A Combined Fit on the Annihilation Corrections in $B_{u,d,s}$ $\to$ $PP$ Decays Within QCDF

Motivated by the possible large annihilation contributions implied by recent CDF and LHCb measurements on nonleptonic annihilation B meson decays, and the refined experimental measurements on hadronic B meson decays, we study the strength of annihilation contributions within QCD factorization (QCDF) in this paper. With the available measurements of two-body B_{u,d,s} ->pi pi, pi K, K K decays, a comprehensive fit on the phenomenological parameters X_A^{i,f} (or rho_A^{i,f} and phi_A^{i,f}) which are used to parameterize the endpoint singularity in annihilation amplitudes is performed with the statistical chi^2 approach. It is found that (1) flavor symmetry breaking effects are hardly to be distinguished between X_{A,s}^i and X_{A,d}^i due to the large experimental errors and theoretical uncertainties, where X_{A,s}^i and X_{A,d}^i are related to the nonfactorization annihilation contributions in B_s and B_{u,d} decays, respectively. So X_{A,s}^i = X_{A,d}^i is a good approximation by now. (2) In principle, parameter X_{A}^f which is related to the factorization annihilation contributions and independent of the initial state can be regarded as the same variable for B_{u,d,s} decays. (3) Numerically, two solutions are found, one is (rho_A^i, phi_A^i) = (2.98^+1.12_-0.86,-105^+34_-24) and (rho_A^f, phi_A^f) = (1.18^+0.20_-0.23,-40^+11_-8), the other is (rho_A^i, phi_A^i) = (2.97^+1.19_-0.90,-105^+32_-24) and (rho_A^f, phi_A^f) = (2.80^+0.25_-0.21,165^+4_-3). Obviously, nonfactorization annihilation parameter X_A^i is generally unequal to factorization annihilation parameter X_A^f, which differ from the traditional treatment. With the fitted parameters, all results for observables of B_{u,d,s} ->pi pi, pi K, K K decays are in good agreement with experimental data.

Theoretically, the branching ratios of pure annihilation nonleptonic B meson decays are formally Λ QCD /m b power suppressed and expected at 10 −7 level, which roughly agrees with the measurements. In the framework of QCD factorization (QCDF) [7], the annihilation amplitudes, together with the chirally enhanced power corrections and possible large strong phase involved in them, play an important role in evaluating the observables of B meson decays. However, due to the endpoint singularities, the amplitudes of annihilation topologies are hardly to be exactly calculated. To estimate the endpoint contributions, phenomenological parameter X A is introduced [8] as where Λ h = 0.5GeV. The QCDF approach itself cannot give some information/or constraint on parameters ρ A and φ A . To simplify the calculation, one usually takes the same parameters ρ A and φ A for factorizable and nonfactorizable annihilation topologies. And as a conservative choice, the values of ρ A ∼ 1 and φ A ∼ −55 • (named scenario S4) [8][9][10] are usually adopted in previous studies on B u,d,s → P P decays, which leads to the prediction 1 B(B d →K + K − ) = (0.10 +0.03+0.03 −0.02−0.03 )×10 −6 [9] and B(B s →π + π − ) = (0.26 +0.00+0.10 −0.00−0.09 ) × 10 −6 [10]. Clearly, the QCDF's prediction on B(B d →K + K − ) agrees well with the current measurements considering the experimental and theoretical errors, while the QCDF's prediction on B(B s →π + π − ) is much smaller than the experimental data Eq.(2) by about 3σ, which implies unexpectedly possible large annihilation corrections and possible large flavor symmetry 1 The second uncertainty comes from parameters ρ A,H and φ A,H . breaking effects between the annihilation amplitudes of B u,d and B s decays [4,5]. Motivated by such mismatch, some works have been done for possible solutions and implications.
Within the QCDF framework, using the asymptotic light-cone distribution amplitudes, the building blocks of annihilation amplitudes are simplified as [7,8] where the superscripts "i" and "f " refer to gluon emission from the initial-and final-states, respectively; the subscripts "1", "2" and "3" correspond to three possible Dirac structures, The explicit expressions of effective annihilation coefficients could be found in Ref. [7,8].
For the annihilation parameters X A in Eqs. (4)(5)(6), although there are no imperative and a priori reasons for it to be the same in the building blocks A i,f k (k = 1, 2, 3), the simplification X i A = X f A = X A is commonly used in many previous works of nonleptonic B decays [7][8][9][10][11], independent of mesons involved and topologies. However, the carefully renewed study in Refs. [5,6] shows that it is hardly to accommodate all available observables of charmless B → P P decays simultaneously with the universal ρ A and φ A . Recently, a refreshing suggestion was proposed in Refs. [4,5] to cope with the parameters X A . The main points of "new treatment" could be briefly summarized as follow:  [8,9] for B(B d →K + K − ), it seems to imply unexpectedly large flavor symmetry breaking effects, then the predictive power of QCD will be rather limited. Thanks to the large ex- [5,13]. Therefore, flavor symmetry might be restored as both aforementioned decays could be accommodated by a common set of (ρ i A , φ i A ). It is interesting and essential to systematically evaluate the exact strength of annihilation contribution and further test the aforementioned points, especially the flavor asymmetry effects.
As it is well known, additional phenomenological parameters X H (or ρ H and φ H ), like to Eq.(3), were introduced to regulate the endpoint singularity in the hard spectator scattering (HSS) corrections involving the twist-3 light cone distribution amplitudes of light final states [7][8][9][10][11]. The phenomenological importance of HSS corrections to the color-suppressed tree contributions which are enhanced by the large Wilson coefficient C 1 has already been recognized by Refs. [9,12,13] in explicating the current experimental measurements on ∆A CP = . Because the B wave functions are also involved in the HSS convolution integral, the flavor symmetry breaking effects might be also embodied in parameter X H .
Following the ansatz in Ref. [4,5], we preform a global fit on the annihilation parameters combining available experimental data on B u,d,s → ππ, πK, KK decays with a statistical χ 2 analysis. Based on our previous analysis [13], the approximation, , is acceptable by current measurements on B u,d decays (see scenario III in Ref. [13] for details), which lessens effectively the unknown variables. Hence, the approximation X H = X i A is assumed for B u,d,s decays in the following analysis. The detailed explanation on the fitting approach could be found in the Appendix C of Ref. [13]. The values of input parameters used in our evaluations are summarized in Table I.
a π 1 = 0, a π 2 (2 GeV) = 0.17, a K 1 (2 GeV) = 0.05, a K 2 (2 GeV) = 0.17 [18] : 68 C. L. : 95 C. L. Firstly, to clarify the flavor symmetry breaking effects on parameters X i,f A , we perform a fit on the annihilation parameters (ρ i,f A , φ i,f A ) for B u,d and B s decays, respectively. For parameters of (ρ i,f A,s , φ i,f A,s ), the constraints come from observables of theB s → π − K + , π + π − , K + K − decays. The fitted results are shown in Fig.1. For parameters of (ρ i,f A,d , φ i,f A,d ), they have been fitted with the constraints from B u,d → πK, ππ, KK decays, especially, focusing on the so-called "πK" and "ππ" puzzles (see Ref. [13] for the details). Their allowed regions (green points) at 68% C.L. are also shown in Fig.1 for a comparison with (ρ Fig.1(a), it is seen clearly that (1) the region of (ρ i A,s , φ i A,s ) cannot be seriously constrained by now, because the current measurements on B s → π − K + , π + π − , K + K − decays are not accurate enough and the theoretical uncertainties are also still large. Moreover, a relatively large ρ i A,s ∼ 3 with φ i A,s ∼ ±100 • in B s system suggested by recent studies [4][5][6] is allowed. (2) The conventional choice of ρ i A,d ∼ 1 and φ i A,d ∼ −55 • [8-10] is ruled out, because the assumption X H = X i A is used in our study to enhance the magnitude and the strong phase of the color-suppressed tree amplitude C via spectator interactions and to solve both "πK" and "ππ" puzzles [13]. Besides, a relatively large ρ which has large experimental error and theoretical uncertainties until now, and is also consistent with Fig.7(a) of Ref. [19] for B d → K + K − decays using the similar statistical fit approach with parameters X i A = X f A . (3) The allowed regions of (ρ i A,d , φ i A,d ) can still overlap with the ones of (ρ i A,s , φ i A,s ) in part, around (3, −100 • ), which implies that the treatment X i A,s = X i A,d from flavor symmetry breaking effects in Ref. [4,5] is not absolutely sure, at least not necessary with current experimental and theoretical precision.
From Fig.1(b), it is seen clearly that (1) Fig.1 (a) with (b), it is seen that (1) generally, the allowed region of (ρ f A , φ f A ) is different from that of (ρ i A , φ i A ), and X i A is not always required to be equal to X f A . So, the "new treatment" on parameters X A according to either factorizable or nonfactorizable annihilation topologies may be reasonable and appropriate for B u,d,s decays.
(2) The flavor symmetry breaking effects on parameters X i,f A could be very small even negligible under the existing circumstances with less available experimental constraints from B s decays.    Based on the above analyses and discussions, we present the most simplified (flaour conserving) scenario for the annihilation parameters that both (ρ f A , φ f A ) and (ρ i A , φ i A ) are universal for both B u,d → P P and B s → P P decay modes to lessen phenomenological parameters, where X i and X f are independent variables. To get their exact values, we perform a global fit by combining available experimental data for B u,d,s → ππ, πK, KK decays, which involve 16 decay modes and 42 observables. In our fit, besides of (ρ i,f A , φ i,f A ), the inverse moment λ B , which is used to parameterize integral of the B meson distribution amplitude and a hot topic by now (see Ref. [20] for details), is also treated as a free parameter and taken into account. We present the allowed parameter spaces in Fig.2 and the corresponding numerical results in Table II. where the first and second theoretical errors are caused by uncertainties of the CKM and the other parameters (including the quark masses, decay constants and form factors) listed in Table I, respectively.

Decay mode
Exp. data This work Cheng [10] B s → π − K + 5.4 ± 0.   As Fig. 2 shows, the allowed spaces of (ρ i,f A , φ i,f A ) and λ B are strongly restricted by combined constraints from B u,d,s → ππ, πK, KK decays, especially for (ρ f A , φ f A ). There are two solutions (named solution A and B, respectively). It is easily found that the allowed regions and the best-fit point of (ρ i A , φ i A ) are so alike that one can hardly distinguish one from these two solutions. For each solution, there is no common overlap at 68% C.L. between the allowed regions of (ρ i A , φ i A ) and (ρ f A , φ f A ) , i.e., the nonfactorizable and factorizable annihilation parameters X i A and X f A should be treated as independent parameters, which confirms the suggestion of Ref. [4,5]. Numerically, as listed in Table II, the fitted result is similar to, but with smaller uncertainties, the results in Ref. [13] where the B s decay modes are not considered. In fact, the two sets of parameters values give the same annihilation contributions. From Table II, it can be seen that a relatively small value of λ B ∼ 0.2 GeV which has been found by, for instance, Refs. [8,10,13,20] and a relatively large value of ρ H ∼ 3 with φ H ∼ −105 • are favored in the phenomenological aspect of B nonleptonic decays.
They will enable the HSS corrections to play an important role in evaluating observables of penguin dominated B → πK decays, and have significant enhancement, assisted with the large Wilson coefficient C 1 , to the color-suppressed tree amplitude with a large strong phase. As noticed and discussed in Refs. [7,8,21], the vertex corrections, including NLO and NNLO contributions, to the color suppressed tree coefficient α 2 exhibit a serious cancellation of the real part of α 2 (for example, see the first line of Eq.(54) in Ref. [21]), but the HSS mechanism can compensate for the destructive interference and enhance the α 2 with a large magnitude. The value of α 2 (ππ) 0.24 − i 0.08 0.25 e −i 18 • including NNLO vertex and HSS corrections [21] obtained with ρ H = 0 and λ B ∼ 0.35 GeV still cannot accommodate the experimental data on branching ratio B d → π 0 π 0 decay. So a relatively large HSS corrections arising from X H might be a crucial key for the "ππ puzzle". The branching rate of B d → π 0 π 0 decay and the CP asymmetry of B u →π 0 K ± decay, they both are sensitive to the choice of coefficient α 2 , and can provide substantial constraints on parameter X H .
With the best-fit values of both (ρ H , φ H ) and λ B in this analysis, one can get α 2 (ππ) 0.28 − i 0.49 0.56 e −i 60 • , which provides a possible solution to the so-called "ππ and πK" puzzles simultaneously. Of course, one can have different mechanism for enhancement of the α 2 in QCDF, for example, the final-state rescattering effect 2 advocated in Ref. [9] and the Principle of Maximum Conformality proposed recently in Ref. [22], where the allowed regions for parameters ρ A,H and φ A,H might be different.
With the inputs in Table I  with the traditional treatment [10]. And B(B s →π + π − ) = (0.61 +0.02+0.07 −0.04−0.06 )×10 −6 is in good agreement with the data within one experimental error. Meanwhile, our result B(B s →π 0 π 0 ) = (0.31 +0.01+0.03 −0.02−0.03 )×0 −6 is twice as large as the traditional result (0.13 +0.0+0.05 −0.0−0.05 ) × 10 −6 . Moreover, there are also some other differences between the two sets of theoretical results more or less. So, the future accurate measurements on the nonleptonic B s meson decays would be helpful to probe the annihilation contributions and to explore the underlying dynamical mechanism.
In summary, we studied the nonfactorizable and factorizable annihilation contributions to B u,d,s → ππ, πK, KK decays with QCDF approach. To clarify the independence of annihilation parameters X i A and X f A and the possible flavor symmetry breaking effects therein, a statistical χ 2 analysis is performed for nonleptonic B u,d and B s decays. It is found that (1) X i A and X f A are independent parameters, which differs from the traditional treatment with annihilation parameters and verifies the proposal of Ref. [4]. (2)  following years, so more information about annihilation contributions could be revealed.