Revisiting Final State Interaction in Charmless $B_q\to P P$ Decays

Various new measurements in charmless $B_{u,d,s}\to PP$ modes, where $P$ is a low lying pseudoscalar meson, are reported by Belle and LHCb. These include the rates of $B^0\to\pi^0\pi^0$, $\eta\pi^0$, $B_s\to\eta'\eta'$, $B^0\to K^+K^-$ and $B^0_s\to\pi^+\pi^-$ decays. Some of these modes are highly suppressed and are among the rarest $B$ decays. Direct CP asymmetries on various modes are constantly updated. It is well known that direct CP asymmetries and rates of suppressed modes are sensitive to final state interaction (FSI). As new measurements are reported and more data will be collected, it is interesting and timely to revisit the rescattering effects in $B_{u,d,s}\to PP$ states. We perform a $\chi^2$ analysis with all available data on CP-averaged rates and CP asymmetries in $\overline B{}_{u,d,s}\to PP$ decays. Our numerical results are compared to data and those from factorization approach. The quality of the fit is improved significantly from the factorization results in the presence of rescattering. The relations on topological amplitudes and rescattering are explored and they help to provide a better understanding of the effects of FSI. As suggested by U(3) symmetry on topological amplitudes and FSI, a vanishing exchange rescattering scenario is considered. The exchange, annihilation, $u$-penguin, $u$-penguin annihilation and some electroweak penguin amplitudes are enhanced significantly via annihilation and total annihilation rescatterings. In particular, the $u$-penguin annihilation amplitude is sizably enhanced by the tree amplitude via total annihilation rescattering. These enhancements affect rates and CP asymmetries. Predictions can be checked in the near future.

It is well known that direct CP asymmetries and rates of suppressed modes are sensitive to final state interaction (FSI) [9,10]. In a study on the effects of FSI on B u,d,s → P P modes [11], the so called (too large) B(π 0 π 0 )/B(π + π − ) ratio and (non-vanishing) ∆A ≡ A(K − π + ) − A(K − π 0 ) direct CP asymmetry puzzles in B u,d decays can both be resolved by considering rescattering among P P states. 1 Several rates and CP asymmetries were predicted. The newly observed B 0 s → K 0 K 0 rate is consistent with the prediction. However, there are some results that are in tension with the recent measurement. In particular, the predicted B s → η η rate is too high compared to data. In fact, its central value is off by a factor of 3. As new measurements are reported and more data will be collected in LHCb, and Belle II will be turned on in the very near future, it is interesting and timely to revisit the subject.
It will be useful to give the physical picture. From the time-invariant property of the Wilson operators in the weak Hamiltonian, one finds that the decay amplitude satisfies [14] where A i is a B q → P P decay amplitude with weak as well as strong phases, A 0 k is a amplitude containing weak phase only, i = 1, . . . , n, denotes all charmless P P states and 1 One is referred to [12,13] for some recent analyses on these puzzles. 2 See Appendix A for a derivation. k = 1, . . . , n, n + 1, . . . , N, denotes all possible states that can rescatter into the charmless P P states through the strong interacting S-matrix, S. Strong phases are encoded in the rescattering matrix. This is known as the Watson theorem [15]. There are two points needed to be emphasised. First, the above result is exact. Every B q → P P decay amplitude should satisfy it. Second, for a typical B q decay, since the B mass is large there is a large number of kinematically allowed states involved in the above equation, i.e. N in the above equation is large. Consequently, the equation is hard to solve.
Although the largeness of the B mass makes it difficult to solve the above equation, it is interesting that on the contrary it is precisely the largeness of m B that makes the problem somewhat trackable. According to the duality argument, when the contributions from all hadronic states at a large enough energy scale are summed over, one should be able to understand the physics in terms of the quark and gluon degrees of freedom. Indeed, several quantum chromodynamics (QCD)-based factorization approaches, such as pQCD [16], QCD factorization (QCDF) [17,18] and soft collinear effective theory (SCET) [19] make use of the large B mass and give predictions on the facrorization amplitudes, A fac . In other words, using the largeness of m B comparing to Λ QCD , the factorization approaches provide solutions to Eq. (1), i.e. A fac i = N k=1 S 1/2 ik A 0 k . In the infinite m B limit, the above program may work perfectly. However, in the physical m B case, power corrections can be important and may not be neglected. In fact, the effects of power corrections are strongly hinted from some unexpected enhancements in rates of several color suppressed modes, such as B 0 → π 0 π 0 decay [6,7], and some unexpected signs of direct CP asymmetries, as in the difference of direct CP asymmetries of B 0 → K − π + and B − → K − π 0 decays [20]. These anomalies lead to the above mentioned ππ and Kπ puzzles. It is fair to say that the factorization approaches can reasonably produce rates of color allowed modes, but it encounters some difficulties in rates of color-suppressed states and CP asymmeties. It is to plausible to assume that factorization approaches do not give the full solution to Eq. (1), some residual rescattering or residual final state interaction is still allowed and needed in B q → P P decays. Note that the group of charmless P P states is unique to B q → P P decays, as P belongs to the same SU(3) multiplet and P P states are well separated from all other states, where the duality argument cannot be applied to these limited number of states [21,22]. Note that residual rescattering among P P modes only slightly affect the rates of color allowed modes, but it can easily change direct CP violation of most modes and the rates of color suppressed modes at the same time. It can be a one stone two birds scenario. It can potentially solve two problems at the same time without affecting the successful results of factorization approach on color allowed rates. In fact, this approach is modest than the factorization approach as it left some rooms for our ignorance on strong dynamics. In the following text, unless indicated otherwise we use rescattering among P P states or rescattering for short to denote this particular type of rescattering, while we assume that FSI contributions from all other states are contained in the factorization amplitudes.
The quark diagram or the so-called topological approach has been extensively used in mesonic modes [9,12,[23][24][25][26][27]. It will be useful and interesting to study the FSI effects on topological amplitudes. For some early works in different approach, one is referred to ref. [9].
The relation on topological amplitudes and rescattering will be explored and it can help to provide a better understanding on the effects of residual rescattering.
The layout of the present paper is as follows: In Sec. II we give the formalism. Results and discussions are presented in Sec. IV. Sec. V contains our conclusions. Some useful formulas and derivations are collected in Appendices A and B.

II. FORMALISM
In this section we will give the rescattering (res.) formulas, topological amplitudes (TA) of B q → P P decays and the relations between res. and TA.

A. Rescattering Formulas
Most of the following formulas are from [11], but some are new. As noted in the Introduction section, in the rescattering we have (see Appendix A) where i, j = 1, . . . , n denote all charmless P P states. To apply the above formula, we need to specify the factorization amplitudes. In this work, we use the factorization amplitudes obtained in the QCD factorization approach [18].
According to the quantum numbers of the final states, which can be mixed under FSI, B q → P P decays can be grouped into 4 groups. Explicit formulas are collected in Appendix A. Here we give an example for illustration. The B 0 d → K − π + decay can rescatter with three other states, namely B 0 d → K 0 π 0 , K 0 η 8 and K 0 η 1 , via charge exchange, singlet exchange and annihilation rescatterings as denoted in Fig. 1 (a)-(c). These states are the group-1 modes. The relevant rescattering formula is given by with S 1/2 res,1 = (1 + iT 1 ) 1/2 and The rescattering parameters r 0,a,e,t ,r 0,a,e,t ,r 0,a,e,t ,r 0,a,e,t andř 0,a,e,t denote 3 rescattering in Flavor symmetry requires that (S res ) m with an arbitrary power of m should also have the same form as S res . More explicitly, from SU(3) symmetry, we should have where T (m) is defined through the above equation and its form is given by for j = 0, a, e, t.
It is useful to note that we have 8 ⊗ 8, 8 ⊗ 1, 1 ⊗ 8 and 1 ⊗ 1 SU(3) products for P 1 P 2 final states, which has to be symmetric under the exchange of P 1 and P 2 in the B → P P decay as the meson pair is in s-wave configuration and they have to satisfy the Bose-Einstein 3 Note thatr andř do not appear in T 1 , but they will contribute to some other P P modes. |p; 1 V m pq q; 1|, (7) where a and b are labels of states within multiplets, and matrices U m and V m are given by respectively. Rescattering parameters r i as the solutions to Eqs. (5) and (6) can be expressed in terms of these angles and phases: with U m ij and V m ij given in Eq. (8). It is interesting to see how the rescattring behaves in a U(3) symmetric case. It is known that the U A (1) breaking is responsible for the mass difference between η and η and U(3) symmetry is not a good symmetry for low-lying pseudoscalars. However, U(3) symmetry may still be a reasonable one for a system that rescatters at energies of order m B . The mass difference between η and η , as an indicator of U(3) symmetry breaking effect, does not lead to sizable energy difference of these particles in charmless B decays. Note that in the literature, some authors also make use of U(3) symmetry in charmless B decays (see, for example [30]). We note that in the U(3) case, we have Consequently, by requiring as required by Eq. (10), one must have [11] r (m) a r (m) There are two solutions, either r (m) To reduce the number of the rescattering parameters and as a working assumption, the above relations will be used in this work, although we are not imposing the full U(3) symmetry to FSI.
After imposing the above relation and factor out a over phase factor, say δ 27 , we are left with two mixing angles and two phase differences: in the scattering matrices. The rescattering formula Eq. (2) now becomes with the overall phase removed. In summary, 4 additional parameters from Res are introduced to the decay amplitudes.
We find that it is useful to incorporate SU(3) breaking effect in the scattering matrix.
The idea is that we try to remove the SU(3) breaking effect in A fac before recattering and put the SU(3) breaking effect back after the rescattering. The underlying reason is as following. In the core of FSI, the rescattering processes are occuring at the m B energy scale, the SU(3) breaking effect cannot be very important at this stage. Hence the amplitudes to be rescattered are taken in the SU(3) limit, but after the rescattering, as the hadronization process takes place, SU(3) breaking cannot be neglected and their effect needs to be included.
In practice we use ratio of decay constants to model the SU(3) breaking effect. For example, other states and is multiplied by (f K /f π ) 2 after rescattering. For convenient these two factors are absorbed in S 1/2 res . These are new to Ref. [11]. The rescattering matrices needed in this work are collected in Appendix A. As we will see in the next section, including these four rescattering parameters will enhance the agreement of theory and data notably.

B. Rescattering and Topological Amplitudes in the SU(3) limit
Topological amplitude approach or flavor flow approach is based on SU(3) symmetry. The amplitudes can contain weak and strong phases. FSI will generate additional strong phases and can potentially mixed up different topological amplitudes. It is therefore interesting to investigate the relation of the FSI and topological amplitudes. We will take a closer look of this issue in the presence of the rescattering among P P states. We will consider the topological amplitudes in the SU(3) limit, rescattering of topological amplitudes in the SU(3) limit and, finally, topological amplitudes and rescattering in the U(3) limit. The discussion will be useful to provide a better understanding of the effect of FSI in B q → P P decays. These are all new to Ref. [11].

Topological Amplitudes in the SU(3) limit
It is well-known that the fields annihilating B − , B 0 d,s and creating π, K, η 8 transform respectively as 3 and 8 under SU(3) (see, for example [28]), For the b → uūd and b → qqd processes, the tree (O T ) and penguin (O P ) operators respectively have the following flavor structure, where we define The flavor structures of |∆S| = 1 tree and penguin operators can be obtained by replacing where the A, P , P A and P EW terms correspond to annihilation, penguin, penguin annihilation and electroweak penguin amplitudes, respectively. and (H eff ) singlet is the hamiltonain involving η 1 , given by 3. Note that we introduce P E EW and P A EW , namely the electroweak penguin exchange and electroweak penguin annihilation terms for completeness. The above With redefinition of the following amplitudes: (H eff ) singlet can be expressed in a more compact form, Using the above approach we can reproduce familiar expressions of decay amplitudes in terms of TA [26,27]. 4 Explicitly, we have the following amplitudes: for group-1 modes, for group-2 modes, for group-3 modes, for group-4 modes, and the following amplitudes: and for B s → P P decays, where the T , C, A, P , P A and P EW terms correspond to color-allowed tree, color-suppressed tree, annihilation, penguin, penguin annihilation and electroweak penguin amplitudes, respectively. Note that P E EW and P A EW , namely the electroweak penguin exchange and electroweak penguin annihilation terms, are introduced for completeness. See Appendix B for details. Those with (without) prime are for ∆S = −1(0) transition.
The one-to-one correspondence of the SU(3) parameters and the topological amplitudes is not a coincidence. It can be understood by using a flavor flow analysis. We take the first term of H eff for illustration. In H W the decays are governed by the O T ∼ (ūb)(du) = Note that we use subscript and superscript according to the field convention. For example, we assign a subscript (superscript) to the i transition with the same spectator anti-quarkq m from B m becoming the final state spectator anti-quarkq m , which ends up in (Π out ) m i . The quark q i from b → q i transition also ends up in (Π out ) m i , while the (Π out ) j k part is responsible for the creation of the meson where the W -emittedq j q k pair ends up with. The above picture clearly corresponds to the external W -emission topology. Similarly, the identification of the other topological amplitudes can be understood similarly.
One can check that all of the above amplitudes can be expressed in terms of the following combinations: For example, we can express the decay amplitude of B 0 → K − π + in the following combinations: It is interesting to compare the amplitudes expressed in terms of the topological amplitudes with the those in the QCDF calculation. We can obtain the following relations in the SU(3) limit: (using formulas in [18] but taking the SU(3) limit) where we use λ ( ) p ≡ V pb V * pd(s) , p = u, c with V pb,pd(s) the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and summation over p is implied. One can find detail definitions of A P P , α and β in [18]. Note that A P P involves a B q → P transition and a P decay constant: It should be note that we have removed an overall i in the definition of A P P . The superscript 0 on TA is denoting the fact that rescattering among P P states has not taken place. In the SU(3) limit, we will use F BP 0 (m 2 P ) = F Bπ 0 (0) and f P = f π in later discussion. For B decays to a final state with η 1 , things are more complicated. For example, A P η 1 is in principle different from A η 1 P . We have in the SU(3) limit: [18] Note that A P η 1 involves a B → P transition, while A η 1 P involves a B → η 1 transition: where in the second equation, we have made use of the approximation from [18]. In fact we Finally comparing our expressions and those in Ref. [18], we havẽ with In the later discussion, we take f η 1 = f P = f π .

Rescattering of Topological Amplitudes in the SU(3) limit
We now turn to the rescattering part. The matrices T 1,2,3,4 can be obtained through a diagrammatic method by matching the Clebsh-Gordan coefficients of scattering mesons (see Fig. 1) or by using an operator method. We have corresponding to r e , r a , r 0 and r t contributions, in the combination of where the remaining terms will be specified in below. The above terms exhaust all possible combinations for Π(8) Π(8) → Π(8) Π(8) scatterings.
To obtain operators involving η 1 , we simply replace Π in the above operators to Π + η 1 1 3×3 / √ 3 and collect terms with different number of η 1 as Note that it is impossible to obtain a term containing three η 1 as is prohibited from SU (3) symmetry. We now have Using Eq. (38), the above equation can be simplified into Note that variousr occur in T (m) only through some very specific combinations. We still preserve the subscripts (i = 0, t, a, e), since theser  It is straightforward to obtain the rescattering effects on topological amplitudes. In analogy to Eq. (15): we have where H eff is given in Eq. (18), T 1/2 in Eq. (40) but with m = 1/2, H 0 eff is the un-scattered effective Hamiltonian with all T A in H eff replaced by T A 0 and the dot in the above equation implies all possible pairing of the P out P out fields in H 0 eff to the P in P in fields in T 1/2 (the P out P out in T 1/2 remains unpaired). As noted previously since the effective Hamiltonian in Eq. (18) is obtained using flavor SU(3) symmetry argument only, its flavor structure will not be changed in the presence of rescattering, i.e. Eq. (42) will not modify the flavor structure of H eff . This feature is indeed verified in the explicit computation. Therefore the expressions of the decay amplitude in term of the TA will remain the same, but now the these TA contain rescattering contributions.
The effect of rescattering on TA can be obtained using the above equation. The computation is straightforward, but tedious. Here we only give the final results, some derivations using the above equation can be found in Appendix B for illustration. We obtain, in the presence of the rescattering, TA will receive corrections in the following ways: and where the superscript 0 denote un-scattered amplitudes and we define r i ≡ r The full topological amplitudes contain the un-scattered and the contribution from the scattering. For example, for the tree amplitude the full amplitude is T ( ) , the un-scattered tree amplitude is T ( )0 . After scattering we have One can check that the above equations are consistent with the topological amplitude expressions Eqs. Note that decay amplitudes can be expressed in terms of several combinations of topological amplitudes, such as T + C, C − E and so on, and FSI affects these combinations only through, We have EW,2 ), and With the help of Eq. (9) (with m = 1/2) we will be able to study the effect of rescattering to the above combinations and give a clearer picture. Note that the above transforma-  9) and (14)], without introducing additional assumption.

Topological Amplitudes and rescattering in the U(3) limit
It is interesting to investigate the above relations in the U(3) limit, where we take Eq. (10) andT Using Eq. (10) and Eqs. (43), (44) and (45), we find that and The above relations can be consistent with the relations in the U(3) limit, Eq. (51), only if we take It is useful to recall that by requiring U(3) symmetry to the rescattering matrix T [Eq. Hence it leads to a more specify relation.

III. NUMERICAL RESULTS
In this section, we will present our numerical results. First, we will give an overview of the results of the fits. We will then discuss the rescattering effects on topological amplitudes.
Finally, numerical results for decay rates and CP asymmetries will be shown.
In summary, 9 hadronic parameters, one CKM phase, γ/φ 3 , involved in the QCDF amplitudes will be fitted from data. The residue rescattering part add 4 more parameters, τ , ν, δ and σ, giving 14 parameters in total. Note that the majority of the fitted parameters are from the factorization part.
In this analysis there are totally 93 measurable quantities, including 34 rates, 34 direct CP asymmetries, 24 mixing induced CP asymmetries and one measurement from semileptonic B decay [Eq. (56)]. Among them we will fit to all available data, including 26 rates, 16 6 It is preferable to use the form factors as inputs instead of variables in the fit, but in the present situation no definite values for these form factors can be found (see for example the collected F Bπ 0 (0) values from [18,[31][32][33][34][35][36]) and we therefore treat them as fitting variables to avoid bias in this work. Hopefully the situation can be improved in future. See also Footnote 5. direct CP asymmetries, 5 mixing induced CP asymmetries and 1 semileptonic decay data, giving 48 in total, and will have prediction on 8 rates, 18 direct CP asymmetries and 19 mixing induced CP asymmetries. The explicit list of these 48 items will be shown later. The total numbers of data in fit and in predictions are roughly the same. The summary of these numbers is shown in Table I.
We perform a χ 2 analysis with all available data on CP-averaged rates and CP asymmetries in B u,d,s → P P decays. In the following study we use two different scenarios: Fac and Res. For the formal we use only factorization amplitudes (i.e. A i = A fac i ), while for the latter we add residue FSI effect as well (i.e. A i = n j=1 (S 1/2 res ) ij A fac j ). Both are fitted to data. The confidence levels and χ 2 s for the best fitted cases in both senarios are shown in Table II ..} in the table denote the χ 2 contribution obtained from 4 CP-average rates and 3 direct CP asymmetries, respectively, of the group-1 modes con- ..} are contributed from the group-2 modes: ..} is contributed from the group-4 modes: ..} only contributed from 3 of the above modes, B 0 → π + π − , π 0 π 0 , K 0 K 0 decays; χ 2

{B(Bs),A(Bs)}
is contributed from 5 CP-averaged rates in B 0 s → K + π − , π + π − , η η , K + K − , K 0 K 0 decays and from 2 direct CP asymmetries in B 0 The semiloptonic data, Eq. (56) is also included in the fit. The above lists are the 26 rates, 16 direct CP asymmetries, 5 mixing induced asymmetries and 1 semileptonic data [Eq. (56)], 48 in totally, that go into the fit. Table II shows the overall performances of the fits. We discuss the factorization case first.
The χ 2 per degree of freedom of Fac is 213.4/(48 − 10). One can compare the χ 2 values and the numbers of data used in the corresponding groups. When the ratio of χ 2 and the number of data is smaller than one, the fit in the group is reasonably well. By inspecting the table, we see that Fac gives a good fit in the direct CP asymmetries of group-1 modes , and produces reasonable fits in the direct CP asymmetries of group-2 modes (B − → K 0 π − , · · ·) and of group-3 modes (B − → π − π 0 , · · ·), but the fits in rates and mixing induced CP asymmetries of all modes (including B s decay modes) and direct CP asymmetries of group-4 modes are poor. In particular, the ratios of χ 2 per number of data used in rates of the group-2 modes (B − → K 0 π − , · · ·), group-4 modes (B 0 → π + π − , · · ·), in the rates and direct CP asymmetries of B s modes and in the semileptonic quantity are as large as 24.7/4, 55.3/9, 64.0/7 and 8.0/1, respectively, indicating the badness of the fit in these sectors.
The fit is significant improved when the rescattering is added. In the best fitted case, the , which is slightly enhanced, are reduced. In particular, the χ 2 per number of data of rates of the group-2 modes ( , the rates and direct CP asymmetries of B s modes and in the semileptonic quantity are 6.3/4, 7.8/9, 0.6/7 and 0.7/1, respectively. The performance of the fit in these sector is improved significantly. Detail results will be shown later.
The fitted parameters are shown in Table III. Uncertainties are obtained by scanning the parameter space with χ 2 ≤ χ 2 min + 1. The parameters consist of those in factorization amplitude and of Res. The Fac fit gives F Bπ = 0.239 ± 0.002, while the Res fit gives

B. Rescattering effects on Topological Amplitudes
In this part, we will show the rescattering effects on topological amplitudes in certain combinations and on some individual topological amplitudes of interest. Note that the discussion in the first part is generic, while we need to impose further assumption in the second part.

Rescattering effects on the Combinations of Topological Amplitudes
It is useful to show the fitted results on residual rescattering parameters r i (or r From the above equation, we see that most of these parameters have large phases (with respect to δ 27 ). Note that ir t + i(4r a + 2r e )/3, i(r e − r a ) and i(2r a +r e ) are three most sizable combinations and they are close to λ, −iλ and −iλ (taking the overall phase δ 27 = 0), respectively, where λ is the Wolfenstein parameter.
In Tables IV and V we show the topological amplitudes of B q → P P and B q → P P decays before rescattering (A 0 ) and after rescattering (A F SI ) in the unit of 10 −8 GeV. 7 These amplitudes are expressed in certain combinations as noted in Eq. (28). Note that without lost of generality the overall phase (δ 27 ) is set to 0 from now on for simplicity. The  of generality the overall phase (δ 27 ) for A F SI is set to 0 for simplicity. shown. Note that we do not use them directly in the fitting. In fact, they can be obtained only after the best fit result is available. Nevertheless they will provide useful information.
From Table IV, we see that before rescattering, we have the following order for B q → P P amplitudes: while the rest are rather small. After rescattering, we have: where |C − E|, |A + C| and |C 1 + 2Ē| are enhanced by 40 ∼ 44%, |P − C + 1 3 P C EW | by 26% and |C +Ẽ +P Note that the orders of |C 2 + 2P − 1 3P C EW,2 | and |P − C + 1 3 P C EW | are switched after turning on Res. Sub-leading tree amplitudes and penguin amplitudes are enhanced. We will return to this shortly. Note that except iñ E EW Res does not introduce sizable phases to these topological amplitude combinations.
Similarly, from Table IV, we see that before rescattering, we have the following order for the conjugated B q → P P decay amplitudes: while the rest are rather small. Note that the above order is different form the one in B q → P P decays. After rescattering, only the first two terms switch order, where |C 2 +2P − 1 3P C EW,2 | is enhanced by 12%, |P −C + 1 3 P C EW | by 17% and |C −E|, |A+C| and |C 1 +2Ē| by 40 ∼ 44%. Note that Res introduces sizable phases to some of these topological amplitude combinations E EW | and |P A − 4 9 C + 13 9 E − 1 3 P C EW | are quite different to those in B q → P P decays. Some comments will be useful. It is useful to see the above enhancements in details. It is clear from Eq. (48) that the effects of Res on T + C and P EW + P C EW are just adding the common phase δ 27 to them. The effects on other combinations of topological amplitudes are more interesting. In B q → P P decays, considering only the dominant contributions in Eq. (48), we have We can estimation the above values by taking the central values of (1 + ir 0 + ir a ), i(r e − r a ) and i(r e + 2r a ) from Eq. (57) and the central values of Similarly from Eq. (50), we have and we find that the T 0 + C 0 and C 0 − E 0 terms give (sizable) destructive contributions, while P 0 − C 0 + 1 3 P C0 EW and P A 0 − 4 9 C 0 + 13 9 E 0 − 1 3 P C0 EW terms give (sizable) constructive contributions via the same Res parameter ir t + i(4r a + 2r e )/3. The final result of the 16% E EW | is the complicate interplay of these contributions.
We now turn to the Res effect on the penguin amplitudes. From [see Eq. (48)] we obtain for B → P P decay: which is close to the value 1.26 e i10.0 • shown in Table IV. where the main contribution is from the r e − r a rescattering term fed from T 0 + C 0 .
We now turn to ∆S = −1 processes. The results are shown in Table V. We see from the table that before rescattering, we have the following order for B q → P P amplitudes: while the rest are rather small. Note that as expected penguin amplitudes dominate over trees. In fact, even the electroweak penguin amplitudes, which were neglected in the ∆S = 0 case, cannot be neglected now. After rescattering, the above orders are rearranged into: We see that the combinations with sub-leading tree amplitudes, C − E and A + C , are enhanced, while the one with the penguin term, P − C + P C EW /3, is slightly reduced. Note that |P A − 4 9 C + 13 9 E − 1 3 P C EW | is enhanced by a factor of 2, but |C +Ẽ +P + 3 E EW | is reduced by about 20%. Similar pattern occurs in the conjugated B q → P P decays.
The effect of rescattering on A +C is similar to the one in A+C. It is enhanced from the exchange and annihilation rescatterings fed from both T 0 + C 0 andT 0 + 2Ā 0 amplitudes.

E EW
We also note that the effect of rescattering on P C EW − P E EW is similar to the one in C − E , but with tree amplitudes replaced by electroweak penguins. Hence P C EW − P E EW is affected most from P 0 EW + P C0 EW and the effect is an enhancement in size. It is useful to see the enhancement and reduction in |P A − 4 9 C + 13 9 E − 1 3 P C EW | and |C +Ẽ +P E EW |, respectively, in more detail. In B q → P P decays, keeping only the (P A 0 − 4 9 C 0 + 13 9 E 0 − 1 3 P C EW ) and the (P 0 − C 0 + 1 3 P C0 EW ) terms in the corresponding formula shown in Eq. (48), we obtain which is close to the value 2.37 e i69.7 • shown in Table V. Similarly using the corresponding formula in Eq. (48) and keep only the (C 0 +Ẽ 0 +P 0 + 3 2 P A E0 EW ) and the (P 0 − C 0 + 1 3 P C0 EW ) terms we obtaiñ which is close to the value 0.83 e i78.5 • shown in Table V. In both cases the most important contributions are from the (P 0 − C 0 + 1 3 P C0 EW ) term.

Rescattering effects on some Individual Topological Amplitudes
The results in Tables IV and V are It is clear that we need the information of r 0 , r a , r e and so on to obtain δE ( ) . From the fit we only have information on some combinations of these rescattering parameters, such as 1 + i(r 0 + r a ), i(r e − r a ) and so on [see Eq. (57)], but not on individual ones. To study the effect of Res on individual topological amplitudes, we make an additional assumption:  Table VI. One should keep in mind of the assumption made. Note that the above assumption will affect our interpretation of the effect of Res on individual topological amplitudes, but not on the interpretation of the effect of Res on the combinations of topological amplitudes as discussed previously. In other words, the above assumption will affect the results stated in Table VI, but not on those in Tables IV and V.   Tables   IV and V. Table VI we see that, before Res, for B q → P P and B q → P P decays, we have while after Res, we have for B q → P P decays, and for B q → P P decays. Note that the positions of |P | and |P A| in the above orders are different in B q → P P and B q → P P decays. We will come to that later.
We see from Table VI that |E|, |E |, |A|, |A |, |P A|, |P A |, |P A,E EW | and |P A,E EW | are enhanced significantly with factors ranging from 2 ∼ 11, while |P | is enhanced by 35% in B q → P P decay, but is suppressed by 35% in B q → P P decay and |P | are suppressed by 6% and 3% in B q → P P and B q → P P decays, respectively. Note that in particular |A| and |A | are enhanced by a factor of 11.5. It is useful to look into the enhancement. From Eq.
Now make use of r e = 0 and Eq. (57), we obtain where the terms in the right hand side of the first equality are from A ( )0 , T ( )0 , C ( )0 , T ( )0 + 2Ā ( )0 contributions, respectively. We see that the T ( )0 , C ( )0 ,T ( )0 + 2Ā ( )0 terms give sizable contributions to A ( ) , via r a , r a andr e + 2r a rescatterings, respectively, and enhance its size significantly. Similarly we have where the terms in the right hand side of the first equality are from A ( )0 , T ( )0 , C ( )0 ,C As noted previously P ( ) and P A ( ) receive different Res contributions in B q → P P and B q → P P decays. It is interesting to investigate the effects of Res on these penguin amplitudes in details. First, we decompose P ( ) into the so-called u-penguin (P ( )u ) and Using these formulas and the best fit parameters, we obtain where the terms in the right hand side of the first equality are from P ( )u0 , T ( )0 , C ( )0 , EW,2 , respectively, and where the terms in the right hand side of the first equality are from P ( )c0 , P  Table VII.
It is useful to note that the ratio of u-penguin and c-penguin in ∆S = 0 process before rescattering is expected to proportional to the CKM factors giving The estimation is close to the ratio |P u0 /P c0 | = 3.51/10.09 0.35 using P u0 and P c0 shown in Table VII. The CKM ratio implies that u-penguin and the c-penguin are not as hierarchical as in the ∆S = −1 case. Furthermore, when rescattering is turned on, the u-penguin and c-penguin receive different contributions as only P u can receive contribution fed from T 0 , see Eq. (74), and, consequently, the above ratio is enhanced to 6.98/9.49 0.74 (see Table VII). These will affect the CP asymmetries of ∆S = 0 modes to be discussed later.
We now turn to P A ( ) . Similarly we decompose P A ( ) into P A ( )u + P A ( )c and from Eq. (43) we have P A ( )u = 1 3 (3 + 3ir 0 − ir e + 16ir a + 12ir t )P A ( )u0 + ir t T ( )0 + 1 9 (2ir e + 4ir a − 3ir t )C ( )0 + 2 9 (ir e + 11ir a + 12ir t )E ( )0 + 2 9 (ir e + 11ir a + 12ir t )P ( )u0 (ir e + 11ir a + 12ir t )P ( )c0 Using these formulas and the best fit parameters, we obtain where the terms in the right hand side of the first equality are from P A ( )u0 , T ( )0 , C ( )0 , contributions, respectively. Note that |P A ( )u | is enhanced by a factor of 18, and the main contributions are from T ( )0 , C ( )0 and P ( )u0 terms via the total annihilation rescattering r t , the annihilation r a and total annihilation r t rescatterings, respectively. In particular, the enhancement from T ( )0 via r t is the most prominent one.
Similarly we have where the terms in the right hand side of the first equality are from P A ( )c0 , P ( )c0 , contributions, respectively. Note that |P A ( )c | is enhanced by a factor of 2.5, while the main contribution is from the P ( )c0 term via the annihilation r a and total annihilation r t rescatterings. The effect of rescattering in P A ( )c is not as prominent as in the P A ( )u case.
We see that in the presence of rescattering, the resulting |P A u | is even greater than |P A c |, while P A ( )u can no longer be neglected (see Table VII). The above observations can shed light on the results in the following discussions.

C. Numerical results for decay rates and CP asymmetries
In this part we will present the numerical results on rates in B 0 and B − decays, direct CP violations in B 0 and B − Decays, rates and direct CP asymmetries in B 0 s decays, and time-dependent CP violations in B 0 and B 0 s decays.

Rates in B 0 and B − Decays
In From the table, we see that, except for rates in B 0 → K − π + , K 0 η and B − → π − η decays, the χ 2 in Res for the other modes are lower than the Fac ones. In particular, the significantly, as Fac encounters difficulties to fit some of these rates well. In fact, in Fac the χ 2 in B 0 → π + π − is as large as 36.1, while it is reduced to 0.7 in Res. We see that in each group the χ 2 is improved in the presence of Res. The total χ 2 from these 21(= 4 + 4 + 4 + 9) modes  Table II as well). Overall speaking rescattering significantly improves the fit in this sector, especially in the last group, and can reproduce all the measured B u,d → P P rates reasonably well.
Note that both Fac and Res can successfully reproduce the newly measured B 0 → π 0 η and K + K − rates [1, 4]. On the other hand, both Fac and Res results on the B 0 → π 0 π 0 rate have tension with the data, while Res is somewhat better as its χ 2 (=3.7) is smaller than the one (5.6) in Fac. It should be note that the uncertainty in the present data is still large and it will be interesting to see the updated measurement. Both Fac and Res fits on the B − → π − π 0 rates are smaller than the experimental result. The χ 2 from Fac on this mode is 5.7, while the Res fit improves it to 2.9 with a slightly large rate, but both results are in tension with data.
We will investigate how rescattering improves the fit in B 0 d → π + π − , π 0 π 0 , K + K − and B − → K − π 0 rates. For simplicity we will concentrate on the dominant contributions to the decay amplitudes in the following discussion. By neglecting the electroweak penguin contributions, the B 0 → π + π − amplitude in Eq. (25) can be expressed as Using the results in Sec. III B, we see that before rescattering and after rescattering, we have (in unit of 10 −8 GeV) 4.7 ± 0.1 4.9 +0.2 −0.1 (4.9) 5.7 2.9 (2.9) b Taken from PDG with an S factor of 1.7 included in the uncertainty.
respectively, where expressions with four terms are given in the order of T , P , E and P A and those in two terms are with the first two terms (T + P ) and the last two terms (E + P A) summed separately. Before we proceed we may compare the above estimation to our full numerical results, where we have ( , respectively, which are close to the above estimation. Note that T +P are dominant contributions, while E +P A are sub-leading contributions, and these two groups interfere destructively. In the presence of rescattering, the sizes of the dominant parts, T + P , are reduced, while the sizes of the destructive and sub-leading parts, E+P A, are enhanced, resulting more effective destructive interferences. From the estimation we see that the B 0 d → π + π − rate is reduced by about 15% bringing B(B 0 → π + π − ) 6×10 −6 down to ∼ 5×10 −6 , which agrees well with the data [(5.1±0.19)×10 −6 ] shown in Table VIII and, consequently, the quality of the fit is improved significantly.
Similarly for B 0 → π 0 π 0 decays, we have which is close to the above B 0 → π + π − amplitudes, but with T replaced by −C. Before rescattering and after rescattering, we have (in unit of 10 −8 GeV) respectively, where terms are given in the order of −C, P , E and P A and the expressions with the first three terms (−C + P + E) combined are also shown. The above estimation is close to the values in the full numerical results with .0 • and 1.08e −i126.0 • in the unit of 10 −8 GeV, respectively.
In the above estimation the first three terms and the last term interfere destructively.
With Res P and E are enhanced giving a larger −C + P + E, while the enhanced P A cannot be neglected anymore, producing a slightly larger decay amplitude and resulting a 35% enhancement in rate, which brings the rate up from B(B 0 → π 0 π 0 ) 0.8 × 10 −6 to ∼ 1.1 × 10 −6 as shown in Table VIII. As noted previously the rate is still smaller than the central value of the data, which however accompanies with large uncertainty.
For the newly observed B 0 d → K + K − mode, we note that as shown in Table VIII rescattering enhances the rate by 0.100/0.03 = 3.33 times. It will be useful to see the enhancement in details. From Tables IV and VII and Eq. (25), we have (in unit of 10 −8 GeV) for the decay amplitudes before and after rescattering, where terms are given in the order of E, P A u , P A c and P A EW /3. Compare the above estimation to the values in our full numerical result, which have 0. 10 −8 GeV, respectively. The discrepancy is mainly from SU(3) breaking effects, which are not included in the above equation. In fact, by scaling the numbers in Eq. (85) by (f K /f π ) 2 , the sizes become 0.199, 0.199, 0.373 and 0.409, which agree better to the above values now.
From the above equation, we see that E, P A u , P A c and P A EW are all enhanced. Note that E interferes destructively with P A u and P A c in The result is an enhancement of 3.8 in the averaged rate, which is close to our numerical result (0.100/0.03 = 3.33) as shown in Table VIII. We will return to this mode again in the discussion of direct CP asymmetry.
Finally we turn to the B − → K 0 π − decay. From Eq. (23) we have which gives before and after rescattering (in unit of 10 −8 GeV) respectively, where terms are given in the order of A , P u , P c and (−P C EW + 2P E EW )/3. Note that in our numerical result, we have 5.17e i167.2 • , 5.29e i171.2 • , 4.86e i179.6 • and 4.98e i175.7 • , respectively. By scaling the values in the Eq. (87) by f K /f π , the sizes become 5.20, 5.17, 4.85 and 4.83, respectively, which are close to the numerical results. In the full numerical result either in the presence of rescattering or without it, the sizes of A B + →K 0 π + is slightly greater than A B − →K 0 π − , but it is the other way around in the estimation. In fact, in the numerical result, we have P u = 0.10e i107.8 • and P c = 5.19e i167.5 • in (A B − →K 0 π − ) 0 and P u = 0.10e −i114.4 • and P c = 5.33e i169.6 • in (A B + →K 0 π + ) 0 . The latter |P c | in (A B + →K 0 π + ) 0 is greater than the one in (A B − →K 0 π − ) 0 . The difference can be traced to the non-vanishing first Gegenbauer moment of the kaon wave function (αK 1 = −α K 1 = 0.2), which will change sign in changing from K to K. This will affect the direct CP asymmetry and such a feature is absent in the above estimation.
From Eq. (87) we see that A + 1 3 (−P C EW +2P E EW ) interferes destructively to the dominating P c term. Since the sizes of A and 1 3 (−P C EW + 2P E EW ) are enhanced, while the size of P c is slightly reduced, the size of the total amplitude is reduced under the rescattering resulting a reduction of 13% in the averaged rate, which brings the rate from B(B − → K 0 π + ) 25 × 10 −6 to ∼ 22 × 10 −6 , which is closer to the data [(23.79 ± 0.75) × 10 −6 ] as shown in Table VIII.

Direct CP Violations in B 0 and B − Decays
Results for direct CP asymmetries (A) in B u,d → P P decays are summarized in Table IX. The Fac and Res fits give similar results in the first group of data, namely the direct CP asymmetries in B 0 → K − π + , K 0 π 0 and K 0 η decays. Both can explain the so-call Kπ CP puzzle by producing positive A(B − → K − π 0 ) and negative A(B 0 → K − π + ), but the Res give a slightly larger A(B − → K − π 0 ). Fac fits better than Res in the B − → π − η and B 0 → K 0 K 0 modes, while Res fits better than Fac in the and π 0 π 0 modes. In particular, the χ 2 in A of B 0 → π + π − is reduced significantly from 11.5 (Fac) to 2.9 (Res). Overall speaking the fit in Res in this sector (see also Table II) is better than Fac, as the corresponding χ 2 are 13.9(= 1.1 + 0.6 + 7.5 + 4.7) and 29.2(= 1.8 + 5.2 + 6.5 + 15.7), respectively.
It is interesting to see how rescattering solve the so-call Kπ CP puzzle, where experimental data gives ∆A ≡ A(K − π + ) − A(K − π 0 ) = (12.2 ± 2.2)%, in details. The B 0 → K − π + and B − → K − π 0 decay amplitudes can be expressed as It is useful to note that these two amplitudes are related by the following relation: Using the values in Table V and the above equation, we have (in unit of 10 −8 GeV and in the corresponding order of the above equation) before and after Res respectively. In our full numerical results, for B 0 → K − π + decay, we have 4.91e i172.0 • , 10 −8 GeV, respectively, which are close to the scaled (by 6.8% and ∆A 13.6% after Res, which are close to the values −8.2%, 4.9% and 13.0% shown in Table IX. As noted in the discussion of the B − → K 0 π − rate in the last subsection, the first Gegenbauer moment of the kaon wave function is the main source of the discrepancies between the estimations and the full numerical results.
As shown in Eq. (90), it is interesting that before rescattering the C and P EW terms are the sources of deviation of A( , while with the presence of Res, the sizes of A and P E EW are enhanced and hence further enlarges the deviation of A(B 0 d → K − π + ) and A(B − → K − π 0 ) producing a larger ∆A. Note that comparing to the discussion in B 0 → π + π − and π 0 π 0 decay rates [see discussion after Eq. (80)], we see that the correlation of the effects of Res on these two sectors is not prominent. Indeed, in the π 0 π 0 mode the most affected TAs under rescattering are P , E and P A, while at here A and P E EW are the most affected and relevant ones.
Note One should be reminded that Res can reproduce the B 0 → K 0 K 0 CP-averaged rate much better than Fac (see Table VIII). We need more data to clarify the situation and to verify these predictions.
It will be useful to see the effect of Res on the B 0 d → K 0 K 0 direct CP asymmetry. From Eq. (25), we can approximate the B 0 d → K 0 K 0 amplitude as From Table VII, before Res and after FSI, we have (in unit of 10 −8 GeV) respectively, where the values of P u , P A u , P c and P A c are shown in the corresponding order. In our full numerical result, we have 1.12e i8. 10 −8 GeV, respectively, which are close to the scaled [by (f K /f π ) 2 ] estimations, 1.16e i8.1 • , 1.31e −i31.6 • , 0.95e −i1.9 • and 1.53e −i27.2 • , from Eq. (92).
In Eq. (92), we see that both P u and the P A u terms are enhanced under Res (mainly through rescattering from T 0 ) and produce richer inference pattern contributing to the direct CP asymmetry. The B 0 d → K 0 K 0 amplitude is reduced, while the amplitude of the conjugated decay mode, B 0 d → K 0 K 0 , is enhanced under Res, producing an enlarged direct CP asymmetry, which is changed from −12% to −45% and hence close to the Belle result.
As shown in Table IX, we see that before Res the direct CP asymmetry of B 0 → K + K − is vanishing. Indeed, as one can infer from Eq. (85) that the rates of B 0 → K + K − and B 0 → K + K − are the same before Res. This can be understood in the following. In QCDF, E, P A and P A EW can be expressed in terms of the so-called A i 1 and A i 2 terms, and these A i 1 and A i 2 terms are identical when the asymptotic distribution amplitudes are used (as in the present case) [18]. Since we have A B 0 →K + K − = E + P A + P A EW /3 and these three topological amplitudes all have a common strong phase resulting a vanishing direct CP asymmetry.
Note that in the presence of Res, E and P A u are enhanced mostly from T 0 [see Eqs. (72) and (78)], while P A c from P c [see Eq. (79)], consequently, the strong phases of these terms are no longer degenerate. In fact, from Eq. (85) one can infer that the direct CP asymmetry is estimated to be −18%, which can be compared to the value of (−7.7 +6.0 −6.2 )% obtained in the full numerical result as shown in Table IX. For prediction, we see that except B 0 → K + K − , the sizes of the predicted direct CP asymmetries from Res are smaller than those in Fac.

Rates and Direct CP asymmetries in B 0 s Decays
We show the CP-averaged rates and direct CP violations of B 0 s → P P decays in Table X. There are five measured B s decay rates, namely K + π − , π + π − , η η , K + K − and K 0 K 0 decay rates. Among them B s → π + π − and η η decays are newly observed by LHCb [3,4]. From the table we see that both Fac and Res can fit the B s → K + π − rate well, but Fac is having difficulties in fitting all other four modes: in particular the χ 2 of π + π − , η η and K + K − are as large as 20.2, 16.0 and 21.3, respectively, while Res can fit all B s decay modes very well and brings down these χ 2 efficiently, giving 0.0, 0.0 and 0.0, respectively. Note that the rates of the two newly measured modes (π + π − and η η ) can be easily reproduced in the Res fit, but not in the Fac fit. For other modes, we see from the table that Res predicts larger rates in B 0 s → K 0 π 0 , K 0 η, π 0 π 0 decays, but gives similar predictions on K 0 η , ηη, ηη , π 0 η and π 0 η rates.
The B 0 s → π + π − rate in the factorization calculation is too small compared to data. As shown in Table X, through Res the rate can be enhanced significantly. It is useful to see the enhancement of the π + π − rate more closely. From Eq. (27),   Tables V and VII, before and after Res, we have (in unit of 10 −9 GeV) respectively, where terms are given in the order of E , P A u , P A c and P A EW /3. In our full numerical result, we have 4.17e −i5.3 • , 4.17e −i11.7 • , 9.19e i66.7 • and 9.04e i64. , which help to enhance the B 0 s → π + π − rate and bring in non-vanishing direct CP asymmetry.
We now compare our results to the data in direct CP asymmetries. There are two reported measurements in direct CP asymmetries of B s modes: A(B 0 s → K + π − ) and A(B s → K + K − ). A better measurement is reported in the K + π − mode with a much reduced uncertainty. From the table we see that Res gives a better fit to this data than Fac with χ 2(Fac) = 4.6 and χ 2(Res) = 0.1. On the other hand both Fac and Res can fit A(B s → K + K − ) well, as the uncertainty in data is still large to accommodate both results, but Res has a smaller χ 2 .
For predictions on direct CP asymmetries, we note that the signs of A(B s → η η ) and A(B s → K 0 K 0 ) are opposite in Fac and Res; Res predicts non-vanishing A(B s → π + π − , π 0 π 0 ) and larger A(B s → π 0 η), while predictions of Fac and Res on other modes are similar. These predictions can be checked in near future. Results on time-dependent CP-asymmetries S are given in Table XI. We fit to data on mixing induced CP asymmetries. There are reported experimental results of mixing induced CP asymmetries in the following 5 modes: Since the measurements are subtle, the experimental progress in this sector is slower than those in rates and direct CP asymmetries. Currently, the B 0 → K 0 π 0 mode was updated up to 2010; the B 0 → K 0 η mode was updated up to 2014; the B 0 → π + π − mode was updated up to 2013, the B 0 → K S K S mode was updated up to 2007 and the B s → K + K − mode was included in these measurement in 2013 [41][42][43][44].
New data are eagerly awaited. Note that for the B 0 → K 0 K 0 mode, the mixing induced CP asymmetry obtained by Belle (−0.38 +0. 69 −0.77 ± 0.09 [43]) and BaBar (−1.28 +0.80+0.11 −0.73−0.16 [44]) are different. As the central value of the latter exceeds the physical range, we only include the former one in our fit.
From Table XI we see that fit in Res for the B 0 → π + π − mode is much better than the one in Fac, where the χ 2 are 1.1 and 9.3 for the former and the latter, respectively. On the contrary, the fit in Fac is better than Res in the B s → K + K − mode, where the χ 2 are 0.6 and 1.4 for the former and the latter, respectively. Note that the uncertainty in the data of the B s → K + K − mode is much larger than the one in the B 0 → π + π − mode. It will be interesting to see the updated data on the B s → K + K − mode. Overall speaking the quality of fit to mixing induced CP asymmetries is improved (χ 2 reduced from 12.9 to 5.2, see also It is useful to look into the mixing induced asymmetry in the B 0 d → K 0 K 0 mode. Recall in Eq. (92) that, before and after Res, we have (in unit of 10 −8 GeV, without SU(3) breaking correction) respectively. Using the well known formula: we obtain S −0.08 and −0.29 without and with Res, respectively, which are close to the values reported in Table XI. As explained previously, although B 0 d → K 0 K 0 is a pure penguin mode, its S is not necessary close to − sin 2β, as the u-penguin contribution is not negligible (|P 0u /P 0c | 0.35, see Table VII). When Res is turned on, the u-penguin and c-penguin receive different contributions, where it is clear that trees can only contribute to the former giving |P u /P c | 0.74 (see Table VII), and, consequently, the value of S can be changed drastically.
We now compare the predictions of Fac and Res on mixing induced CP asymmetries.

IV. CONCLUSION
Various new measurements in charmless B u,d,s → P P modes are reported by Belle and LHCb. These include the rates of B 0 → π 0 π 0 , ηπ 0 , B s → η η , B 0 → K + K − and B 0 s → π + π − decays. Some of these modes are highly suppressed and are among the rarest B decays.
Direct CP asymmetries on various modes are constantly updated. It is well known that direct CP asymmetries and rates of suppressed modes are sensitive to final state interaction.
As new measurements are reported and more data will be collected, it is interesting and timely to studied the rescattering on B u,d,s → P P decays. We perform a χ 2 analysis with all available data on CP-averaged rates and CP asymmetries in B u,d,s → P P decays. Our numerical results are compared to data and those from factorization approach. The quality of the fit is improved significantly in the presence of Res, especially in the decay rates in the B 0 ∆S = 0 sector and in rates and direct CP asymmetries in the B 0 s decay modes. Indeed, the χ 2 in the B 0 → K 0 π 0 , π + π − , K 0 K 0 , B − → K 0 π − , K − η, π − π 0 , π − η and B 0 s → π + π − , η η and K + K − rates, and in B 0 → π + π − and B 0 s → K + π − direct CP asymmetries are improved significantly. Res also fit. better to the semileptonic data on |V ub |F Bπ (0) [see Eq. (56)].
The relations on topological amplitudes and rescattering are explored and they help to provide a better understanding of the effects of rescattering. As suggested by U(3) symmetry on topological amplitudes and FSI, a vanishing exchange rescattering scenario is considered.
The exchange, annihilation, u-penguin, u-penguin annihilation and some electroweak penguin amplitudes are enhanced significantly via annihilation and total annihilation rescatterings. In particular, the u-penguin annihilation amplitude is sizably enhanced by the tree amplitude via total annihilation rescattering. These enhancements affect rates and CP asymmetries. For example, the enhanced P A u changes the B 0 d → K 0 K 0 direct CP asymmetry significantly; the enhanced P , E and P A produce (through complicate interference) a slightly larger B 0 → π 0 π 0 decay amplitude and resulting a 35% enhancement in rate; A and P E EW are enhanced and enlarges the deviation of A(B 0 d → K − π + ) and A(B − → K − π 0 ) producing a larger ∆A; the B 0 s → π + π − rate is sizably enhanced through the enhancement in P A c ; the |P u /P c | ratio is enhanced from 0.35 to 0.74 and can change mixing induced CP asymmetries drastically.
For the comparison of the predictions of Fac and Res, we observed the following points.
(i) Belle and BaBar give very different results in A(B 0 → K s K s ) mode, namely Belle gives A(B 0 → K s K s ) = −0.38 ± 0.38 ± 0.5 [43], while BaBar gives 0.40 ± 0.41 ± 0.06 [44]. The result of Res prefers the Belle result, while Fac prefers a negative but less sizable direct CP asymmetry. (ii) Except B 0 → K + K − , the sizes of the predicted direct CP asymmetries of B − , B 0 → P P modes from Res are smaller than those in Fac. (iii) For B s decay rates, Res predicts larger rates in B 0 s → K 0 π 0 , K 0 η, π 0 π 0 decays, but gives similar predictions on K 0 η , ηη, ηη , π 0 η and π 0 η rates. (iv) For predictions on direct CP asymmetries, we note that the signs of A(B s → η η ) and A(B s → K 0 K 0 ) are opposite in Fac and Res; Res predicts non-vanishing A(B s → π + π − , π 0 π 0 ) and larger A(B s → π 0 η), while predictions of Fac and Res on other modes are similar. (v) Finally, Fac and Res have different predictions on the mixing induced CP asymmetries of B 0 → ηη, ηη , π 0 η, π 0 η , B s → π 0 η, π 0 η , K s π 0 , K s η and K S η modes. In particular, the signs of central values of the asymmetries of B 0 → π 0 η, π 0 η , B s → π 0 η, π 0 η , and K s η modes are opposite. These predictions can be checked in the future.
where S 1 is a non-singular n × n matrix with n the total number of charmless P P states and S 2 is defined through the above equation, i.e. S 2 ≡ S −1 1 S. As mentioned previously (in the introduction) the factorization amplitudes contain a large portion of rescattering effects as encoded in S 2 , while some residual rescattering among a small group of states is still allowed and needs to be explored: with N the total number of states entering Eq. (1), A fac j the factorization amplitude and S res the rescattering matrix to govern rescattering among P P states.
We collect the rescattering formulas used in this work. We have for group-1 modes, for group-2 modes, for group-3 modes and Note that for identical particle final states, such as π 0 π 0 , factors of 1/ √ 2 are included in the amplitudes and the corresponding S res matrix elements. The rescattering formulas for To include the SU(3) breaking effect, we proceed as outlined in the main text. First we remove the SU(3) breaking effect in A fac before recattering and put it back after the rescattering. For the reasoning one is referred to the main text. For convenient we absorb these two action into the rescattering matrices. We use ratios of decay constants to model the SU(3) breaking effect. For example, in the group-3 modes, in the π − π 0 -K 0 K − -π − η q -π − η s basis, we have In the numerical study we follow [46] to use f ηq /f π = 1.07 and f ηs /f π = 1.34. It is clear that when FSI is turned off the above S where the identical particle factor of 1/ √ 2 is properly included in the mixing matrix. Note that the above formulas can be easily used to transform the η q -η s basis into the η 8 -η 1 basis by replacing the above ϑ by tan −1 √ 2.
If the full U(3) symmetry is a good symmetry, it requires: for each i = 0, a, e, t. We are constrained to have .
It is interesting to note that in both solutions of the U(3) case, a common constraint has to be satisfied.
H 0 eff + iT 1/2 · H 0 eff , where H eff is given in Eq. (18), T 1/2 in Eq. (40), H 0 eff is the un-scattered effective Hamiltonian with all T A in H eff replaced by T A 0 and the dot in the above equation implies all possible pairing of the P out P out fields in H 0 eff to the P in P in fields in T 1/2 . It is useful to use H ik i = H k , H ik k = 0, (H EW ) ik k = 0, (H EW ) ik i = − 1 3 H k , (Π in ) a a = (Π out ) a a = 0 and the fact that the paring of creation and annihilation fields gives the following flavor structure: (Π out ) j k (Π in ) a b → δ j b δ a k − 1 3 δ j k δ a b . In bellow we work out the contribution from T 0 via the rescattering among P P states for illustration. We shall concentrate on the flavor structures after the pairings in (iT 1/2 · H 0 eff ) and compare them to the operators in H eff . 8 2. Pairing T 0 B m H ik j (Π out ) j k (Π out ) m i and ir e T r(Π in Π out Π in Π out )/2