Reinvestigating the $B$ ${\to}$ $PP$ decays by including the contributions from ${\phi}_{B2}$

Considering the $B$ mesonic distribution amplitude ${\phi}_{B2}$, we reinvestigated the $B$ ${\to}$ $PP$ (where $P$ $=$ ${\pi}$ and $K$) decays with the perturbative QCD (pQCD) approach based on the $k_{T}$ factorization for three scenarios. It is found that the contributions of ${\phi}_{B2}$ to formfactors $F_{0}^{B{\to}P}(0)$ and branching ratios are comparable with those from the NLO corrections. The $B$ ${\to}$ $K{\pi}$ decays could be well explained by considering the ${\phi}_{B2}$. Hence, when the nonleptonic $B$ decays are studied withe the pQCD approach, the ${\phi}_{B2}$ should be taken into account seriously.

(0) and branching ratios are comparable with those from the NLO corrections. The B → Kπ decays could be well explained by considering the φ B2 . Hence, when the nonleptonic B decays are studied with the pQCD approach, the φ B2 should be taken into account seriously.
It is well known that many breakthrough discoveries have come from precise experiments.
B physics is on the bleeding edge and one of hot topics of current particle physics, because of the renewed impetus from the successive CLEO, BaBar, Belle, LHCb and Belle-II experiments. Various B meson decay modes with branching ratio larger than 10 −6 have been extensively studied by the BaBar and Belle Collaborations with 0.56 ab −1 and 1.02 ab −1 data samples in the past years [1,2]. A few phenomena of inconsistencies between experimental measurements and theoretical expectations from the standard model (SM) are emerging.
More and more B meson data are expected in the near future, about 50 ab −1 by the Belle-II detector at the e + e − SuperKEKB collider [3] and about 300 f b −1 by the LHCb detector at the High Luminosity LHC (HL-LHC) hadron collider [4]. Besides some new phenomena, the much more precise measurements of B meson weak decays will offer a much more rigorous test on SM. When looking for a smoking gun of new physics and settling the temporary differences between experimental and theoretical results, a more careful calculation on B meson decays within SM is very necessary and important. In this paper, we will reinvestigate the B → P P decays (here P = π and K) based on the perturbative QCD approach within SM, by considering the contributions from B mesonic wave function φ B2 which usually attract less attention in previous calculation.
For clarity, we will sketch the phenomenological study of nonleptonic B → P P decays, although they have been extensively studied, for example, in Refs. [5][6][7][8][9][10][11][12][13]. Because of our inadequate comprehension of the flavor mixing and possible glueball components, the final states of η and η ′ mesons are not considered here for the moment.
At the quark level, based on the operator product expansion and renormalization group (RG) method, the effective Hamiltonian responsible for B → P P decays is written as [14], where G F ≃ 1.166×10 −5 GeV −2 [1] is the Fermi weak coupling constant. With the Wolfenstein parametrization, the related Cabibbo-Kobayashi-Maskawa (CKM) factors are written as follows.
and the latest values of the four Wolfenstein parameters (A, λ, ρ and η) from data with the CKMfitter method [1] are listed in Table I are local four-quark interactions and expressed as follows.
The hadronic matrix elements (HMEs), O i ≡ P 1 P 2 |O i |B , describe the transformations from the quarks to hadrons. The calculation of HMEs is on the one hand very complicated due to the entanglements between perturbative and nonperturbative contributions, and on the other hand very sensitive to phenomenological models because of our limited knowledge of dynamics of hadronization and final state interactions. One of the main challenges is to calculate HMEs as properly as possible. Theoretically, the radiative corrections to HMEs should be appropriately included so that the strong phase angles closely related to CP violation could be obtained. For nonleptonic B decays, the HMEs are usually written as the product of the rescattering amplitudes of quarks (which are calculable order by order with perturbation theory in principle) and wave functions of participating hadrons (where nonperturbative contributions are housed) with the fashionable QCD-inspired phenomenological models, either the QCD factorization (QCDF) approach [15][16][17][18][19][20] based on the collinear approximation or the perturbative QCD (pQCD) approach [21][22][23][24][25][26][27] retaining the effects of transverse momentum k T . Hadronic wave functions (WFs) or distribution amplitudes (DAs) are independent of specific process and determined from data, which enable evaluating HMEs to simplify greatly.
WFs and/or DAs are the essential ingredients of the master formulas for evaluating HMEs with the QCDF and pQCD approaches. The B mesonic WFs are generally composed of two scalar functions [28,29] and written as follows with the convention of Refs. [30,31].
where the coordinate of the light quark is on the light cone i.e., z 2 = 0 and z + = 0. n µ + = (1, 0, 0) and n µ − = (0, 1, 0) are the light cone vectors. f B is the decay constant. φ + B and φ − B are respectively the leading-and sub-leading-twist WFs. The properties and relations of φ ± B are listed as follows.
where x is the longitudinal momentum fraction carried by the light quark in the B meson.
φ + B and φ − B have different asymptotic behaviors as x → 0, φ + B ∼ x but φ − B will not vanish. So they do not coincide, i.e., φ + B = φ − B or φ B2 = 0. In many actual calculations of nonleptonic B decays, only the contributions of φ B1 are considered appropriately, while those of φ B2 are assumed to be power suppressed and almost completely neglected. However, studies of Refs. [30][31][32][33][34] have shown that contributions of φ B2 to the B → π transition formfactors with the pQCD approach could have a large proportion rather than negligible. For example, the share could reach up to ∼ 30% for some specific cases [30,31]. Clearly, the contributions of φ B2 will have some impacts on branching ratios of B meson decays. We should pay due attention to contributions of φ B2 in pace with the improvements of measurement precision, which is one main motivation of this work. The contributions of φ B2 to the B → P P decays have been studied with the QCDF approach [35]. The study of Ref. [35] showed that φ B2 only contributed to nonfactorizable annihilation amplitudes, and is helpful in explaining pure annihilation B decays. Different from the QCDF case, φ B2 will contribute to both factorizable and nonfactoizable emission amplitudes with the pQCD approach, besides the nonfactorizable annihilation amplitudes. That is to say, φ B2 would have much more influence on nonleptonic B decays with the pQCD approach when compared with the QCDF approach.
However, the contributions of φ B2 to the B → P P decays have not been studied with the pQCD approach, which is the focus of this paper.
One candidate of the most often used leading B mesonic WF φ + B in earlier studies with the pQCD approach is written as [25] where b is the conjugate variable of the transverse momentum k T .x = 1 − x. ω B is the shape parameter. N is the normalization constant.
The corresponding sub-leading B mesonic WF φ − B [31] can be obtained by solving the equation of motion given by Eq. (20).
In addition, according to the convention of Refs. [36,37], WFs of the final pseudoscalars π + and K + are generally written as follows.
where f M is the decay constant. x is the longitudinal momentum fraction of the anti-quark.
µ M = 1.6±0.2 GeV [37] is the chiral mass. DA φ a M is the leading twist (twist-2), and φ p,t M is the twist-3. Their explicit expressions are given in Ref. [37].
where the variable ξ = x −x = 2 x − 1. The normalization conditions are Other parameters are expressed as [37]: The Gegenbauer polynomials are written as follows.
wB =0.43GeV The curves of the normalized DAs φ + B (x, 0) and φ − B (x, 0) for B meson in Eq.(21) and Eq.(23) are displayed in Fig.1. It can be clearly seen from Fig.1 that (1) DAs φ ± B are very asymmetric, and peak at small x region. This fact is generally consistent with the plausible suspicion that the light quark shares a small momentum fraction in B meson. In addition, DAs φ ± B vanish as x → 1, and thus offer a natural cutoff on the seemingly counterintuitive contributions from large x domain. (2) φ − B and φ B2 do not vanish as x → 0, thus the integral dx φ B2 x and dx φ B2 x 2 corresponding to the factorizable emission topologies (form factors) diverge at the endpoint x = 0, as discussed in Ref. [29] with the collinear approximation.
This implies that, on the one hand, the contributions of φ B2 might be important at small x regions and should be given due consideration in calculation, although φ − B is sub-leading twist; on the other hand, it seems reasonable and necessary to retain the contributions of the transverse momentum to regulate the singularities at the endpoint with the pQCD approach. The SU(3) breaking effects on kaonic DAs are considered. The quark-mass corrections modify the asymptotic behaviors of φ p,t M and induce the logarithmic endpoint singularities, as analyzed in Ref. [37].
With the above mesonic DAs, we can obtain the hadron transition formfactors and amplitudes of the B → P P decays with the pQCD approach. There are some conventions in our calculation. In the rest frame of B meson, the light-cone kinematic variables of participating particles in the heavy quark limit are defined as follows.
where k 1 and x 1 are respectively the momentum and longitudinal momentum fraction of light quark in the B meson; k 2,3 and x 2,3 are respectively the momentum and longitudinal momentum fraction of anti-quark in final hadrons. k iT is the transverse momentum. It is The formfactors for the B → P transition are defined as [38] Gegenbauer moments at the scale of µ = 1 GeV [37] a π 1 = 0, a π 2 = 0.25 (15), (15).
The lowest order Feynman diagrams for the B → M transition formfactors are shown in Fig.3. The formfactors F i are written as the convolution integrals of the quark scattering amplitudes T and hadron WFs Φ i with the pQCD approach.
where b i is the conjugate variable of transverse momentum k iT . The Sudakov factors e −s B and e −s M are introduced for WFs Φ B and Φ M , respectively. The Sudakov factor is a char-acteristic element and highly recommended by the pQCD approach to effectively regulate the nonperturbative contributions, so that a dominant share of formfactor would come from hard gluon exchange, and the perturbative calculation would be reasonable and practicable.
The expressions for formfactors including the φ B2 contributions are listed in Appendix A.
Our results of formfactors are shown in Fig. 4, 5, 6 and Table II.  The dependences of formfactor F B→π 0 (0) on some input parameters are shown in Fig.4.
It is seen clearly that (1) formfactors obtained with the pQCD approach are sensitive to the shape parameter ω B for B mesonic WFs. This phenomenon is basically analogical with that of Ref. [31].
(2) the effects of the chiral mass µ M indicate the importances of the twist-3 contributions. It is shown in Ref. [31] that the contributions from twist-3 φ p,t π to     Fig.3 (a) and Fig.3  share of formfactors is from B mesonic WFs φ B1 , and the share of φ B2 is relatively small. This is why the contributions from B mesonic WFs φ B2 were usually not considered in most of previous works. Our results in Table II show that the contributions from φ B2 to formfactors F B→π 0 (0) and F B→K 0 (0) are about 17%, which is much larger than 7% from the next-to-leading order (NLO) contributions [40]. (3) More than half of the formfactors is from the contributions of topology Fig. 3 (a), about 67% shown in Table II. In Fig. 6, more details about the contributions from φ B1 and φ B2 , from topology Fig. 3 (a) and (b) to formfactor F B→π 0 (0) at q 2 = 0 are displayed bin by bin with respect to the distributions of α s π . It shows that about 90% of formfactor comes from the region of α s π ≤ 0.2, where the contributions from φ B1 and φ B2 account for more than 70% and 15%, the contributions from Fig. 3 (a) and (b) account for more than 55% and 30%, respectively.
These results may imply that the quark scattering amplitudes are dominated by hard gluon The values of formfactors in Table II are less than those of Refs. [30,31], due to different DAs models and different values of input parameters. As is shown in Fig. 4, the formfactors decrease with the increase of shape parameter ω B . A large shape parameter ω B for B mesonic WFs is uesd in our calculation, compared with that in Ref. [31]. It should be pointed out that a relatively small value of formfactor F B→π 0 (0) has recently been obtained by fitting the Bourrely-Lellouch-Caprini parametrization [41] with the available experimental data and theoretical informations and then extrapolating to the point of q 2 = 0, for example, 0.254 +0.023 −0.022 in Ref. [41] a , 0.248±0.082 in Ref. [42], 0.20±0.14 in Ref. [43], 0.254±0.081 in Ref. [44]. Our results of F B→π 0 (0) are basically consistent those of Refs. [41][42][43][44] within uncertainties. In addition, from the definition of formfactor in Eq. (40), it is clear that there should be a relation between formfactors and decay constants, The numbers in Table II hold this relation well. The small violation arises from the SU (3) flavor breaking effects.
The Feynman diagrams for two-body nonleptonic B meson decays are shown in Fig. 7.
The amplitudes A with the pQCD approach are usually divided into three parts : the shortdistance contributions encoded in the Wilson coefficients C i , the quark scattering amplitudes a F B→π T i , and hadron WFs Φ i . The general form of decay amplitude is In the rest frame of the B meson, the CP -averaged branching ratios are defined as : where τ B is the lifetime of the B meson. p cm is the common momentum of final states. The decay amplitudes including the φ B2 contributions are listed in Appendix B. For the charged B u meson decays, the CP violating asymmetries arises from the interference between tree and penguin amplitudes. The direct CP violating asymmetry is defined as follows.
For the neutral B d meson decays into final state f with f =f , the time-dependent CP violating asymmetry is defined as follows.   The numerical results on the CP -averaged branching ratios together with experimental data are presented in Table III and IV, CP asymmetries in Table V and VI. Using the minimum χ 2 method, three optimal scenarios (S1, S2 and S3) of parameters ω B and µ M are obtained when the contributions of φ B2 are considered. For the ten concerned B decay modes, χ 2 /d.o.f =  Table. III.   Fig. 8. The followings are our comments.
(1) From Table III and IV, it is seen that the contributions of WF φ B2 are more than 25% of total branching ratios, except for the pure annihilation B 0 → K + K − decay. That is because WF φ B2 contributes nothing to the factorizable annihilation amplitudes of Eqs.(C13)-(C18). From Table V and VI, it is seen that the contributions of WF φ B2 result in a small  Table III. reduction of direct CP asymmetries.
(2) From appendix B, it is clearly seen that for the B → ππ and KK decays, the CKM factors of the tree and penguin amplitudes are respectively V ub V * ud and V tb V * td , and have the same order of magnitude ∝ λ 3 . For the B → πK decays, the tree amplitudes being proportional to the CKM factor V ub V * us are suppressed by λ 2 compared with the penguin  Table III.  amplitudes being proportional to the CKM factor V tb V * ts . In addition, the theoretical and experimental results in Table III and IV show that branching ratios for B → πK decays are in general larger than those for B → ππ and KK decays. These facts confirm previous studies [6,10] that penguin contributions are dynamically enhanced and essential for explaining the B → πK decays. What's more, our studies show that the nonfactorizable annihilation amplitudes mainly from WF φ B1 rather than φ B2 provide large strong phases for the B → πK decays, as analyzed in Ref. [10].
(3) From Fig. 8, it is seen that (i) for the B → πK decays, when the contributions of WF φ B2 are included, theoretical results of branching ratio can give a satisfactory explanation on experimental data. Compared the numbers in Table III and IV with the NLO results of Refs. [11,12,45] (see Table VII), it is seen that the contributions of φ B2 to branching ratios at the leading order (LO) is roughly equivalent to the NLO corrections without the participation of φ B2 . (ii) The consideration of WF φ B2 cannot well settle the so-called CP asymmetries "Kπ" puzzle, i.e. the discrepancy between theoretical and experimental results The studies of Refs. [12,45] showed that the NLO corrections including the glauber effects could flip the sign of A CP (B − →π 0 K − ). It should be noted that the NLO and NLOG theoretical uncertainties of branching ratios are still large, and the current measurement accuracy of A CP (B − →π 0 K − ) needs to be improved.
(4) From Fig. 8 and Table III and IV, it is seen that for the B → ππ decays, the pQCD results of branching ratios deviate from the current experimental measurement. The contributions of WF φ B2 can enhance the branching ratios and reduce these deviations.
Compared the numbers in Table III and IV with the NLO results of Refs. [11,12,46,47] (see Table VIII), it is seen that (i) for the B 0 → ππ decays, the LO contributions of φ B2 to branching ratios is roughly equivalent to the NLO corrections without the participation of φ B2 . (ii) Besides the large theoretical uncertainties, the pQCD results, including either WF φ B2 or the NLO contributions, cannot well explain data on branching ratio for the B 0 → π 0 π 0 decay and CP asymmetries for the B 0 → π + π − decay.
The glauber phases S e = S e1 = S e2 = − π 2 is assumed.  Table VI, there is a particularly interesting phenomenon that the direct CP asymmetries C f is in general larger than the mixing-induced CP asymmetries S f for the B → P P decays, but the opposite is true for the pure annihilation In summary, the B mesonic WF φ B2 can contribute to emission amplitudes and nonfactorizable annihilation amplitudes with the pQCD approach. The enhancements from φ B2 to hadronic transition formfactors and branching ratios for the nonleptonic B → P P decays are comparable with those from the NLO corrections without taking the φ B2 into account.
The relations among formfactors are Using the pQCDF formula of Eq.(41), the formfactors can be written as follows.
where N c = 3 is the color number. The color factor . The parameterization of S t (x) can be found in Ref. [27]. Other parameters are written as follows.
where I 0 and K 0 are Bessel functions. The expression of s(x, b, Q) can be found in Ref. [23].
γ q = − α s π is the quark anomalous dimension. α g and β a(b) are the virtualities of gluon and quarks, respectively. The amplitude of B meson nonleptonic weak decay is written as where H eff is given in Eq.(1).