Revisiting CP violation in $D\to P\!P$ and $V\!P$ decays

Direct CP violation in the hadronic charm decays provides a good testing ground for the Kobayashi-Maskawa mechanism in the Standard Model. Any significant deviations from the expectation would be indirect evidence of physics beyond the Standard Model. In view of improved measurements from LHCb and BESIII experiments, we re-analyze the Cabibbo-favored $D \to P\!P$ and $V\!P$ decays in the topological diagram approach. By assuming certain SU(3)-breaking effects in the tree-type amplitudes, we make predictions for both branching fractions and CP asymmetries of the singly Cabibbo-suppressed decay modes. While the color-allowed and -suppressed amplitudes are preferred to scale by the factor dictated by factorization in the $P\!P$ modes, no such scaling is required in the $V\!P$ modes. The $W$-exchange amplitudes are found to change by 10\% to 50\% and depend on whether $d\overline{d}$ or $s\overline{s}$ pair directly emerges from $W$-exchange. The predictions of branching fractions are generally improved after these SU(3) symmetry breaking effects are taken into account. We show in detail how the tree-type, QCD-penguin, and weak penguin-annihilation diagrams contribute and modify CP asymmetry predictions. Future measurements of sufficiently many direct CP asymmetries will be very useful in removing a discrete ambiguity in the strong phases as well as discriminating among different theory approaches. In particular, we predict $a_{CP}(K^+K^-)-a_{CP}(\pi^+\pi^-) = (-1.14 \pm 0.26) \times 10^{-3}$ or $(-1.25 \pm 0.25) \times 10^{-3}$, consistent with the latest data, and $a_{CP}(K^+K^{*-})-a_{CP}(\pi^+\rho^-) = (-1.52 \pm 0.43) \times 10^{-3}$, an attractive and measurable observable in the near future. Moreover, we observe that such CP asymmetry differences are dominated by long-distance penguin-exchange through final-state rescattering.

The time-integrated asymmetry can be further decomposed into a direct CP asymmetry a dir CP and a mixing-induced indirect CP asymmetry a ind CP a CP (f ) = a dir where t is the average decay time in the sample, τ is the D 0 lifetime and y CP is the deviation from unity of the ratio of the effective lifetimes of D 0 meson decays to flavor-specific and CP-even final states. To a good approximation, a ind CP is independent of the decay mode. Hence, Based on the LHCb averages of y CP and a ind CP , it is known that ∆A CP is primarily sensitive to direct CP violation.
Since ∆a dir CP in the Standard Model (SM) is naively expected to be at most of order 1 × 10 −3 , many new physics (NP) models [2-4, 6-15, 35] had been proposed to explain the measurement of large ∆A CP , although it was also argued in [16][17][18][19][20][21][22][23] that large CP asymmetries in singly Cabibbosuppressed (SCS) D decays were allowed in the SM due to some nonperturbative effects or unexpected strong dynamics and the measured ∆a dir CP could be accommodated or marginally achieved. On the experimental side, the large ∆A CP observed by LHCb in 2011 was subsequently confirmed by CDF [24] and by Belle [25]. However, the effects disappeared in the muon-tag LHCb analyses in 2013 and 2014 [26,27] and were not seen in the subsequent pion-tag analysis in 2016 [28]. Finally, in this year LHCb announced the measurements based on pion and muon tagged analyses [29]. Combining these with previous LHCb results in 2014 and 2016 leads to [29] ∆A CP = (−1.54 ± 0.29) × 10 −3 , (LHCb2019).
which yields ∆a dir CP = (−1.56 ± 0.29) × 10 −3 . This is the first observation of CP violation in the charm sector! It is most important to explore whether the first observation of CP violation in the charm sector (4) is consistent with the standard model or not. 1 A common argument against the SM interpretation of Eq. (4) goes as follows. Consider the tree T and penguin P contributions to D 0 → K + K − and D 0 → π + π − . A simplified expression of the CP asymmetry difference between them is given by (for a complete expression of ∆a dir CP , see Eq. where θ KK is the strong phase of (P/T ) KK and likewise for θ ππ . Since |P/T | is naïvely expected to be of order (α s (µ c )/π) ∼ O(0.1), it appears that ∆a dir CP is most likely of order 10 −4 even if the strong phases are allowed to be close to 90 • . Indeed, using the results of |P/T | obtained from lightcone sum rules, the authors of [36] claimed an upper bound in the SM, |∆A SM CP | ≤ (2.0±0.3)×10 −4 . The notion that this would imply new physics was reinforced by a recent similar analysis [31].
In 2012, we have studied direct CP violation in charmed meson decays based on the topological diagram approach for tree amplitudes and QCD factorization for penguin amplitudes [37,38]. We have pointed out the importance of a resonantlike final-state rescattering which has the same topology as the QCD-penguin exchange toplogical graph. Hence, penguin annihilation receives sizable long-distance contributions from final-state interactions. We have shown that ∆a dir CP arises mainly from long-distance weak penguin annihilation. Moreover, we predicted that ∆a dir CP is about (−0.139 ± 0.004)% and (−0.151 ± 0.004)% for the two solutions of W -exchange amplitudes [38]. Those were the main predictions among others made in 2012. Since the world average during that time was ∆a dir CP = (−0.645 ± 0.180)% [39], we concluded that if this CP asymmetry difference continues to be large with more statistics in the future, it will be clear evidence of physics beyond the standard model in the charm sector. Nowadays, we know that the LHCb new measurement almost coincides with our second solution. This implies that one does not need New Physics at all to understand the first observation of ∆a dir CP by LHCb! 2 The purpose of this work is twofold. First, we would like to improve the analysis of CP asymmetries in D → PP decays. For example, it is well known that the penguin-exchange amplitude PE and the penguin-annihilation one PA evaluated in the approach of QCD factorization is subject to the end-point divergence. We need to address this issue. Also in our previous study of the long-distance contribution to PE, we did not consider the uncertainties connected with final-state rescattering [38]. This will be improved in this work. Secondly, although we have studied CP asymmetries in D → VP decays before in [37], we focused only to the neutral charmed meson ones.
Owing to the lack of information on W -annihilation amplitudes, no prediction was attempted for D + → VP and D + s → VP decays. Thanks to the BaBar's measurement of D + s → π + ρ 0 [41], the amplitudes A V,P can be extracted for the first time in [42]. Consequently, in this work we are able to complete the analysis of CP violation in the VP sector.
The layout of the present paper is as follows. After a brief review of the diagrammatic approach, we study various mechanisms responsible for the large SU(3) violation in the branching fraction ratio of D 0 → K + K − to D 0 → π + π − and fix the SU(3) breaking effects in weak annihilation amplitudes in Section II. Penguin amplitudes are studied in the framework of QCD factorization as illustrated in Section II C. We then discuss direct CP violation in SCS D → PP decays in Section III and compare our results with other works in the literature. Section IV is devoted to D → VP decays and their direct CP asymmetries. Finally, Section V comes to our conclusions. 2 A similar result of ∆a dir CP based on a variant of the diagrammatic approach was obtained in [40].

II. D → PP DECAYS
It is known that a reliable theoretical description of the underlying mechanism for exclusive hadronic D decays based on QCD is still not yet available as the mass of the charm quark, being about 1.3 GeV, is not heavy enough to allow for a sensible heavy quark expansion. It has been established sometime ago that a more suitable framework for the analysis of hadronic charmed meson decays is the so-called topological diagram approach [43][44][45]. In this diagrammatic scenario, the topological diagrams can be classified into three distinct groups (see Fig. 1 of [37]). The first two of them (see [46] for details) are: 1. Tree and penguin amplitudes: color-allowed tree amplitude T ; color-suppressed tree amplitude C; QCD-penguin amplitude P ; singlet QCD-penguin amplitude S involving flavor SU(3)-singlet mesons; color-favored electroweak-penguin (EW-penguin) amplitude P EW ; and colorsuppressed EW-penguin amplitude P C EW . 2. Weak annihilation amplitudes: W -exchange amplitude E; W -annihilation amplitude A; QCD-penguin exchange amplitude PE; QCD-penguin annihilation amplitude PA; EW-penguin exchange amplitude PE EW ; and EW-penguin annihilation amplitude PA EW .
In this approach, the topological diagrams are classified according to the topologies in the flavor flow of weak decay diagrams, with all strong interaction effects included implicitly in all possible ways. Therefore, analyses of topological graphs can provide valuable information on final-state interactions.

A. Topological amplitudes
The topological amplitudes T, C, E, A are extracted from the Cabibbo-favored (CF) D → PP decays [47] to be (in units of 10 −6 GeV) for φ = 43.5 • [48], where φ is the η − η ′ mixing angle defined in the flavor basis with η q = 1 √ 2 (uū + dd) and η s = ss. The fitted χ 2 value is 0.135 per degree of freedom. Comparing with the amplitudes obtained in a previous fit in [49] T = 3.14 ± 0.06, we see that the errors in T , C, E and A are substantially reduced, especially for the annihilation amplitude A, thanks to the improved data precision from 2019 PDG [47]. We note in passing that since we will only fit to the observed branching fractions, the results will be the same if all the strong phases are subject to a simultaneous sign flip. Throughout this paper, we only present one of them. Presumably, such a degeneracy in strong phases can be resolved by measurements of sufficiently many CP asymmetries.
One of the most important moral lessons we have learnt from this approach is that all the topological amplitudes except the tree amplitude T given in Eq. (6) are dominated by nonfactorizable long-distance effects. For example, in the naïve factorization approach, the topological amplitudes T and C in CF D →Kπ decays have the expressions with a 1 = c 1 + c 2 /3 and a 2 = c 2 + c 1 /3. It turns out that a 1 (Kπ) ≈ 1.22 and a 2 (Kπ) ≈ 0.82e −i(151) • [49] extracted from the experimental values of T and C given in Eq. (6) and the phenomenological model for the D to K and π transition form factors. Since c 1 (m c ) ≈ 1.274 and c 2 (m c ) ≈ −0.529, it is evident that a 1 = c 1 + c 2 /3 ≈ 1.09 is close to a 1 (Kπ), while a 2 = c 2 + c 1 /3 ≈ −0.11 expected from naïve factorization is far off from a 2 (Kπ), including its size and phase. This implies that the short-distance contribution to C is very suppressed relative to the long-distance one. In the topological approach, the long-distance color-suppressed C is induced from the color-allowed T through final-state rescattering with quark exchange. The nontrivial relative phase between C and T indicates that final-state interactions (FSI's) via quark exchange are responsible for this. Likewise, short-distance weak annihilation diagrams are helicity suppressed, whereas data imply large sizes of them. This is because they receive large 1/m c power corrections from FSI's and large nonfactorizable contributions for a 2 . For example, the topological amplitude E receives contributions from the tree amplitude T via final-state rescattering with nearby resonance effects. The large magnitude and phase of weak annihilation can be quantitatively and qualitatively understood as elaborated in Refs. [50,51].
As emphasized in [37], one of the great merits of the topological approach is that the magnitude and the relative strong phase of each individual topological tree amplitude in charm decays can be extracted from the data. Consequently, direct CP asymmetries in charmed meson decays induced at the tree level can be reliably estimated as we shall discuss in Sec. III A.

B. Flavor SU(3) symmetry breaking
Using the topological amplitudes in Eq. (6) extracted from the CF modes, we can predict the rates for the SCS decays (see the second column of Table II below). It is known that there exists significant SU(3) breaking in some of the SCS modes from the flavor SU(3) symmetry limit. For example, the rate of D 0 → K + K − is larger than that of D 0 → π + π − by a factor of 2.8 [47], while the magnitudes of their decay amplitudes should be the same in the SU(3) limit. This is a long-standing puzzle since SU(3) symmetry is expected to be broken roughly at the level of 30%. Also, the decay D 0 → K 0 K 0 is almost prohibited in the SU(3) symmetry limit, but the measured branching fraction is of the same order of magnitude as that of D 0 → π 0 π 0 .
Since SU(3) breaking effects in D → PP decays have been discussed in detail in [38], in this section we will recapitulate the main points and update some of the results.
As stressed in [52], a most natural way of solving the above-mentioned long-standing puzzles is that the overall seemingly large SU(3) symmetry violation arises from the accumulation of several small and nominal SU(3) breaking effects in the tree amplitudes T and E. We will illustrate this point. Following [21], we write where λ p ≡ V * cp V up (p = d, s, b), the subscript refers to the quark involved in the associated penguin loop, and Likewise, As far as the rate is concerned, we can neglect the term with the coefficient λ b which is much smaller than (λ d − λ s ). SU(3)-breaking effects in the tree amplitudes T can be estimated in the factorization approach as where T is the tree amplitude in CF D → Kπ decays given in Eq. (9). Using the form-factor q 2 dependence determined experimentally from Ref. [53], we find SU(3) symmetry should be also broken in the W -exchange amplitudes. This can be seen from the observation of the decay D 0 → K 0 K 0 whose decay amplitude is given by with E q referring to the W -exchange amplitude associated with cū → qq (q = d, s). In the SU(3) limit, the decay amplitude is proportional to λ b and hence its rate is negligibly small, while experimentally B(D 0 → K 0 K 0 ) = (0.282 ± 0.010) × 10 −3 [47]. This implies sizable SU (3) symmetry violation in the W -exchange and QCD-penguin annihilation amplitudes. Neglecting PA and λ b terms and assuming that the T and E amplitudes are responsible for the SU(3) symmetry breaking, we can fix the SU(3) breaking effects in the W -exchange amplitudes from the following four D 0 decay modes: K + K − , π + π − , π 0 π 0 and K 0 K 0 [38]. A fit to the data yields two possible solutions: The corresponding χ 2 vanishes as these two solutions can be obtained exactly. If the SU(3)-breaking effects in the T and C topologies are ignored, we find that χ 2 will become very large, of order 340. This is understandable because the large rate disparity between K + K − and π + π − cannot rely solely on the nominal SU(3) breaking in the tree or W -exchange amplitudes. When considering SU(3)-breaking effects in T , we find that B(D 0 → π + π − ) is reduced slightly from 2.27 (in units of 10 −3 ) to 2.11, while B(D 0 → K + K − ) is increased substantially from 1.91 to 3.15 (see Eq. (14)). When E is replaced by E d = 1.10e i15 • E in the amplitude of D 0 → π + π − , the magnitude of (0.96T + E d ) in A(D 0 → π + π − ) becomes smaller than that of (0.96T + E) as the phase of E is about 121 • , so that B(D 0 → π + π − ) is decreased further from 2.11 to 1.47 . Likewise, with E being replaced by E s = 0.62e −i20 • E or E s = 1.42e −i14 • E in the amplitude of D 0 → K + K − , the magnitude of (1.27T + E s ) is enhanced relative to (1.27T + E). It follows that B(D 0 → K + K − ) is increased further from 3.15 to 4.03 or 4.05 . This shows that the seemingly large SU(3) symmetry violation in Γ(D 0 → K + K − ) and Γ(D 0 → π + π − ) simply follows from the accumulation of several smaller and nominal SU(3) breaking effects in the tree amplitudes T and E.
At the hadron level, flavor SU(3) breaking due to the strange and light quark differences will manifest in the decay constants, form factors, wave functions and hadron masses, etc. That is how we evaluate the SU(3)-breaking effect in the T amplitude via Eq. (13). Since the W -exchange is governed by long-distance effects, we do not know how to estimate its SU(3) symmetry violation.
Hence, we rely on the four modes: K + K − , π + π − , π 0 π 0 and K 0 K 0 to extract E d and E s .
Different mechanisms have been proposed in the literature for explaining the large rate difference between D 0 → π + π − and D 0 → K + K − . For example, it has been argued that ∆P dominated by the difference of s-and d-quark penguin contractions of 4-quark tree operators is responsible for the large SU(3) breaking in K + K − and π + π − modes [21]. However, this requires that |∆P/T | ∼ 0.5 . This mechanism demands a large penguin which is comparable or even larger than T . Moreover, it requires a large difference between s-and d-quark penguin contractions. In Sec. III B, we shall see that |∆P/T | is estimated to be of order 0.01 for the short-distance ∆P . Because of the smallness of ∆P , we need to rely on SU(3) violation in both T and E amplitudes to explain the large disparity in the rates of D 0 → K + K − and π + π − .
Another scenario in which the dominant source of SU(3) breaking lies in final-state interactions was advocated recently in [54]. To fit the data, several large strong phases such as δ 0 , δ 1 and δ 1/2 from final-state interactions are needed [54]. They deviate substantially from the SU(3) limit, namely, δ 0 = δ 1 = δ 1/2 . SU(3) breaking effects in the topological amplitudes for SCS D → P P decays are summarized in Table I. For simplicity, flavor-singlet QCD penguin, flavor-singlet weak annihilation and electroweak penguin annihilation amplitudes have been neglected in subsequent numerical analyses. The reader is referred to Refs. [38,49] in which we have illustrated SU(3) breaking effects in some selective SCS modes. The predicted and measured branching fractions are given in Table II. 3 While the agreement with experiment is improved for most of the SCS modes after taking into account SU(3) breaking effects in decay amplitudes, there are a few exceptions. For example, the predicted rate for D 0 → π 0 η ( ′ ) becomes slightly worse compared to the prediction based on SU(3) symmetry even though D + → π + η ( ′ ) works better in the presence of SU(3) breaking.

C. Penguin amplitudes in QCD factorization
Although the topological tree amplitudes T, C, E and A for hadronic D decays can be extracted from the data, information on penguin amplitudes (QCD penguin, penguin annihilation, etc.) is still needed in order to estimate CP violation in the SCS decays. To calculate the penguin contributions, we start from the short-distance effective Hamiltonian 3 Throughout this paper, predictions are made by sampling 10 4 points in the parameter space, assuming that each of the parameters has a Gaussian distribution with the corresponding central value and symmetrized standard deviation. Then the predicted values are the mean and standard deviation of data computed using the 10 4 points.  (3) shows the predictions based on our best-fitted results in Eq. (6) with exact flavor SU(3) symmetry, while SU(3) symmetry breaking effects are taken into account in the column denoted by B SU(3)−breaking . The first (second) entry in D 0 → ηη, ηη ′ , K + K − and K 0 K 0 modes is for Solution I (II) of E d and E s in Eq. (16). Experimental results of branching fractions are taken from PDG [47].
with O 3 -O 6 being the QCD penguin operators and (q 1 q 2 ) V ±A ≡q 1 γ µ (1 ± γ 5 )q 2 . We shall work in the QCD factorization (QCDF) approach [55,56] to evaluate the hadronic matrix elements, but keep in mind that we employ this approach simply for a crude estimate of the penguin amplitudes because the charm quark mass is not heavy enough and 1/m c power corrections are so large that a sensible heavy quark expansion is not allowed.
Let us first consider the penguin amplitudes in D → P 1 P 2 decays where p = d, s and is a chiral factor. Here we have followed the conventional Bauer-Stech-Wirbel definition for the form factor F DP 0 [57]. The explicit expressions of the flavor operators a p 4 and a p 6 will be given in Eq. (41) below. The annihilation operators b p 3,4 are given by where the annihilation amplitudes A i,f 1,2,3 are defined in Ref. [56]. In practical calculations of QCDF, the superscript 'p' can be omitted for a 3 , a 5 , b 3 and b 4 . Hence, we have PE s = PE d , for instance. For a p 4 and a p 6 , the terms dictating the 'p' dependence are G M 2 (s p ) andĜ M 2 (s p ), respectively, defined in Eq. (43) below.

III. DIRECT CP VIOLATION IN D → PP DECAYS
In Ref. [38], we have discussed direct CP violation in D → PP decays. Here we will update and improve the results. For example, we will discuss the issue of end-point divergences with the penguin-exchange and penguin-annihilation amplitudes. We will also consider the uncertainties connected with long-distance contribution to the penguin-exchange amplitude. We shall keep some necessary formula presented in [38] for ensuing discussions.

A. Tree-level CP violation
Direct CP asymmetry in hadronic charm decays defined by can occur even at the tree level [58]. As stressed in [37,38], the estimate of the tree-level CP violation a (tree) dir should be trustworthy since the magnitude and the relative strong phase of each individual topological tree amplitude in charm decays can be extracted from the data. The predicted tree-level CP asymmetries for SCS modes are shown in Table III. We see that larger CP asymmetries can be achieved in those decay modes with interference between T and C or C and E. For example, a where δ ds is the strong phase of E s relative to E d . From the two solutions of E d and E s given in Eq. (16), we find 4 For comparison, various predictions available in the literature are discussed here. a (tree) dir (K S K S ) = 1.11 × 10 −3 was predicted in [40]. It ranges in (0.38 − 0.43) × 10 −3 according to [54] (see also the last column of Table III). Both predictions are of the opposite sign from ours. As explained in [38], the positive sign of a (tree) dir (K S K S ) given in [40] can be traced back to the phase of the W -exchange amplitude. In our case, the W -exchange amplitude is always in the second quadrant, while it lies in the third quadrant in [40] due to a sign flip. As noticed in passing, all the strong phases extracted from a fit to branching fractions are equivalent to those with a simultaneous sign flip. This explains why the strong phases of C and E in [40] are simultaneously opposite to ours in sign, and the sign difference between this work and [40] for a (tree) will resolve the discrete phase ambiguity. If it is measured to be negative as predicted by us, then the W -exchange amplitude should be in the second quadrant.
In [23], the direct CP violation in Taking a dir CP (D 0 → K + K − ) to be (−0.48 ± 0.09) × 10 −3 from Table III and the measured branching fractions, the obtained result a dir CP (D 0 → K S K S ) ≈ −1.8 × 10 −3 is in agreement in magnitude and sign with ours. a dir CP (D 0 → K S K S ) was estimated to be 0.6% in [21], while an upper bound |a dir (D 0 → K S K S )| ≤ 1.1% was set in [59].
The current experimental measurements are Since LHCb has measured ∆A CP to the accuracy of 10 −3 , it is conceivable that an observation of CP violation in the decay D 0 → K S K S will be feasible in the near future. denotes CP asymmetry arising from purely tree amplitudes. The superscript (t+p) denotes tree plus QCDpenguin amplitudes, (t+pa) for tree plus weak penguin-annihilation (PE and PA) amplitudes and "tot" for the total amplitude. The first (second) entry in D 0 → ηη, ηη ′ , K + K − and K S K S is for Solution I (II) of E d and E s [Eq. (16)]. For QCD-penguin exchange PE, we assume that it is similar to the topological E amplitude [see Eq. (33)]. For comparison, The predicted results of a (tot) dir in [54] for both the negative (former) and positive (latter) solutions for the phase δ i are also presented.

B. Penguin-induced CP violation
Direct CP violation does not occur at the tree level in D 0 → K + K − and D 0 → π + π − . In these two decays, the CP asymmetry arises from the interference between tree and penguin amplitudes. From Eq. (10), we obtain where δ ππ is the strong phase of (P s + PE s + PA s ) ππ relative to (T + E + ∆P ) ππ and likewise for a dir CP (K + K − ). Hence, with δ KK being the strong phase of ( Using the input parameters for the light-cone distribution amplitudes of light mesons, quark masses and decay constants from Refs. [63,64] and form factors from Refs. [49,65], we find to the leading order in Λ QCD /m b in QCDF that It is obvious that ∆P = P d −P s arising from the difference in the d-and s-loop penguin contractions [see Eq. (41)] is very small compared to the tree amplitude. It is straightforward to show for Solutions I and II of W -exchange amplitudes E d and E s (see Eq. (16)). It follows from Eq. (28) that a dir CP (π + π − ) = 0.029 × 10 −3 , and Evidently, CP asymmetries in D 0 → π + π − , K + K − induced by QCD penguins are very small mainly due to the strong phases δ ππ and δ KK being not far from 180 • . So far we have only discussed leading-order QCDF calculations except for the chiral enhanced penguin contributions, namely, the a 6 terms in Eq. (19). For QCD-penguin power corrections, we shall consider weak penguin annihilation, namely, QCD-penguin exchange PE and QCD-penguin annihilation PA which are formally of order 1/m c . However, it is well known that the weak penguin annihilation amplitudes in QCDF derived from Eq. (19) involve troublesome endpoint divergences [55,56]. Hence, subleading power corrections generally can be studied only in a phenomenological way. For example, the endpoint divergence is parameterized as [55,56] with Λ h being a typical hadronic scale of order 500 MeV, and ρ A , φ A being unknown real parameters. In hadronic B decays, the values of ρ A and φ A can be obtained from a fit to B → PP, VP and VV decays [66]. However, this is not available in charmed meson decays since penguin effects manifest mainly in CP violation. Therefore, we will not evaluate PE and PA in this way in the charm sector. Nevertheless, if we borrow typical values of ρ A and φ A from the B system, we find weak penguin annihilation contributions smaller than QCD penguin; for instance, (PE/T ) ππ ∼ 0.04 and (PA/T ) ππ ∼ −0.02. Therefore, it is safe to neglect short-distance contributions to weak penguin annihilation amplitudes.
As pointed out in [37], long-distance contributions to SCS decays, for example, D 0 → π + π − , can proceed through the weak decay D 0 → K + K − followed by a resonant-like final-state rescattering as depicted in Fig. 2 of [37]. It has the same topology as the QCD-penguin exchange topological graph PE. Since weak penguin annihilation and FSI's are both of order 1/m c in the heavy quark limit, this means FSI's could play an essential role in charm decays. Hence, it is plausible to assume that PE is of the same order of magnitude as E. In [37], we took (PE) LD = 1.60 e i115 • (in units of 10 −6 GeV). In this work we will assign by choice the same magnitude and phase as E with 20% and 30 • uncertainties, respectively, so that For simplicity, we shall assume its flavor independence, that is, (PE) LD d = (PE) LD s . Including the long-distance contribution to penguin exchange PE, we get As shown in Table III, we see that the predicted CP violation denoted by a (tot) dir or a (tree) dir is at most of order 10 −3 in the SM. Specifically, we have 5 Theoretical uncertainties are dominated by that of (PE) LD . Hence, the CP asymmetry difference between D 0 → K + K − and D 0 → π + π − is given by Although our new results of ∆a dir CP are slightly smaller than the previous ones in [38], they have more realistic estimates of uncertainties and are consistent with the LHCb's new measurement in Eq. (4) within 1σ. Here we note in passing that the CP asymmetry predictions are very sensitive to (PE) LD . Had we chosen to use the value of 1.60 × 10 −6 e i121 • GeV, as done in [37], ∆a dir CP would become (−1.24 ± 0.26) × 10 −3 for Solution I and (−1.34 ± 0.25) × 10 −3 for Solution II. 5 Since Eqs. (34) and (27) lead to a dir CP (π + π − ) = 0.91 × 10 −3 and a dir CP (K + K − ) = −0.40 × 10 −3 for Solution I and −0.51 × 10 −3 for Solution II, the reader may wonder why they are slightly larger in magnitude than the final results presented in Table III. Such a difference is related to the fact that the predictions are made, as alluded to in Footnote 3, statistically and the fact that CP asymmetries are not linear in the parameters.

C. Comparison with Li et al. [40]
Based on the so-called factorization-assisted topological-amplitude approach, an estimate of ∆a CP = −1.00 × 10 −3 in the SM was made in [40]. In this work, the topological amplitudes in units of 10 −6 GeV are given by 6 As a result, This leads to the aforementioned value of ∆a CP . For comparison, in our case we have There are three crucial differences between this work and [40]: (i) the phase of E amplitudes is in the second quadrant in the former while in the third or fourth quadrant in the latter, (ii) the phase of the penguin amplitude P is in the third quadrant in our work while in the second quadrant in [40], and (iii) our PE amplitude comes from long-distance final-state rescattering as we have neglected short-distance contributions to weak penguin annihilation amplitudes PE and PA. As discussed in passing, there is a discrete phase ambiguity for the phases of C, E and A topological amplitudes in our analysis. Presumably, a measurement of a dir CP (D 0 → K S K S ) will resolve the discrete phase ambiguity for the E amplitude. However, the phase of the penguin amplitude is calculated in theory. Let us examine this issue as follows.
Consider the penguin amplitude P p P 1 P 2 given in Eq. (19). Within the framework of QCDF, the flavor operators a p 4,6 are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex corrections V i , penguin contractions P i and hard spectator interactions H i : where the explicit expressions of V i and H i can be found in [56]. The order α s corrections from penguin contraction read [56] and G(s, x) = −4 In [40], the flavor operators a 4,6 and a 1,2 are taken to be Comparing Eq. (44) with Eq. (41), we see that the source of the QCD penguin's strong phase is assumed to be the same as that of a 2 in [40], while it arises from nonfactorizable contributions in QCDF. In other words, while we consider the effects of vertex corrections, penguin contractions and hard spectator interactions for the QCD penguin amplitude, these effects are parameterized in [40] in terms of χ nf and φ, which are determined from a global fit to the measured branching fractions. Since the color-suppressed C amplitude in [40] is in the second quadrant, so is the penguin amplitude. This explains the difference between our work and [40] for the QCD penguin amplitudes.

D. Comparison with Chala et al. [31]
Based on the light-cone sum rule calculations of Khodjamirian and Petrov [36] argued an upper bound in the SM, |∆a SM CP | ≤ (2.0 ± 0.3) × 10 −4 . Including higher-twist effects in the operator product expansion for the underlying correlation functions which are expected to be Chala et al. [31] claimed a modification of the SM bound, |∆a SM CP | ≤ (2.0 ± 1.0) × 10 −4 . This conclusion seems to be very naïve. First, as stated in [36], Khodjamirian and Petrov have neglected the contributions from the penguin operators O i=3,··· ,6,8g due to their small Wilson coefficients. This means they only considered the penguin contraction from the tree operators O 1,2 . Consequently, Secondly, penguin-exchange and penguin-annihilation contributions have not been considered, not mentioning the possible final-state resattering effect on PE. They play an essential role in understanding the LHCb measurement of ∆a CP . Otherwise, it is premature to claim the necessity of New Physics in this regard.

IV. D → VP DECAYS
In the treatment of D → VP decays, we continue to use the same topological diagram notation as in the PP decays, except that a subscript of V or P is attached to the flavor amplitudes and the associated strong phases to denote whether the spectator quark in the charmed meson ends up in the vector or pseudoscalar meson in the final state. The V -type and P -type parameters are completely independent a priori, though certain relations can be established under the factorization assumption.

A. Topological amplitudes
The partial decay width of the D meson into a vector and a pseudoscalar mesons are usually expressed in two different ways: and Even though both formulas have the same cubic power dependence on p c (as required for a P-wave configuration), a main difference resides in the fact that the latter has incorporated an additional SU(3)-breaking factor for the phase space, resulting from the sum of possible polarizations of the vector meson in the final state. By performing a χ 2 fit to the CF D → V P decays, we extract the magnitudes and strong phases of the topological amplitudes T V , C V , E V , A V and T P , C P , E P , A P from the measured partial widths through Eq. (48) or (49) and find many possible solutions with local χ 2 minima. Here we take the convention that all strong phases are defined relative to the T V amplitude. In 2016 we have performed a detailed analysis and obtained some best χ 2 fit solutions (A) and (S) through Eqs. (48) and (49), respectively [42]. It turns out that solutions (S) give a better description for SCS decays such as D 0 → π + ρ − , π 0 ρ 0 and D + → π + ρ 0 , possibly because the additional SU(3)-breaking factor in phase space has been taken care of, as mentioned above. Hence, we will confine ourselves to using Eq. (49) and thus solutions (S) in this work. The six best χ 2 -fit solutions (S1)-(S6), with χ 2 min < 10, are listed in Table V, where we have chosen the convention such that the central values of strong phases to fall between 0 and 360 degrees, while noting again that a simultaneous sign flip of all strong phases is equally viable. The flavor amplitudes of all these solutions respect the hierarchy pattern, [42], the decay D + s → ρ 0 π + plays an essential role in the determination of the annihilation amplitudes A V,P . Its large error in the branching fraction reflects in the large uncertainties in the magnitudes and strong phases of A V,P , which will be improved once we have a better measurement of D + s → ρ 0 π + . While the size of each topological amplitude is similar across all solutions, the strong phases vary among the solutions except for those of C P and E P . We find (δ C P , δ E P ) to be either (201 • , 108 • ) or (159 • , 252 • ). A close inspection tells us that Solutions (S1) and (S4) are close to each other in the sense that the corresponding amplitudes are similar in size, except for |A V | and |A P |, and the corresponding strong phases add up to roughly 360 • . So are Solutions (S2) and (S5).
Although solutions in set (S) generally fit the Cabibbo-favored modes well (see Table VI for results based on Solutions (S3) and (S6)), there are two exceptions, namely, D + s → K 0 K * + and ρ + η ′ , where the predictions are smaller than the experimental results. The first mode was measured three decades ago with a relatively large uncertainty [67], and the experimental result was likely  to be overestimated. The second mode has a decay amplitude respecting a sum rule [42]: Assuming this relation, the current data of B(D + s → π + ω) and B(D + s → ρ + η) give the bounds 1.6% < B(D + s → ρ + η ′ ) < 3.9% at 1σ level, significantly lower than the current central value. A better determination of these branching fractions will be very helpful in settling the issues.
Various (S) solutions lead to very different predictions for some of the SCS decays. Especially, the D 0 → π 0 ω and D + → π + ω decays are very useful in discriminating among different solutions. We first consider the π 0 ρ 0 , π 0 ω and ηω modes. Their topological amplitudes are given by Since the magnitude of C V is comparable to that of C P , the smallness of B(D 0 → π 0 ω), the sizable B(D 0 → ηω) and the large B(D 0 → π 0 ρ 0 ) imply that the strong phases of C V and C P should be close to each other. An inspection of Table V indicates that the phase difference between C V and C P is large for Solutions (S1), (S2) and (S5). It turns out that (S2) and (S5) are definitely ruled out as they predict too large B(D 0 → π 0 ω), with the central values of 4.65 and 3.91 (in units of 10 −3 ), respectively, while the measured value is 0.117 ± 0.035 (see Table VII). Solution (S1) gives a relatively better prediction of B(D 0 → π 0 ω) = 0.62 ± 0.13 among the three solutions. We next turn to the π + ρ 0 and π + ω modes. Neglecting the penguin contributions, their topo-

Meson Mode
Representation 11.3 ± 0.7 11.4 ± 0.6 11.7 ± 0.8 1.30 ± 0.12 1.35 ± 0.13 3.92 ± 0.14 3.94 ± 0.18 3.94 ± 0.18 ± 0.8 9.02 ± 0.37 8.86 ± 0.38 3.25 ± 0.12 2.92 ± 0.11 0.020 ± 0.012 0.023 ± 0.014 0.024 ± 0.016  [68] has been taken into account in the world average. logical amplitudes read (see Table VII) It is well known that the CF decays D + s → π + ρ 0 and π + ω can only proceed through the Wannihilation topology The extremely small branching fraction of D + s → π + ρ 0 compared to D + s → π + ω (see Table V) implies that A V and A P should be comparable in magnitude and roughly parallel to each other with a phase difference not more than 30 • . At a first glance, it is tempting to argue from Eq. (52) that D + → π + ω should have a rate larger than D + → π + ρ 0 . Experimentally, it is the other way around [47]: Since C P is comparable to T V in magnitude, there is a large cancellation between T V and C P . As a consequence, the rates of π + ρ 0 and π + ω become sensitive to the strong phases of the small annihilation amplitudes A V and A P . It turns out that A V should be in the 4th quadrant while A P in the third quadrant in order to satisfy the experimental constraints from Eq. (54). We find only Solutions (S3) and (S6) in line with this requirement (see Table V) and yielding predictions in agreement with experiment for π + ρ 0 and π + ω (see Table VII). For Solutions (S1), (S2), (S4) and (S5), the branching fractions of D + → π + ρ 0 and D + → π + ω (in units of 10 −3 ) are found to have the central values (0.45, 1.06), (0.87, 0.98), (0.67, 1.05), (0.96, 1.76), respectively. All these solutions imply that the latter is larger than the former in rates, in contradiction with experiment. Finally, we comment on two of the D + s decay modes: K + ρ 0 and K + ω. From Table VII, we see that Since |C P | ≫ |A P |, it is expected that the two modes have similar branching fractions of order 2×10 −3 . However, the recent BESIII experiment yields B(D + s → K + ω) = (0.87±0.25)×10 −3 [68]. The ρ − ω mixing effect to be mentioned below in Eq. (63) in principle can push up (down) the rate of K + ρ 0 (K + ω). For the mixing angle ǫ = −0.12 (see Eq. (63) and note a sign difference from [69]), we find B(D + s → K + ρ 0 ) = (2.63 ± 0.11) × 10 −3 and B(D + s → K + ω) = (1.65 ± 0.09) × 10 −3 . The former is now in better agreement with experiment, but the latter is still too large compared to the data. The ω − φ mixing also does not help much. Moreover, in our framework we do not need ρ − ω mixing to explain the smallness of D 0 → π 0 ω and D + → π + ω. Therefore, the issue with D + s → K + ω remains to be resolved.

B. Flavor SU(3) symmetry breaking
As noted in passing, a most noticeable example of SU(3) breaking in the PP sector lies in the decays D 0 → K + K − and D 0 → π + π − . Experimentally, the rate of the former is larger than that of the latter by a factor of 2.8 . More precisely, |T + E| KK /|T + E| ππ ≈ 1.80, implying a large SU(3) breaking effect in the amplitude of T + E. However, it is the other way around for the counterparts in the VP sector where we have Γ(K + K * − ) < Γ(π + ρ − ) and Γ(K − K * + ) < Γ(π − ρ + ). Since the available phase space is proportional to p 3 c /m 2 V in the convention of Eq. (49), this explains why Γ(D 0 → KK * ) < Γ(D 0 → πρ) owing to the fact that p c (πρ) = 764 MeV and p c (KK * ) = 608 MeV. From the measured branching fractions, we find by ignoring the penguin amplitudes that This implies that SU(3) breaking in the amplitudes of T V + E P and T P + E V is small, contrary to the PP case.
In Table VII, we show the calculated branching fractions of SCS D → V P decays using Solutions (S3) and (S6). It is clear that Solution (S6) is slightly better, though the predicted K 0 K * 0 and K 0 K * 0 branching fractions are too large compared to the data in both solutions. SU(3) breaking effects in the color-allowed and color-suppressed amplitudes can be estimated provided they are factorizable: Hence, .
Assuming that a 1 (ρπ) is similar to a 1 (K * π) and a 1 (Kρ), we find Similar relations can be derived for the C V and C P amplitudes as well. These lead to two difficulties: (i) The sizable SU (3) breaking in the ratios |T V | π + ρ − /|T V | K + K * − ≃ 0.64 and |T P | π − ρ + /|T V | K − K * + ≃ 0.72 are not consistent with Eq. (56), and (ii) the branching fractions of D 0 → π + ρ − and D 0 → π − ρ + will become smaller, while B(D 0 → K + K * − ) and B(D 0 → K − K * + ) become larger. Hence, the discrepancy becomes even worse. In other words, the consideration of SU(3) breaking in the tree amplitudes T V,P and C V,P alone will render even larger deviations from the data in both Solutions (S3) and (S6). A way out is to consider SU(3) breaking in the W -exchange amplitudes. Indeed, the too large rates predicted for K 0 K * 0 and K 0 K * 0 modes call for SU(3) breaking in the W -exchange amplitudes as both modes proceed through E P and E V . In the P P sector, we need SU(3) breaking in Wexchange in order to induce D 0 → K S K S . Here we need SU(3) breaking again for a different reason, otherwise, the calculated D 0 → K 0 K * 0 and K 0 K * 0 will be too large in rates. Since |E P | ≫ |E V |, it is natural to expect that |E P | (|E V |) has to be reduced (increased) after SU(3) breaking in order to accommodate the data. Writing and with we are able to determine the eight unknown parameters e d V , e d P , e s V , e s P and δe d V , δe d P , δe s V , δe s P from the branching fractions of these eight modes. In the SU (3) Table IX using the topological amplitudes given in Solution (S6). For SU(3) breaking effects in W -exchange amplitudes E V and E P , we specifically choose solution (iv) for SU(3) breaking parameters given in Table VIII, though the results are very similar in other schemes. The decays D 0 → π 0 φ and D + → π + φ are special as they proceed only through the internal W -emission diagram C P . Its SU(3) breaking can be estimated from Eq. (58) to be For the q 2 dependence of the form factor we use with F Dπ 1 (0) = F Dπ 0 (0) = 0.666 and α Dπ 1 = 0.24, and find C πφ P = 1.37C P . The resulting B(D 0 → π 0 φ) = (1.22 ± 0.04) × 10 −3 and B(D + → π + φ) = (6.29 ± 0.21) × 10 −3 are consistent with experiment, though the latter is slightly large in the central value.
Comparison with the work of Qin et al. [69] In Table IX, we have compared our results of SCS D → VP branching fractions with that in the factorization-assisted topological approach [69] without and with the ρ − ω mixing, denoted by FAT and FAT[mix], respectively. The predicted B(D 0 → π 0 ρ 0 ) = 0.85 (in units of 10 −3 ) in FAT is far too large compared to the data of 0.117 ± 0.035. In order to resolve this discrepancy, Qin et al. considered the ρ − ω mixing defined by where |ρ 0 I and |ω I denote the isospin eigenstates. Using the mixing angle ǫ = 0.12, the predicted branching fraction of D 0 → π 0 ω ia reduced to 0.18, while B(D 0 → π 0 ρ 0 ) is increased from 3.55 to  Table VIII). For QCD-penguin exchanges PE V and PE P , we assume that they are similar to the topological E V and E P amplitude, resepctively [see Eq. (64)]. The results from [69] in the factorizationassisted topological approach without and with the ρ − ω mixing (denoted by FAT  3.83. However, the calculated B(D + → π + ω) = 0.80 after taking into account of ρ − ω mixing is still too large compared to the experimental value of 0.28 ± 0.06. As for the D + s → K + ω mode, it appears that the predicted branching fraction of 0.6 before ρ − ω mixing agrees with the data of 0.87 ± 0.25 [68], while the predicted value of 0.07 after the mixing effect is far too small. Therefore, irrespective of ρ − ω mixing, D 0 → π 0 ω and D + s → K + ω cannot be explained simultaneously in the FAT or modified FAT approach.

C. Direct CP violation
It has been noticed that weak penguin annihilation will receive sizable long-distance contributions from final-state rescattering. We shall assume that the long-distance PE V and PE P are of the same order of magnitude as E V and E P in Solution (S6), respectively. For concreteness, we take (in units of 10 −6 ) The calculated results are shown in Table X. In comparison, the predictions given in [69] in general are substantially smaller than ours in magnitude. We find several golden modes for the search of CP violation in the VP sector: These modes are "golden" in the sense that they have large branching fractions and sizeable CP asymmetries of order 1 × 10 −3 . It is interesting to notice that the CP asymmetry difference defined by in analogy to ∆A PP CP defined in Eq. (1), is predicted to be (−1.52 ± 0.43) × 10 −3 , which is very similar to the recently observed CP violation in D 0 → K + K − and D 0 → π + π − . It is thus desirable to first search for CP violation in the aforementioned golden modes.

V. DISCUSSIONS AND CONCLUSIONS
In this analysis, we have revisited two-body hadronic charmed meson decays to PP and VP final states, where P and V denote light pseudoscalar and vector mesons, respectively. Taking flavor SU(3) symmetry as our working assumption for the Cabibbo-favored decays, we extract tree-type flavor amplitudes through a global fit to the latest experimental data. We then discuss whether and how SU(3) symmetry breaking factors should be taken into account when moving on to the singly Cabibbo-suppressed decay modes. We have made predictions for the branching fractions as well as the CP asymmetries for these decay modes where we observe that the importance of penguin-type amplitudes, if present, often significantly modify the latter.
In the P P sector, several SU(3) breaking effects are crucial in explaining the measured branching fractions of singly Cabibbo-suppressed decay modes, as already noticed in Ref. [38]. The T and C denotes CP asymmetry arising from purely tree amplitudes. The superscript (t+p) denotes tree plus QCD-penguin amplitudes, (t+pa) for tree plus weak penguin-annihilation (PE and PA) amplitudes and "tot" for the total amplitude. amplitudes should be scaled by a factor given under the factorization assumption. We acknowledge that the E amplitude is governed mainly by long-distance rescattering effects and, therefore, the associated symmetry breaking factors need to be obtained via a fit to the eight D 0 decays. In particular, one has to distinguish between two types of W -exchange amplitudes: E d and E s , depending upon whether it is dd or ss pair coming out of the exchange diagram. While |E d | is about 10% larger in magnitude than |E| of the Cabibbo-favored modes, |E s | has two possibilities: either larger or smaller than |E| by about 40%, as given in Eq. (16). The above-mentioned SU(3) symmetry breaking effects are most notably successful in explaining the large disparity between B(D 0 → π + π − ) and B(D 0 → K + K − ).
To test among different theory models, we have proposed to have a better precision in the measurement of a CP (D 0 → K S K S ), which is primarily due to interference between E d and E s amplitudes. We also revisit the CP asymmetry difference between D 0 → K + K − and D 0 → π + π − and find two results: ∆a dir CP = (−1.14 ± 0.26) × 10 −3 for Solution I and (−1.25 ± 0.25) × 10 −3 for Solution II. Both of them are consistent with the latest LHCb result [29] within 1σ. We have also observed that these predictions are sensitive to the assumed contribution from weak penguin annihilation diagrams. Comparisons with a few other works are made to highlight the distinctive features of our approach.
In contrast to the PP sector, a global fit to the Cabibbo-favored modes in the VP sector gives many solutions with similarly small local minima in χ 2 (six of them, as listed in Table V, when we restrict ourselves to χ 2 min < 10), revealing significant degeneracy in the current data. These solutions can explain the Cabibbo-favored decay branching fractions well except for the D + s → K 0 K * + and ρ + η ′ modes. For the former, we urge the experimental colleagues to update the figure. For the latter, an amplitude sum rule confines its branching fraction in the range (1.6, 3.9)% at the 1σ level. The above-mentioned solution degeneracy is lifted once we use them to predict for the singly Cabibbo-suppressed modes. In the end, we find that only Solution (S3) and (S6) which have a common feature that C V and C P are close in phase in order to simultaneously explain the small B(D 0 → π 0 ω) and large B(D 0 → π 0 ρ 0 ). Another common feature is that A V and A P are comparable in size and similar in phase, in order to simultaneously explain the small B(D + → π + ω) and large B(D + → π + ρ 0 ). We note that the recent BESIII result of B(D + s → K + ω) is a factor of two to three smaller than our prediction, and remains an issue to be resolved.
Unlike the PP sector, the singly Cabibbo-suppressed decay data in the VP sector do not call for an introduction of SU(3) breaking for the T V,P and C V,P amplitudes dictated by the factorization assumption. Instead, SU(3) breaking in E V,P is still required and, analogous to the P P sector, one should consider different long-distance effects on diagrams with dd and ss emerging from the W exchange. A fit to eight singly Cabibbo-suppressed D 0 decays shows that the symmetry breaking effects are often large. We have identified the set with the smallest SU(3) breaking in the E-type amplitudes (< 50%) of Solution (S6) as our best solution and make predictions for the branching fractions and CP asymmetries of the singly Cabibbo-suppressed decays. In particular, we point out that the D 0 → π + ρ − , K + K * − , D + → K + K * 0 , ηρ + , and D + s → π + K * 0 , π 0 K * + modes have sufficiently large branching fractions and CP asymmetries at per mille level. Interestingly, ∆a VP CP ≡ a CP (K + K * − ) − a CP (π + ρ − ) ≃ (−1.52 ± 0.43) × 10 −3 , very similar to the recently observed CP violation in D 0 → K + K − and D 0 → π + π − .