Constraints on hard spectator scattering and annihilation corrections in $B_{u,d}$ ${\to}$ $PV$ decays within QCD factorization

In this paper, we investigate the contributions of hard spectator scattering and annihilation in $B\to PV$ decays within the QCD factorization framework. With available experimental data on $B\to \pi K^{\ast}$, $\rho K$, $\pi \rho$ and $K\phi$ decays, comprehensive $\chi^2$ analyses of the parameters $X_{A,H}^{i,f}({\rho}_{A,H}^{i,f},{\phi}_{A,H}^{i,f})$ are performed, where $X_A^f$ ($X_A^i$) and $X_H$ are used to parameterize the endpoint divergences of the (non)factorizable annihilation and hard spectator scattering amplitudes, respectively. Based on $\chi^2$ analyses, it is observed that (1) The topology-dependent parameterization scheme is feasible for $B\to PV$ decays; (2) At the current accuracy of experimental measurements and theoretical evaluations, $X_H=X_A^i$ is allowed by $B\to PV$ decays, but $X_{H}\neq X_A^f$ at $68%$ C. L.; (3) With the simplification $X_H=X_A^i$, parameters $X_A^f$ and $X_A^i$ should be treated individually. The above-described findings are very similar to those obtained from $B\to PP$ decays. Numerically, for $B\to PV$ decays, we obtain $(\rho_{A,H}^i,\phi_{A,H}^i[^{\circ}]) =(2.87^{+0.66}_{-1.95}, -145^{+14}_{-21})$ and $(\rho_A^f,\phi_A^f[^{\circ}]) = (0.91^{+0.12}_{-0.13}, -37^{+10}_{-9})$ at $68%$ C. L.. With the best-fit values, most of the theoretical results are in good agreement with the experimental data within errors. However, significant corrections to the color-suppressed tree amplitude $\alpha_2$ related to a large $\rho_H$ result in the wrong sign for $A^{dir}_{CP}(B^- \to \pi^0 K^{{\ast}-})$ compared with the most recent BABAR data, which presents a new obstacle in solving"$\pi\pi$"and"$\pi K$"puzzles through $\alpha_2$. A crosscheck with measurements at Belle (or Belle II) and LHCb, which offer higher precision, is urgently expected to confirm or refute such possible mismatch.

Nonleptonic decays of hadrons containing a heavy quark play an important role in testing the Standard Model (SM) picture of the CP violation mechanism in flavor physics, improving our understanding of nonperturbative and perturbative QCD and exploring new physics beyond the SM. For charmless B meson decays, experimental studies have been successfully carrying out at B factories ( BABAR and Belle) and Tevatron ( CDF and D0) in the past and will be continued by running LHCb and upgrading Belle II experiments. These experiments provide highly fertile ground for theoretical studies and have yielded many exciting and important results, such as measurements of pure annihilation B s → π + π − and B d → K + K − decays reported recently by CDF, LHCb and Belle [1][2][3], which may suggest the existence of unexpected large annihilation contributions and have attracted much attention, for instance, Refs. [4][5][6][7][8][9].
Theoretically, to calculate the hadronic matrix elements of hadronic B weak decays, some approaches, including QCD factorization (QCDF) [10], perturbative QCD (pQCD) [11] and soft-collinear effective theory (SCET) [12], have been fully developed and extensively employed in recent years. Even though the annihilation contributions are formally power suppressed in the heavy quark limit, they may be numerically important for realistic hadronic B decays, particularly for pure annihilation processes and direct CP asymmetries.
Unfortunately, in the collinear factorization approximation, the calculation of annihilation corrections always suffers from end-point divergence. In the pQCD approach, such divergence is regulated by introducing the parton transverse momentum k T and the Sudakov factor at the expense of modeling the additional k T dependence of meson wave functions, and large complex annihilation corrections are presented [13]. In the SCET approach, such divergence is removed by separating the physics at different momentum scales and using zero-bin subtraction to avoid double counting the soft degrees of freedom [14,15]; thus, the annihilation diagrams are factorable but real to the leading power term of O(α s (m b )Λ QCD /m b ).
The absence of strong phases from SCET's annihilation amplitudes differs with the pQCD's estimation and the QCDF expectation [16].
Within the QCDF framework, to estimate the annihilation amplitudes and regulate the endpoint divergency, the logarithmically divergent integral is usually parameterized in a model-independent manner [16] and explicitly expressed as with the typical scale Λ h = 0.5 GeV. Moreover, a similar endpoint singularity also appears in the hard spectator scattering (HSS) contributions of higher twist distribution amplitudes that are also formally power suppressed but chirally enhanced; therefore, a similar parameterization ansatz is used to cope with HSS endpoint divergency, and quantity X H (ρ H , φ H ), similar to the definition of Eq.(1), is introduced. As discussed in Ref. [16], is expected because the effects of the intrinsic transverse momentum and off-shellness of partons would be to modify x → x + with ∼ O(Λ QCD /m b ) in the denominator of Eq.(1). The factor (1 + ρe iφ ) summarizes the remaining unknown nonperturbative contributions, where φ, which is related to the strong phase, is important for direct CP asymmetries. In such a parameterization scheme, even though the predictive power of QCDF is partly weakened due to the incalculable parameters ρ and φ that are introduced, it also provides a feasible way to evaluate the effects and the behavior of annihilation and HSS corrections from a phenomenological view point, which is helpful for understanding and exploring possible underlying mechanisms.
Although the magnitude of and constraints on parameter ρ are utterly unknown based on the first principles of QCD dynamics for now, an excessively large value of ρ would significantly enhance the subleading 1/m b contributions, and hence, a conservative choice of ρ A ∼ 1 has typically been used in previous phenomenological studies [16][17][18][19]. In practice, different values of (ρ A , φ A ) chosen according to various B meson decay types (P P , P V , V P and V V ) have been used to fit experimental data [16,19]. However, with the favored "Scenario S4", in which ρ A 1 and φ A −55 • [16] for B → P P decay, the QCDF prediction B(B s →π + π − ) = (0.26 +0.00+0.10 −0.00−0.09 )×10 −6 [19] is about 3.4σ less than the experimental data (0.73±0.14)×10 −6 [20].
Motivated by this possible mismatch, detailed analyses have been performed within the QCDF framework [6][7][8][9]. In Refs. [6,7], a "new treatment" for endpoint parameters is presented in which the flavor dependence of the annihilation parameter X A on the initial states should be carefully considered, and hence, X A is divided into two independent parameters X i A and X f A , which are responsible for parameterizing the endpoint divergences of nonfactorizable and factorizable annihilation topologies, respectively. Following the proposal of Refs. [6,7] and combining available experimental data for B u,d,s → πK, ππ and KK decays, the comprehensive χ 2 analyses of X i,f A and X H in B → P P decays were performed in Refs. [8,21]. It was found that • Theoretically, there is neither a compulsory constraint nor a priori reason for both X i A = X f A = X A and X A being universal for all hadronic B decays; Phenomenologically, it is required by available measurements regarding B → P P decays that X i A and X f A should be treated individually; in addition, the simplification X H = X i A is allowed by data, which effectively reduces the number of unknown variables, but X H = X f A (see scenario III in Ref. [8] for detail); • The effect of flavor symmetry breaking on parameter X i,f A is tiny and negligible for the moment due to large experimental errors and theoretical uncertainties; • A slightly large ρ H ∼ 3 with φ H ∼ −105 • and a relatively small inverse moment parameter λ B ∼ 200 MeV for B meson wave functions are required to enhance the color-suppressed coefficients α 2 with a large strong phase, which is important in accommodating all available observables of B u,d,s → πK, ππ and KK decays simultaneously, even the so-called "πK" and "ππ" puzzles (see Refs. [8,21] for detail); • Numerically, in the most simplified scenario in which X H = X i A is assumed, combining the constraints from B u,d,s → πK, ππ and KK decays, two solutions responsible for B → P P decays are obtained [21], Solution A : Solution B : which yield similar HSS and annihilation contributions.
In recent years, many measurements of B → P V decays have been performed anew at higher precision [20]. Thus, with the available experimental data, it is worth reexamining the agreement between QCDF's predictions and experimental data on B → P V decays, investigating the effects of HSS and annihilation contributions, and further testing whether the aforementioned findings regarding B → P P decays still persist in B → P V decays. In this paper, we would like to extend our previous studies on B → P P decays [8,21] to B → P V decays with the same χ 2 fit method and similar treatment of annihilation and HSS parameters; the details of the statistical χ 2 approach can be found in the appendix of Refs. [8,22].
For B → P V decays, the decay amplitudes and relevant formulae have been clearly listed in Ref. [16]. The parameters X i,f A under discussion appear in the basic building blocks of annihilation amplitudes, which can be explicitly written as follows [16]: and change the sign of A f 3 . Further explanation and information on QCDF's annihilation amplitudes can be found in Ref. [16].
Before entering further discussion, we would like to note the following: (1) In previous studies, the annihilation parameters were assumed to be process-dependent [16][17][18][19] where were introduced to describe nonleptonic B decay into the final states P V and V P decays, respectively; sometimes, additional values of (ρ A , φ A ) for B → Kφ decays [18] were required. In our analysis, parameters (ρ i A , φ i A ) and (ρ f A , φ f A ) are topology-dependent. (2) As discussed in Refs. [6][7][8], In this paper, only B u,d → P V decays are considered (most B s → P V decays have not been measured), and the isospin symmetry is assumed to be held. (4) Unlike in the case of B → P P decays, in which both final states are pseudoscalar mesons, the wave functions of the vector mesons are also required to evaluate the hadronic matrix elements of B → P V decays. Therefore, following the treatment of annihilation parameters presented in Refs. [16][17][18][19], the parameters for B → P V decays are generally different from those for B → P P decays. As is well known, for the b → s transition, the tree contributions are strongly suppressed by the CKM factor |V * us V ub | ∼ O(λ 4 ), whereas the penguin contributions are proportional to the CKM factor |V * cs V cb | ∼ O(λ 2 ) [23]. In addition, the nonfactorizable contributions between vertex and HSS corrections largely cancel each other out [16]. Therefore, the weak annihilation amplitudes are important for the b → s nonleptonic B decays. Large annihilation contributions are derived from the coefficient b 3 , because b 3 is proportional to the CKM factor |V * cs V cb | and sensitive to the annihilation building block A f 3 , which is always accompanied by N c C 6 . Hence, it is expected that precise observables of b → s nonleptonic . For the b → d transition, the tree contributions are dominant if they exist, whereas the penguin contributions are suppressed due to the cancellation between the CKM factor V * ud V ub and V * cd V cb [23]. Large annihilation contributions are derived from the coefficient b 1,2 , which is always accompanied by large Wilson coefficients C 1,2 . For color-suppressed tree-dominated hadronic B decays, the contributions of HSS and factorizable annihilation corrections are particularly important, for example, the resolution of the so-called "ππ" puzzle [8]. Therefore, severe restrictions on parameters X i A,H could be derived from many precise observables of the b → d nonleptonic B decays.
The decay modes considered in this paper include the penguin-dominated B → πK * , ρK decays induced by the b → sqq (q = u, d) transition, the penguin-dominated B → φK decays induced by the b → sss transition, the tree-dominated B → πρ decays induced by the b → dqq transition, and the penguin-and annihilation-dominated B → KK * decays induced by the b → dss transition. For the observables of the above-mentioned decay modes, the available experimental data are summarized in the "Exp." columns of Tables I, II and III, in which most of data are the averaged results given by HFAG [20], except for the branching fractions and direct CP asymmetries of B − → π −K * 0 , π 0 K * − andK 0 ρ − decays. Recently, using the full dataset of 470.9 ± 2.8 million BB events, the BABAR collaboration reported the latest results from an analysis of B + → K 0 π + π 0 (and the combined results from this and previous BABAR analyses) [24] B − →π −K * 0 : in which, in particular, the first evidence of a CP asymmetry of B − →π 0 K * − is observed at the 3.4σ significance level. In our following analysis, such (combined) results for B − →π 0 K * − andK 0 ρ − decays in Eqs. (9) and (10) are used. For B − →π −K * 0 decay, its branching fractions and direct CP asymmetry are also measured by Belle collaboration [25]; therefore, we adopt the weighted averages of observables, which are presented in Table I.
The data listed in Tables I, II and III demonstrate that the first three sets of decay modes are well measured; therefore, experimental data of these decay modes are used in our fitting.
In addition, the theoretical inputs are summarized in the Appendix. Our following analyses and fitting are divided into three cases for different purposes.
(1) For case I, five parameters, (ρ i,f A , φ i,f A ) and λ B , are treated as free parameters, and the simplification X H = X i A , which is allowed in B → P P decays [8], is assumed. Moreover, the constraints from B → πK * , ρK decays, B → πρ decays, and B → φK decays are considered separately. The fitted results are shown in Fig. 1. Eq. (3)). Moreover, these two regions overlap with the blue and green dotted regions, which implies that the two solutions of (ρ f A , φ f A ) are also allowed by B → πρ, φK decays. As shown in Fig. 1 (b), under the constraints from B → πρ decays, the parameters (ρ i A , φ i A ) are loosely restricted into two wide bands (blue points) around φ i A ∼ −130 • and ∼ −300 • because the experimental precision of the observables, especially the direct CP asymmetries, on B → πρ decays is still very rough. Under the constraints from B → πK * and ρK decays, (ρ i A , φ i A ) are restricted around φ i A ∼ −200 • (red points) and overlap partly with the blue pointed region, which implies that the allowed spaces of (ρ i A , φ i A ) would be seriously shrunken under the combined constraints.
From Fig. 1 (c), parameter λ B cannot be determined exclusively, although an additional phenomenological condition 115 MeV ≤ λ B ≤ 600 MeV is imposed during our fit based on the studies of Refs. [16,[26][27][28][29][30]. In principle, parameter λ B is only related to the B wave function and independent of any decay modes. Therefore, in our following analyses, the result λ B = 0.19 +0.09 −0.04 GeV fitted from B → P P decays [21] will be adopted.

Case II
: (2) For Case II, to determine whether the simplification are treated as free parameters. Combining all available constraints from B → πK * , ρK, πρ, φK decays, the allowed parameter spaces at 68% C.L. are shown in Fig. 2. Fig. 2 clearly shows that (i) Similarly to Case I, two solutions of (ρ f A , φ f A ) with very small uncertainties (red points) are obtained, which are denoted "solution A" for φ f A ∼ −40 • and "solution B" for φ f A ∼ −200 • for convenience; Meanwhile, the spaces of (ρ i A , φ i A ) are still hardly well bounded (blue points) as in Case I; (ii) The allowed spaces of (ρ f A , φ f A ) are small and tight, whereas those of (ρ i A , φ i A ) are big and loose; thus, they generally differ from each other. This finding implies that X f A and X i A may be treated individually, as in the case for B → P P decays discussed in Refs. [8,21], which provides further evidence to support the speculation regarding the topology-dependent annihilation parameters reported in Ref. [6,7]; (iii) Interestingly, the spaces of (ρ H , φ H ) (green points) are significantly separated from those of (ρ f A , φ f A ) but overlap partly with the regions of (ρ i A , φ i A ), which implies that the simplification X H X i A is roughly allowed for B → P V decays as in the case of B → P P decays [21].

Case III Solution A
: best fit point : 68 C. L. : 95 C. L.  (3) For Case III, based on the above-described analysis, we will present the most simplified scenario with four free parameters, i.e., (ρ f A , φ f A ) and (ρ i A , φ i A ) = (ρ H , φ H ). Combining the constraints from 35 independent observables of B → πK * , ρK, πρ, φK decays, our fitted results are shown in Fig. 3, where "solution A and B" corresponds to the minimal values From Fig. 3, it is observed that (i) Similarly to Case II, the parameters (ρ f A , φ f A ) are severely restricted to two small and tight spaces. (ii) In contrast with Case II, the allowed regions of parameters (ρ i A , φ i A ) at 68% C.L. shrink notably due to the simplification X H = X i A . (iii) The allowed regions of parameters (ρ f A , φ f A ) are completely separated from those of (ρ i A , φ i A ), which implies that the factorizable annihilation parameters X f A should be different from the nonfactorizable annihilation parameters X i A . (iv) The spaces of (ρ i,f A , φ i,f A ) for B → P V decays are separated from the spaces of B → P P decays (yellow points in Fig. 3), which implies that parameters X A for B → P P and P V decays should be introduced and treated individually.
Using the best-fit (central) values of solution A in Eq.(11), we present the theoretical results for the branching fractions and CP asymmetries of B → P V decays in the "Case III" columns of Tables I, II and III. For comparison, the theoretical results of "Scenario S4" , are also listed in the "S4" columns of the tables. It is observed that most of our theoretical results are consistent with the experimental data except for a few contradictions in the B − →π 0 K * − decay, which will be discussed later, and are similar to the "S4" results.
For the well-measured observables, such as the branching ratios B(B→φK), Table I), the direct CP asymmetry forB 0 → π + K * − decay (see Table I) and ∆C for B → π ± ρ ∓ decay (see Table   III) with a significance level ≥ 4σ, compared with the traditional "S4" results, our results are more in line with the experimental data. In particular, compared with the measurement ∆C = (27±6)% for B → π ± ρ ∓ decay, the difference between the "S4" results and ours is clear and notable, which may imply that a relatively large ρ i A,H ∼ 3 rather than the conventionally used small ρ i A ∼ 1 [16][17][18][19] may be necessary for nonfactorizable annihilation corrections. In addition, evidence of a large ρ A for B → Kρ, K * π decays is also presented in Fig. 3 (11), a small ρ H is also allowed at 68% C.L., which would yield a better agreement but result in a relative larger χ 2 ). One interesting and important problem   Here we adopt the same definition of direct CP asymmetry as HFAG [20].
is that a relatively large ρ H ∼ 3 in B → P P decays, which is similar to the best-fit value for B → P V decays in this work, is always required to enhance α 2 contributions in resolving the "ππ" and "πK"puzzles [8,21] but clearly leads to a wrong sign for A dir CP (B − →π 0 K * − ) when confronted with BABAR data, as indicated herein and in Ref. [32]. Therefore, if a large neg- The mixing-induced CP asymmetries (in units of 10 −2 ). The explanation for the uncertainties is the same as that indicated in Table I.

Decay modes
Exp. Case III S4 Here we adopt the same definition of mixing-induced CP asymmetries as HFAG [20].  Table I. Here we adopt the same definition for the parameters C ff , S ff , ∆C ff , ∆S ff and A ff CP as HFAG [20] and choose the final states f = ρ + π − , K * + K − and K * 0K 0 .
ative A dir CP (B − →π 0 K * − ) is confirmed by Belle (or future Belle II) and LHCb collaborations, resolving the "ππ" and "πK" puzzles through color-suppressed tree amplitude α 2 will be challenging. If so, a large complex electroweak amplitude α 3,EW is probably required [32], which may hint possible new physics effects. In addition, the measurements for observables of B s → φπ 0 decay, whose amplitude is related to α 2 and α 3,EW only, may provide a clue even though such decay mode is not easily to be measured soon. [33], and it is eagerly expected that these decays can be precisely measured, which should be useful in probing the annihilation corrections and the corresponding mechanism. Of course, one can use different mechanisms for enhancing the nonfactorizable contributions in QCDF, for example, the final state rescattering effects advocated in Ref. [17][18][19], in which the allowed regions for parameters (ρ i A,H , φ i A,H ) might be different. In summary, we studied the contributions of HSS and annihilation in B → P V decays within the QCDF framework. Unlike the traditional treatment of annihilation endpoint divergence with process-dependent parameters (ρ P V A , φ P V A ) and (ρ V P A , φ V P A ) in previous studies [16][17][18][19], the topology-dependent parameters (ρ i,f A , φ i,f A ) based on a recent analysis of B → P P decays [6][7][8]21] were used in this paper. Combining available experimental data, we performed comprehensive χ 2 analyses of B → P V decays and obtained information and constraints regarding the parameters (ρ i,f A , φ i,f A ). It is observed that most of the measurements on observables of B → P V decays, except for some contradictions in B − →π 0 K * − decay, could be properly interpreted with the best-fit values presented in Eq. (11), which suggests that the topology-dependent parameterization of annihilation and HSS corrections may be suitable. The other findings of this study are summarized as follows: • The relatively small value of the B wave function parameter λ B ∼ 0.2 GeV, which is only related to the universal B wave functions and plays an important role in providing a possible solution to the so-called "ππ" and "πK" puzzles [8], is also allowed by the constraints from B → P V decays.
• As used extensively in phenomenological studies on hadronic B decays [16][17][18][19], generally, parameters X i,f A,H for B → P P and P V decays should be independent of each other and be treated individually.
• The allowed regions of parameters (ρ f A , φ f A ) are strictly constrained by available experimental data, whereas the accessible spaces of parameters (ρ i A , φ i A ) are relatively large. Generally, there is no common space between (ρ f A , φ f A ) and (ρ i A , φ i A ) with the approximation of X i A = X H , which implies that factorizable annihilation parameters X f A should be different from nonfactorizable annihilation parameters X i A . Moreover, a relatively large ρ i A ∼ 3 is required by the considerable fine-tuning of X i,f A to reproduce most of the measurements on hadronic B decays. The above-described evidence and features have been clearly observed in both B → P P decays [6][7][8]21] and B → P V decays.
• Unfortunately, a relatively large ρ H ∼ 3 with φ H ∼ −145 • related to significant HSS corrections to color-suppressed tree amplitude α 2 , which is helpful for resolving the "ππ" and "πK" puzzles and allowed by most B → P P and P V decays, result in a wrong sign for A dir CP (B − →π 0 K * − ) when confronted with recent BABAR data (−39±12)%. This finding suggests a large, complex electroweak amplitude attributed to possibly new physics or an undiscovered mechanism [32], which deserves much attention. Before we know for sure, the crosscheck based on refined measurements conducted at Belle (Belle II) and LHCb is urgently awaited.
Overall, the annihilation and HSS contributions in nonleptonic B decays should be and have been attracting much attention and careful study. For B → P V decays, a comparative advantage is that there are more decay modes and more observables than those for B → P P decays, and hence more information and more stringent constraints on parameters X A,H can be obtained, which represents an opportunity as well as a challenge in the rapid accumulation of data on B events at running LHCb and forthcoming Belle II/SuperKEKB. Theoretically, these results will surely help us to further understand the underlying mechanism of anni- For other inputs, such as the masses and lifetimes of mesons et al., we adopt the values given by PDG [23].