2})\) relative to the possible tree contributions associated with \(V_{ub}V_{us}^{*}\) \({\sim }\) \({\mathcal {O}}({\lambda }^{4})\), their branching ratios can reach up to even \({\mathcal {O}}(10^{-5})\) for the S2 scenario. These flagship decay modes should get priority in the future experimental research program for searching for the B \({\rightarrow }\) SS decays.

  5. (5)

    Experimentally, more than ten years ago, a hint of the \(B^{0}\) \({\rightarrow }\) \(K_{0}^{{*}0}\overline{K}_{0}^{{*}0}\) decays with a significance of \(0.8\,{\sigma }\) has been reported by the Belle Collaboration with the \(K^{+}K^{-}{\pi }^{+}{\pi }^{-}\) final states [77], and branching ratio \({\mathcal {B}}\) \(=\) \((3.21^{+2.89+2.31}_{-2.85-2.32}) {\times }10^{-6}\) and the upper limit at the \(90\%\) confidence level \({\mathcal {B}}\) < \(8.4{\times }10^{-6}\). Our results are marginally consistent with data when considering the large experimental errors. Theoretically, besides the small Wilson coefficients and the CKM elements \(V_{tb}V_{td}^{*}\) \({\sim }\) \({\mathcal {O}}({\lambda }^{3})\), the relatively smaller branching ratios might arise from the Gegenbauer moments, which result in a flatter shape line of the scalar mesonic DAs, and further leads to a milder overlap among the participating mesonic DAs, and finally give a more modest decay amplitudes. Experimentally, it is entirely necessary and desirable to improve the accuracy of measurements and investigate more and more B \({\rightarrow }\) SS decays in the future in order to verify various theoretical models and explore the properties the scalar mesons.

  6. (6)

    The weak annihilation amplitudes are thought to be power suppressed with the QCDF approach [50, 51]. The purely WA B decays should in principle have very small branching ratios. The evidences have been demonstrated in the B \({\rightarrow }\) \(K^{{\pm }}K^{{*}{\mp }}\) and \(B_{s}\) \({\rightarrow }\) \({\pi }{\pi }\) decays theoretically [57,58,59,60, 66,67,68,69,70] and experimentally [1]. The similar phenomena or/and patterns also appear in Table 4 for the A-class B \({\rightarrow }\) SS decays, with branching ratios of \({\mathcal {B}}\) \({\mathcal {O}}(10^{-7})\). The impressive and amazing thing is that in some cases, branching ratios of the A-class B \({\rightarrow }\) SS decays with appropriate parameters can catch up with or overtake those of the T-class decays, which is very unlike the hadronic B \({\rightarrow }\) PP, PV decays. These typical characteristics for the B \({\rightarrow }\) SS decays are closely related with the properties of the scalar mesons, such as the decay constants, DAs and so on. Additionally, branching ratios of the A-class B \({\rightarrow }\) SS decays are very sensitive to the parameter \(X_{A}\), with both the S1 and S2 scenarios. It is clear that with the topologically dependent parameters, i.e., \(X_{A}^{i}\) \({\ne }\) \(X_{A}^{f}\) for the C2 case, the corresponding branching ratios are relatively larger, due to the larger value of \({\rho }^{i}\). Although the endpoint singularities in the twist-2 contributions can be successfully avoided by using the strategy of Ref. [73], but the ones in the higher twist contributions need to be further studies. Our understanding of the WA contributions to the nonleptonic B decays with the QCDF approach is not comprehensive enough. Albeit very challenging, the experimental measurements on the A-class B \({\rightarrow }\) SS decays are interesting and helpful to explore the underlying dynamical mechanism and the higher power corrections to HMEs.

  7. (7)

    In Table 4, the theoretical uncertainties are very large, especially those from the hadronic parameters. Usually, the ratio of branching ratios are defined to reduce theoretical uncertainties on one hand, and on the other hand to check some potential symmetry or conservation quantities, for example, the observables \(R_{K,D}\) for the universality of the inherited electroweak couplings to all charged leptons. Here, we give some ratios of branching ratios with the universal parameter \(X_{A}\) for the C1 case, for example,

    $$\begin{aligned} R_{1}&= \frac{{\mathcal {B}}( B^{-} {\rightarrow } a_{0}^{-} \overline{K}_{0}^{{*}0} ) }{ 2\,{\mathcal {B}}( B^{-} {\rightarrow } a_{0}^{0} K_{0}^{{*}-} ) }\nonumber \\&\approx 1.04^{+0.00+0.16}_{-0.00-0.14} ~(\text {S1}),\ 1.06^{+0.00+0.11}_{-0.00-0.11} ~(\text {S2}); \end{aligned}$$
    (65)
    $$\begin{aligned} R_{2}&= \frac{{\mathcal {B}}( \overline{B}^{0} {\rightarrow } a_{0}^{+} K_{0}^{{*}-} ) }{ 2\,{\mathcal {B}}( \overline{B}^{0} {\rightarrow } a_{0}^{0} \overline{K}_{0}^{{*}0} ) }\nonumber \\&\approx 0.96^{+0.00+0.15}_{-0.00-0.14} ~(\text {S1}), \ 0.94^{+0.00+0.11}_{-0.00-0.10} ~(\text {S2}); \end{aligned}$$
    (66)
    $$\begin{aligned} R_{3}&= \frac{ {\mathcal {B}}( \overline{B}_{s}^{0} {\rightarrow } K_{0}^{{*}0} \overline{K}_{0}^{{*}0} ) }{ {\mathcal {B}}( \overline{B}_{s}^{0} {\rightarrow } K_{0}^{{*}+} K_{0}^{{*}-} ) }\nonumber \\&\approx 1.08^{+0.00+0.02}_{-0.01-0.02} ~(\text {S1}), \ 1.07^{+0.00+0.01}_{-0.00-0.01} ~(\text {S2}). \end{aligned}$$
    (67)

    All these ratios are expected to be \(R_{1,2,3}\) \(=\) 1 by applying the SU(3) flavor symmetry.

  8. (8)

    It is clear in Table 5 that the CP violating asymmetries depend on the parameter \(X_{A}\) which contains the strong phases. It is known that with the QCDF approach, the strong phases necessary for the direct CP violation arise from the NLO contributions, which are the order \({\alpha }_{s}\) or \({\Lambda }_{\textrm{QCD}}/m_{b}\) and suppressed compared with the LO contributions. However, as noted earlier, the LO contributions are seriously suppressed by the factor \(N_{i}(S_{2})\) in Eq. (13), which will indirectly result in the larger strong phases from the NLO contributions. These effects will have more influences on the direct CP violating asymmetries for the T- and C-class B \({\rightarrow }\) SS decays than the P-class ones, because of the larger Wilson coefficients for the T- and C-class decays. The larger direct CP asymmetries for the \(B_{d}\) \({\rightarrow }\) \(a_{0}a_{0}\) and \(B_{s}\) \({\rightarrow }\) \(a_{0}K_{0}^{*}\) are expected for both the S1 and S2 scenarios. And by comparison, the absolute values of the direct CP violating asymmetries in the T- and C-class B \({\rightarrow }\) SS decays are generally larger than those in the corresponding B \({\rightarrow }\) PP, PV decays [57,58,59,60, 66,67,68,69,70]. In addition, to contract with the so-called \({\pi }K\) puzzle, the difference between the direct CP asymmetries for the \(B^{-}\) \({\rightarrow }\) \(a_{0}^{0} \overline{K}_{0}^{{*}-}\) and \(\overline{B}_{0}\) \({\rightarrow }\) \(a_{0}^{+} K_{0}^{{*}0-}\) decays is estimated to be,

    $$\begin{aligned} {\Delta }A_{CP}&\equiv A_{CP}( { B^{-} {\rightarrow } a_{0}^{0} \overline{K}_{0}^{{*}-} } ) -A_{CP}(\overline{B}_{0} {\rightarrow } a_{0}^{+} K_{0}^{{*}0-})\nonumber \\&= \left\{ \begin{array}{ccc} (5.96^{+0.18+1.11}_{-0.17-1.29})\%~(\text {S1}), (5.48^{+0.16+0.52}_{-0.17-0.73})\% ~(\text {S2}), &{} \text {C1} \\ (3.71^{+0.11+2.46}_{-0.11-1.95})\%~(\text {S1}),\ (5.05^{+0.15+1.66}_{-0.15-1.40})\% ~(\text {S2}), &{} \text {C2} \\ { (5.83^{+0.18+0.89}_{-0.18-1.23})\%~(\text {S1}),\ (5.36^{+0.14+0.52}_{-0.14-0.91})\% ~(\text {S2}), } &{} \text {C3} \end{array}\right. \end{aligned}$$
    (68)

The future measurement on the CP asymmetries for the B \({\rightarrow }\) SS decays are expected for testing the possible puzzles.

5 Summary

To meet the coming high precision measurements on the B meson decays based on the huge amount of data, and provide a ready and helpful reference in clarifying the open questions related to the scalar mesons, the hadronic charmless B \({\rightarrow }\) SS decays are studied with the QCDF approach, where the symbol S denotes the scalar mesons \(K_{0}^{*}(1430)\) and \(a_{0}(1450)\). It is found that the LO contributions are proportional to the mass difference of the two valence quarks embedded in the scalar mesons, and thereby seriously suppressed. This causes two consequences. (1) The branching ratios for the B \({\rightarrow }\) \(a_{0}a_{0}\) and \(B_{s}\) \({\rightarrow }\) \(a_{0}K_{0}^{*}\) decays belonging to the T- and C-class are very small, about the order of \({\mathcal {O}}(10^{-7})\). (2) The NLO contributions become necessary and predominant for the B \({\rightarrow }\) SS decays. With the updated values of hadronic parameters obtained from CLFQM, including the transition form factors, the decay constants and Gegenbauer moments in mesonic DAs for the two scenarios where the scalar mesons in question are the 1P and 2P triplet states, the CP-averaged branching ratios and CP violating asymmetries are given with the universal end-point parameters \(X_{A}\) and topology-dependent parameters \(X_{A}^{i}\) \({\ne }\) \(X_{A}^{f}\), besides, a new strategy proposed in Ref. [73] for treating the end-point contributions is employed. The numerical results show that (1) theoretical uncertainties of both the branching ratios and the direct CP asymmetries come mainly from hadronic parameters. (2) Branching ratios for the \(B_{s}\) \({\rightarrow }\) \(K_{0}^{*}\overline{K}_{0}^{*}\) decays and the purely weak annihilation decays \(B_{s}\) \({\rightarrow }\) \(a_{0}a_{0}\) and \(B_{d}\) \({\rightarrow }\) \(K_{0}^{{*}+}K_{0}^{{*}-}\), and the direct CP asymmetries for the \(B_{d}\) \({\rightarrow }\) \(a_{0}a_{0}\) decays are very sensitive to the parameter \(X_{A}\). (3) For the B \({\rightarrow }\) \(a_{0}K_{0}^{*}\) and \(B_{s}\) \({\rightarrow }\) \(K_{0}^{*}\overline{K}_{0}^{*}\) decays, branching ratios for the S2 scenario are about one order of magnitude larger than those for the S1 scenario, and can reach up to the order of \({\mathcal {O}}(10^{-5})\). These decays should first be searched and investigated experimentally. (4) Theoretical uncertainties come mainly from the hadronic parameters. More focus and effort are needed to improve the theoretical calculation precision. Some ratios of branching ratios are given based on the SU(3) flavor symmetry. In addition, there is too little available data to draw any conclusions on whether the scalar mesons \(K_{0}^{*}(1430)\) and \(a_{0}(1450)\) are the 1P or 2P states. Hope more and more B \({\rightarrow }\) SS decays can be measured with higher and higher precision at the high-luminosity colliders in the future.