Anatomy of $B\rightarrow D\bar{D}$ Decays

The decays $B_d^0\rightarrow D_d^-D_d^+$ and $B_s^0\rightarrow D_s^-D_s^+$ probe the CP-violating mixing phases $\phi_d$ and $\phi_s$, respectively. The theoretical uncertainty of the corresponding determinations is limited by contributions from penguin topologies, which can be included with the help of the $U$-spin symmetry of the strong interaction. We analyse the currently available data for $B^0_{d,s}\rightarrow D_{d,s}^-D_{d,s}^+$ decays and those with similar dynamics to constrain the involved non-perturbative parameters. Using further information from semileptonic $B_d^0\rightarrow D_d^-\ell^+\nu_{\ell}$ decays, we perform a test of the factorisation approximation and take non-factorisable $SU(3)$-breaking corrections into account. The branching ratios of the $B_d^0\rightarrow D_d^-D_d^+$, $B_s^0\rightarrow D_s^-D_d^+$ and $B_s^0\rightarrow D_s^-D_s^+$, $B_d^0\rightarrow D_d^-D_s^+$ decays show an interesting pattern which can be accommodated through significantly enhanced exchange and penguin annihilation topologies. This feature is also supported by data for the $B_s^0\rightarrow D_d^-D_d^+$ channel. Moreover, there are indications of potentially enhanced penguin contributions in the $B_d^0\rightarrow D_d^-D_d^+$ and $B_s^0\rightarrow D_s^-D_s^+$ decays, which would make it mandatory to control these effects in the future measurements of $\phi_d$ and $\phi_s$. We discuss scenarios for high-precision measurements in the era of Belle II and the LHCb upgrade.


Introduction
CP-violating effects offer important tools to search for new physics (NP) beyond the Standard Model (SM). In this endeavour, B 0 q -B 0 q mixing (q = d, s) is a key player. This phenomenon does not arise at the tree level in the SM and may induce interference effects between oscillation and decay processes, resulting in "mixing-induced" CP violation. The BaBar and Belle experiments at the e + e − B-factories and the LHCb experiment at the Large Hadron Collider (LHC) have already performed high precision measurements of the B 0 d -B 0 d and B 0 s -B 0 s mixing phases φ d and φ s , respectively [1,2]. In the era of the Belle II [4] and LHCb upgrade [3], the experimental analysis will be pushed towards new frontiers of precision.
In this paper, we present an analysis of the decays B 0 which are related to each other through the U -spin symmetry of strong interactions [5,6]. With the help of this flavour symmetry, penguin effects can be included in the determination of φ d and φ s from the mixing-induced CP asymmetries of these decays. The theoretical precision is limited by non-factorisable U -spin-breaking effects. The impact of these contributions can be probed in a clean way by comparing branching ratio measurements of the non-leptonic decays with data from semileptonic B 0 d → D − d + ν and B 0 s → D − s + ν decays. The use of the latter two modes is a new element in this strategy. We also explore the role of exchange and penguin annihilation topologies, which govern the decays B 0 d → D − s D + s and B 0 s → D − d D + d [7]. These modes are also related to each other by the U -spin symmetry of strong interactions.
The analysis of the B 0 d → D − d D + d , B 0 s → D − s D + s system complements the determination of φ d and φ s from the decays B 0 d → J/ψK 0 S and B 0 s → J/ψφ, respectively, where penguin effects have to be included as well [5,[8][9][10][11][12][13][14][15][16][17]. The dynamics of the B → DD modes differs from those of the B → J/ψX channels. In the latter case, the QCD penguins require a colour-singlet exchange and are suppressed by the Okubo-Zweig-Iizuka (OZI) rule [18][19][20], while this feature does not apply to the electroweak (EW) penguins, which are colour-allowed and hence contribute significantly to the decay amplitudes [21]. On the other hand, the QCD penguins are not OZI suppressed in the B 0 s → D − s D + s system, whereas the EW penguins contribute only in colour-suppressed form. The EW penguin sector offers an interesting avenue for NP to enter weak meson decays [22][23][24][25], such as in models with extra Z bosons [26,27], and would then lead to discrepancies in the determined values of φ d and φ s should the Z bosons have CP-violating flavour-changing couplings to quarks.
The outline of this paper is as follows: in Section 2, we discuss the decay amplitude structure of the B 0 d → D − d D + d , B 0 s → D − s D + s decays and their observables, while we turn to the picture emerging from the current data in Section 3. There, we include additional decay modes, which have dynamics similar to the B 0 d → D − d D + d , B 0 s → D − s D + s system, to address the importance of exchange and penguin annihilation topologies, and probe non-factorisable effects by means of the differential B 0 d → D − d + ν rate. We perform a global analysis of the penguin parameters a and θ, which allows us to extract φ d and φ s from measurements of CP violation in the B 0 d → D − d D + d and B 0 s → D − s D + s modes, respectively. The current uncertainties of these measurements are unfortunately still very large. In Section 4, we focus on the era of the Belle II and LHCb upgrade, and explore the prospects by discussing different scenarios. Finally, we summarise our conclusions in Section 5. In an appendix, we give a summary of the various parameters and observables used in our analysis. 2 Decay Amplitudes and Observables

Amplitude Structure
The B 0 d → D − d D + d mode is caused byb →ccd quark-level transitions, and in the SM receives contributions from the decay topologies illustrated in Fig. 1. The decay amplitude takes the following form [5]: where γ serves as a CP-violating weak phase and is the usual angle of the unitarity triangle (UT) of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [28,29], while and are CP-conserving hadronic parameters. Here T and P (q) denote the strong amplitudes of the (colour-allowed) tree and penguin topologies (with internal q-quark exchanges), respectively, which can be expressed in terms of hadronic matrix elements of the corresponding low-energy effective Hamiltonian. We have also included the amplitudes describing exchange E and penguin annihilation P A (q) topologies, which are naively expected to play a minor role [30]. However, we find that the current data imply sizeable contributions for E + P A (c) − P A (t) with respect to T + P (c) − P (t) . The parameter (4) measures the side of the UT originating from the origin of the complex plane with the angle γ between the real axis, while λ ≡ |V us | = 0.22548 ± 0.00068 is the Wolfenstein parameter of the CKM matrix [31], and A ≡ |V cb |/λ 2 = 0.806 ± 0.017; the numerical values refer to the analysis of Ref. [32]. The U -spin partner B 0 s → D − s D + s of the B 0 d → D − d D + d channel originates from thē b →ccs quark-level processes. Its SM transition amplitude can be written as where the hadronic parameters A and a e iθ are given by expressions which are analogous to those in Eqs. (2) and (3), respectively. The key difference in the structure of the B 0 s → D − s D + s decay amplitude with respect to Eq. (1) is the suppression of the a e iθ e iγ term by the tiny CKM parameter ≡ λ 2 1 − λ 2 = 0.0536 ± 0.0003 .
Moreover, the overall factor of λ is absent, thereby enhancing the decay rate with respect to B 0 d → D − d D + d . Therefore, sizeable penguin effects in B 0 d → D − d D + d could be probed and subsequently used to estimate the penguin effects in B 0 s → D − s D + s , applying U -spin symmetry.
The U -spin symmetry of strong interactions implies the following relations between the hadronic parameters: ae iθ = a e iθ , It is important to emphasise that hadronic form factors and decay constants cancel within factorisation in ae iθ and a e iθ , since these quantities are defined as ratios of hadronic amplitudes, as can be seen in Eq. (3). Consequently, factorisable U -spinbreaking corrections to the relation in Eq. (7) vanish [5,6]. On the other hand, the U -spin relation in Eq. (8) is affected by SU (3)-breaking effects 1 in B q → D q form factors and D q decay constants (q = d, s). We discuss these effects in more detail later.

CP-Violating Asymmetries
Due to B 0 q -B 0 q oscillations (q = d, s), an initially present B 0 q -meson state evolves in time into a linear combination of B 0 q and B 0 q states. CP violation in the B 0 q → D − q D + q decays, which are characterised by CP-even final states, is probed through the following time-dependent rate asymmetries [33]: where ∆M q ≡ M H denote the mass and decay width difference between the two B q mass eigenstates, respectively. The three CP observables are given by 2 where we have to make the following replacements for the decays at hand [5]: The parameter η q denotes the CP eigenvalue of the final state and is given by +1. While the direct CP asymmetries A dir CP (B q → D − q D + q ) are caused by interference between tree and penguin contributions, the mixing-induced CP asymmetries A dir CP (B q → D − q D + q ) originate from interference between B 0 q -B 0 q mixing and decay processes, and depend on the mixing phases φ d and φ s . These quantities take the general forms where β is the usual angle of the UT. The SM value of φ s , which is given by −2β s = −2λ 2 η and hence doubly Cabibbo suppressed, can be determined with high precision from SM fits of the UT [32]: The CP-violating phases φ NP q vanish in the SM and allow us to take NP contributions to B 0 q -B 0 q mixing into account. It is useful to introduce "effective mixing phases" through the following expression [12,16]: 2 Whenever information from both B 0 q → f and B 0 q → f decays is needed to determine an observable, as is the case for CP asymmetries or untagged branching ratios, we use the notation B d and B s . Figure 2: Illustrations of the rescattering processes in Eqs. (23) and (24).
where the hadronic "penguin phase shifts" ∆φ tan ∆φ D − s D + s s = 2 a cos θ sin γ + 2 a 2 sin 2γ 1 + 2 a cos θ cos γ + 2 a 2 cos 2γ = 2 a cos θ sin γ + O( 2 a 2 ) . (19) In the limit a = a = 0, we simply have The penguin parameter ae iθ cannot be calculated reliably within QCD. Since this quantity is governed by the ratio of a penguin amplitude to a colour-allowed tree amplitude, it is plausible to expect a ∼ 0.1-0.2. Applying the Bander-Silverman-Soni mechanism [34] and the formalism developed in Refs. [35,36] yields the following estimate [6]: In the corresponding calculation, form factors and decay constants cancel as the parameter ae iθ is actually defined as a ratio of hadronic amplitudes, as we emphasised after Eq. (8). However, incalculable long-distance contributions, such as processes of the kind and which can be considered as long-distance penguins with up-and charm-quark exchanges [37], respectively, as illustrated in Fig. 2, may have an impact on ae iθ .
In this paper, we discuss strategies to control these effects by means of experimental data. In the B 0 (20)), the penguin effects have to be taken into account for the determination of φ d . In the B 0 s → D − s D + s case (Eq. (21)), the parameter a is associated with the tiny factor and is hence doubly Cabibbo-suppressed. However, in view of the experimental precision in the LHCb upgrade era, also these effects have to be controlled. 5

Untagged Decay Rate Information
For the analysis of the experimental data later on, it is useful to introduce another observable, containing the untagged rate information. It is defined as [5,6]: where is the well-known B → P P phase-space function. Due to the sizeable lifetime difference in the B s -meson system, y s ≡ ∆Γ s /2Γ s = 0.0608 ± 0.0045 [2], a difference arises between the "theoretical" branching ratio defined through the untagged decay rate at time t = 0 [5] and the "experimental" branching ratio which is extracted from the time-integrated untagged rate [38]. They can be related as [39] B where the numerical estimate uses extracted from the measurement of the effective B 0 s → D − s D + s lifetime [6,40]. The observable H takes the following form in terms of the penguin parameters [5]: Moreover, the U -spin relation in Eq. (7) implies .
Using Eq. (7) and keeping a and θ as free parameters, the expression in Eq. (29) results in the following lower bound [41,42]: Moreover, H allows us to put a lower bound on the penguin parameter a: The signs have been chosen in such a way that this expression applies to the current experimental situation discussed in Section 3.4. Figure 3: Flow chart illustrating the new strategy to determine H using data from semileptonic B 0 q → D − q + ν decays.

Information from Semileptonic Decays
The experimental determination of H through Eq. (25) requires information on the amplitude ratio |A /A|, which is affected by U -spin-breaking corrections to the relation in Eq. (8). To avoid the limitations this brings, we propose a new method to determine H using data from semileptonic B 0 q → D − q + ν decays, which is illustrated by the flow chart in Fig. 3. To this end, we introduce the ratio where the parameters b q and ρ q are given in Eq. (13); V cq is the relevant CKM matrix element, f Dq denotes the D q -meson decay constant defined through and the factor X Dq is given by where the form factors are defined through with q ≡ p − k. Ratios between non-leptonic decay rates and differential semileptonic rates as in Eq. (33) are well-known probes for testing factorisation [43][44][45][46][47][48].
The parameter a (q) NF measures non-factorisable effects in the amplitudes defined through Eq. (2), which we may write as where T is the amplitude of the colour-allowed tree topology in factorisation, with G F denoting Fermis constant. In naive factorisation, a (q) NF = a 1 , where a 1 represents the appropriate combination of Wilson coefficient functions of the current-current operators 3 . We have A ≡ A d and A ≡ A s , and shall suppress the label q in the following discussion for simplicity. Introducing the abbreviations we obtain where measures the importance of the penguin topologies with respect to the colour-allowed tree amplitude. On the other hand, probes the importance of the exchange and penguin annihilation topologies. We will return to x in Subsection 3.3.2, where we determine |x| and the CP-conserving strong phase σ from experimental data. The parameter a T NF describes the non-factorisable corrections to the "tree" diagram (Eq. (39)), i.e. we have with ∆ T NF = 0 for exact factorisation. Finally, the observable H can be expressed as follows: In comparison with Eq. (25), the advantage is that the theoretical precision is now only limited by non-factorisable U -spin-breaking effects. Moreover, as the R Dq are ratios of B q rates, the dependence on the ratio of fragmentation functions f s /f d , which is needed for normalisation purposes [50], drops out in this expression. It is instructive to have a closer look at the non-factorisable U -spin-breaking effects entering Eq. (45), although we will constrain them through experimental data. We obtain the following expression: Using heavy-meson chiral perturbation theory and the 1/N C expansion, non-factorisable SU (3)-breaking corrections to the colour-allowed tree amplitudes of B 0 q → D q D p decays were found at the level of a few percent in Ref. [51], suggesting small corrections from the last factor. The U -spin relation between the B 0 s → D − s D + s and B 0 d → D − d D + d decays is reflected by the one-to-one correspondence of the r (q) P and x (q) terms. These contributions enter Eq. (46) only in ratios of terms with structures 1 + Λ (q) , where we expect the Λ (q) to be at most O(0.2). Assuming SU (3)-breaking at the 30% level for the Λ (q) terms yields a correction of only O(5%) for the ratios, i.e. a robust situation.
Let us now exploit experimental data to probe these effects. Using the ratio R Ds , we may actually determine |a where ∆ (s) NF is now -by definition -a real parameter and the corrections due to the a term are at most at the level of a few percent. Assuming Consequently, the information for the semileptonic differential rate allows us to quantify the non-factorisable U -spin-breaking corrections to the determination of H (Eq. (45)).
Let us now return to discuss the remaining quantities entering Eq. (45). The D qmeson decay constants (q = d, s) can be extracted from leptonic decays: The current experimental status has been summarised in Ref. [52]: A detailed overview of the status of lattice QCD calculations has been given by the FLAG Working Group in Ref. [53].
In the infinite quark-mass limit, the following consistency relation arises [54]: For a discussion on QCD and Λ QCD /m Q corrections to this relation we refer the reader to Refs. [54][55][56]. There has recently been impressive progress in the calculation of hadronic form factors within lattice QCD, where now the first unquenched calculations of thē B →D ν form factors at nonzero recoil are available [57,58]. Using the results of Ref. [58], we obtain F while the expression in Eq. (52) gives 0.924, thereby indicating small corrections. In the numerical analysis in this paper, we will use the result in Eq. (53). Experimental data for the semileptonic decay B 0 s → D − s + ν is not yet available. Although it is experimentally challenging to disentangle the semileptonic B 0 s → D − s + ν and B 0 s → D * − s + ν decays, it might be feasible to distinguish them due to the shifted invariant mass spectrum of the D + s µ − combinations, and the difference in the missing reconstructed mass, which is correlated to the "corrected mass" as illustrated in Ref. [59]. Combined with a fit to the angular distributions, this gives information on the different form factors. We encourage to add this channel to the experimental agenda of the LHCb and Belle II experiments and perform detailed studies for the upgrade era. On the other hand, the differential rate of the B 0 d → D − d + ν mode has already been measured and will actually be used in the next section to estimate the non-factorisable effects in B 0 d decays.
The lack of experimental data on semileptonic B 0 s decays can be circumvented by studying the ratio of other B 0 d and B 0 s decays, discussed in the next Section, and applying SU (3)  3 Picture Emerging from the Current Data

Overview
The main objective of this analysis is the determination of the B 0 q -B 0 q mixing phases High precision determinations of these phases require us to control not only the contributions from penguin topologies, but also the impact of additional decay topologies and nonfactorisable effects. The latter two aspects cannot be quantified using information from s system therefore need to be studied. An overview of the different decay modes discussed in this section and their applications is given in Table 1.

Preliminaries
The direct and mixing-induced CP asymmetries of the B 0 d → D − d D + d decay and the H observable, when using the U -spin relation in Eq. (7), depend on the four parameters a, θ, φ d and γ. In 1999, when this decay was originally suggested by one of us, the determination of the UT angle γ was the main goal. The proposed strategy therefore assumed and H [5]. However, at present it is possible to extract γ in a powerful way through pure B → D ( * ) K ( * ) tree decays. Using current data for these channels, the CKMfitter and UTfit collaborations have obtained the following averages: For the numerical analysis in this paper, we shall use the CKMfitter result. By the time of the Belle II and LHCb upgrade era, much more precise measurements of γ from pure tree Decay A Topologies Used for: x x x . . . and consistency of a NF,c /a NF,c Table 1: Overview of the various topologies contributing to the B → DD decays. The naming convention is indicated in the second column.
decays will be available (see Section 4). Using γ as an input, we may instead determine φ d and the penguin parameters from H and the CP asymmetries . The penguin parameters thus determined allow us to take their effects into account in the determination of φ s from the mixing-induced CP asymmetry

Comparing B → DD Branching Fractions
To quantify the contributions from additional decay topologies and the impact of nonfactorisable effects in the B 0 s , which are characterised by the following decay amplitudes: whereÃ A andã e iθ take analogous expressions. If we use the U -spin flavour symmetry, we obtain the following relations:ã Figure 4: Illustration of the annihilation topology contributing to Moreover, there are the charged decays B + → D 0 D + d and B + → D 0 D + s , which are again related to each other through the U -spin symmetry. These modes also do not receive contributions from exchange and penguin annihilation topologies. However, there are additional contributions from annihilation topologies, as illustrated in Fig. 4, which enter with the same CKM factor as the penguin contributions with up-quark exchanges.

Probing Annihilation Topologies with Charged B decays
The decay amplitudes take the following forms: withÃ c denoting the annihilation amplitude; the expressions for the primed amplitudes are analogous. The penguin parameter satisfies with It is useful to introduce the following ratios: Using the expressions for the decay amplitudes given above yields If we apply the SU (3) flavour symmetry (actually the V -spin subgroup), we obtaiñ while the isospin symmetry of strong interactions implies If we neglect the annihilation contribution in Eq. (64) and assume the same penguin contributions in Eq. (68), i.e.ã =ã c , we obtain A deviation from unity of this ratio would therefore imply either the presence of non-zero annihilation contributions or large SU (3)-breaking effects through Eq. (70). In the case of (69), the penguin parameters are suppressed by the tiny factor and hence play a negligible role. Consequently, the ratio essentially relies on the strong isospin symmetry. The current experimental results compiled by the Particle Data Group (PDG) read as follows [1]: and correspond to 4 For the last decay combination, we may also employ the direct measurement of the ratio of the relevant branching fractions [61], which is given by This leads to which has a significantly smaller uncertainty with respect to Eq. (79) thanks to a cancellation of uncertainties in the directly measured ratio of branching fractions. We note the deviation from one at the 2.4 σ level, which is unexpected.

Probing Exchange and Penguin Annihilation Topologies
The current PDG results for the CP-averaged branching ratios of the s decays are given as follows [1]: In comparison with (75) and (77), the branching ratio in (83) is surprisingly small. A similar pattern -although not as pronounced in view of the larger uncertainties -is observed if we compare (82) with (74) and (76).
decays receive contributions from exchange and penguin annihilation topologies, which have no counterparts in the B 0 modes, respectively (see Table 1), it is possible that the puzzling pattern of the data is actually due to the presence of these exchange and penguin annihilation contributions. Let us first have a closer look at the ratio of the amplitudes of the B 0 wherex measures, in analogy to the parameter x introduced in Eq. (43), the importance of the exchange and penguin annihilation topologies with respect to the dominant tree topology; we use abbreviations as in Eq. (40). If we neglect the terms with the penguin parameters, which enter with the tiny , and introduce the SU (3)-breaking parameter we obtain the relation The parameter is only affected by SU (3)-breaking effects entering at the spectatorquark level. Applying factorisation, where P (ct) /T =P (ct) /T (see the comment after Eq. (22)), we obtain .
Here we have taken into account the restrictions following for the corresponding B q → D q form factor from the heavy-quark effective theory [46]: where ξ q (w q (q 2 )) is the Isgur-Wise function with Studies of the light-quark dependence of the Isgur-Wise function were performed within heavy-meson chiral perturbation theory, indicating an enhancement of ξ s /ξ d at the level of 5% [62]. Applying the same formalism to f Ds /f D d leads to estimates for the value of this ratio of about 1.2 [63], which are in agreement with the experimental results in Eq. (51).
Since 1992, when these calculations were pioneered, there has been a lot of progress in lattice QCD (for an overview of the state-of-the-art analyses, see Ref. [64]). The most recent result for the SU (3)-breaking effects in the form factors reads as follows [65]: which is in excellent agreement with the picture from heavy-meson chiral perturbation theory. Using this result as an input yields The error quantifies only the uncertainties related to the form factors. We cannot quantify the non-factorisable effects. However, as they enter only at the level of different spectator quarks and as already the leading SU (3)-breaking effects are small, we expect a minor impact. The ratio 5 15 then takes the following form: thereby fixing a circle forx in the complex plane. The numerical value in Eq. (94) refers to a direct measurement of the corresponding ratio of branching ratios [61].
For the other decay combination, we obtain withx In analogy to Eq. (86), we introduce a parameter which is given in factorisation by neglecting the tiny difference between the form-factor ratios for q 2 = m 2 D d and m 2 Ds . As in Eq. (93), the error quantifies only the form factor uncertainties.
The penguin parameters do not enter Eq. (96) with the tiny . However, if we use the SU (3) relation where decay constants and form factors cancel in factorisation, we get with the parameter x was introduced in Eq. (43). Consequently, we have where the second-order terms are expected to give small corrections at the few-percent level. Introducing the ratio Penguin Annihilation (PA) Figure 5: Illustration of exchange and penguin annihilation topologies contributing to we obtain in analogy to Eq. (95). It is interesting to consider the double ratio where we have neglected the |x ( ) | terms and have used Eq. (99). The experimental results in Eqs. (94) and (104) give which is in agreement with the expectation in Eq. (106). The current uncertainties are unfortunately too large to draw any further conclusions.

Probing Exchange and Penguin Annihilation Topologies Directly
The exchange and penguin annihilation topologies can be probed in a direct way by means of the decays B 0 These modes receive only contributions from exchange and penguin annihilation topologies [7,30], as illustrated in Fig. 5, and are related to each other through the U -spin symmetry. The current experimental information on the corresponding CP-averaged branching ratios is given as follows [1]: The experimental signal for the B 0 s → D − d D + d decay is in accordance with the picture emerging from the discussion given above.
Let us now have a closer look at these decays. Their amplitudes can be written as where the primed parameters are defined in an analogous way. We obtain then where we have neglected the terms proportional to the tiny factor and introduced the parameter In the estimate of this SU ( (116) takes then the simple form which fixes a circle with radius |ς x | around the origin in the complex plane.
where we have neglected the penguin annihilation contributions on the right-hand side, and have introduced 6 We assume Im which takes the simple form Also in this case it is interesting to consider the double ratio which allows us to test the relatioñ Let us now convert the experimental results in Eqs. (94) and (116) into constraints on |x | andσ . While the latter observable simply fixes a circle around the origin in the Im Excluded at 90 % C.L.
with the corresponding confidence-level contours shown in Fig. 6. The exchange and penguin annihilation topologies play hence a surprisingly prominent role in the decays at hand, pointing towards large long-distance strong interaction effects. An example of such a contribution to the exchange topology is given by as illustrated in the right panel of Fig. 2.
A similar analysis can be performed for the observables in Eqs. (104) and (120), which allow the determination of |x| andσ. In contrast to the determination of |x | and σ , the penguin effects in the amplitude ratios do not enter with the tiny and lead to additional uncertainties. In Fig. 7, we show the constraints from the current data, which are still pretty weak. Here we may have long-distance rescattering contributions from processes of the kind In the future, following these lines, the comparison between the values ofx andx will offer yet another test of the relation in Eq. (123), going beyond Eq. (122) through information on the strong phases. 20

Information from Branching Ratios and Non-factorisable Effects
Let us now use the currently available data to constrain the penguin parameters. Unfortunately, a measurement of the differential semileptonic B 0 s → D − s + ν rate is not available. Consequently, we may not yet apply Eq. (45) and have to follow a different avenue, involving larger theoretical uncertainties. In analogy to the H observable defined in Eq. (25), we introduce the following quantities: If we complementH with direct CP violation in B 0 s → D − s D + d (and the suppressed CP asymmetry in B 0 d → D − d D + s ) and H c with direct CP violation in B ± → D u D ± d (and the suppressed CP asymmetry in B ± → D u D ± s ), we may determine the penguin parameters (ã,θ) and (ã c ,θ c ), respectively. These determinations are analogous to the determination of (a, θ) from H and the direct CP asymmetry in B 0 d → D − d D + d (and the suppressed CP asymmetry in B 0 s → D − s D + s ). The hence determined parameters offer insights into the r A and r P A parameters introduced in Eqs. (65) and (102), respectively, and allow for a comparison of the relative non-factorisable contributions.
In contrast to the measurements of the CP asymmetries, the extraction of the H,H and H c observables from the data requires knowledge of the following amplitude ratios: These quantities are governed by U -spin-breaking effects in the ratio of the colour-allowed tree contributions, which we may write as where the parameters a T NF describe non-factorisable contributions affecting the colourallowed tree amplitude (see Eq. (44)). If we assume that all the a T NF parameters are equal to one another due to the SU (3) flavour symmetry, the following relation can be derived: where the decay constants and form factors cancel. In terms of branching ratios, using Eqs. (132)-(134), this relation implies The current data give G = 0.85 ± 0.16 , which is consistent with Eq. (139) within the uncertainties. Using data for the semileptonic B 0 d → D − d + ν decay, the non-factorisable effects can be probed through where X Ds andã Figure 8: Flow chart illustrating the classic strategy to determine H using data from B → DD branching ratio measurements.
defined in analogy to Eq. (42). Experimentally, we find corresponding to a non-factorisable contribution where Eq. (53) has been used for the ratio of form factors. In this result, the penguin effects suppressed by in Eq. (141) were neglected. It is plausible to interpret the deviation from one at the 2.9 σ level as footprints of sizeable penguin effects. We shall discuss this parameter below. The ratios in Eqs. (132)-(134) can be written in the following forms: which is also graphically illustrated in Fig. 8. In the above equation, a NF differs from a NF introduced in Eq. (41) through the (1 + x) term, to account for the contributions from exchange and penguin annihilation topologies, which are absent in the decays of the other two ratios. We thus have 23 As in the discussion in Subsection 2.4, it is convenient to write where the parameterξ SU (3) describes SU (3)-breaking effects in the non-factorisable contributions, yielding Consequently,∆ NF = −0.244 ± 0.085, as determined from Eq. (146), allows us to take non-factorisable corrections to Eqs. (147)-(149) into account, assuming In these relations, SU (3)-breaking effects enter only through different spectator quarks and are expected to be small. In order to determine the SU (3)-breaking effects in the ratio of the colour-allowed tree amplitudes |T /T |, we use again Eq. (90) to derive the following expression [5]: For the calculation of the numerical value, we have used Eq. (92) and the values of the decay constants in Eq. (51). In analogy, we get In the case of the charged decays, a particularly simple situation arises in factorisation as the form factors cancel: T as is also evident from Eq. (137). Using the amplitude ratio in Eq. (147), we can now determine the observable H (see Eq. (25)). For this, we also have to quantify the uncertainty from the SU (3)-breaking corrections to the term involving the x and x parameters. The analysis discussed in Subsection 3.3.2 allows us to accomplish this task. Using the results forx andσ in Eq. (125) and the relations 25 Finally, we obtain and correspondingly In Fig. 9 we show the corresponding uncertainty budgets. Should the relations in Eq. (153) actually receive corrections, they would affect this error budget. In the future, implementing the strategy proposed in Section 2.4, this approximation/assumption is no longer needed. The result on H allows us to put first constraints on the penguin parameters a with the help of the lower bound in Eq. (32): where we have used the lower value of H at one standard deviation.

Information from CP Asymmetries
Let us now add experimental information on CP violation to our analysis. Concerning the decay B 0 d → D − d D + d , the current status of the measurement of the direct and mixinginduced CP asymmetries is given as follows: .06 +0.14 −0.21 ± 0.08 (Belle [68]) .

(167)
The measurements by the BaBar and Belle collaborations are not in good agreement with one another, in particular for the mixing-induced CP asymmetry. HFAG gives the following averages [2]: and Following Refs. [11,16], we illustrate the various constraints entering the fit through contour bands of the individual observables in Fig. 10.
For the fit, we use the expressions for the CP asymmetries in Eqs. (10) and (11), and the expression for H in Eq. (29) with the U -spin relation in Eq. (7). Since a enters Eq. (29) with the tiny , U -spin-breaking corrections to Eq. (7) have a very minor impact. Although the uncertainties are large, the results from the fit may indicate significant penguin contributions. Should this actually be the case, long-distance effects, such as those illustrated in Fig. 2, would be at work in the penguin sector of the B 0 q → D − q D + q decays. It is interesting to have a closer look at the result of non-factorisable contributions Assuming that |ã T NF | = 1, i.e. that the colour-allowed tree topologies have negligible non-factorisable contributions, and that all parameters are real, this results iñ Note that the uncertainty onr P only reflects the uncertainty on the input quantity (146), but does not take into account further theoretical uncertainties associated with the made approximation. Further assumingP (ut) =P (ct) , which leads to the approximation ae iθ ≈ R br P /(1 +r P ), we obtain which is in the ballpark of the theoretical estimate in Eq. (22). Consequently, it is still premature to draw definite conclusions on anomalously enhanced penguin contributions at this point and future analyses are required to shed light on this issue. Large penguins would have important consequences for the determination of the B 0 s -B 0 s mixing phase φ s from the mixing-induced CP violation of the B 0 s → D − s D + s decay. The LHCb collaboration has recently presented the first analysis of CP violation in this channel [69], yielding which can be converted into

Im[a]
A dir Re   These results are in good agreement with the predictions based on a global SU (3) analysis of the B → DD system [70]. Moreover, we obtain which should be compared with Eq. (28) corresponding to the direct measurement of the effective lifetime τ eff If we generalise the U -spin relation in (7) as with the assumed U -spin-breaking parameters ξ = 1.00 ± 0.20 and δ = (0 ± 20) • (which are of similar size as the corresponding parameters in Ref. [16]) and use the expression in Eq. (19), the penguin parameters in Eq. (170) determined from the fit can be converted into Finally, we can extract φ s form the effective mixing phase in Eq. (176), which yields Despite the suppression through the parameter , penguins may have a significant impact on the extraction of φ s and have to be taken into account. This will be particularly relevant for the LHCb upgrade era. In this new round of precision, we will also get valuable insights into the validity of the U -spin symmetry, parameterised through Eq. (179). Unfortunately, there is no measurement of CP violation in B 0 s → D − s D + d available, which would be very interesting, in particular in view of the situation for B 0 d → D − d D + d . Consequently, we may not yet determineã andθ from the data.
However, for the charged B + → D 0 D + d decay, the PDG gives Re In Fig. 11, we illustrate the corresponding situation, which complements Fig. 10.
It is interesting to compare the penguin parameter ae iθ with its charged decay counterpartã c eθ c . We obtain where we have used Eqs. (64) and (101). The precision that can be obtained with the current data does not yet allow us to draw any conclusions regarding r A P A . However, in the future it will be interesting to monitor this quantity as the experimental precision improves. Moreover, a measurement of the direct CP violation in the B 0 s → D − s D + d channel will allow us to determineãe iθ from the information fromH. The comparison withã c eθ c will yield the r A parameter from Eq. (64), so that Eq. (185) will then allow the determination of r P A . Consequently, following these lines, we may reveal the impact of the annihilation and penguin annihilation topologies in the decays at hand.

Prospects for the LHCb Upgrade and Belle II Era
Let us conclude the discussion on the B → DD decays by exploring the potential of these decay modes in the Belle II era and at the LHCb upgrade. We do this using several scenarios, examined in Section 4.3, that reflect the different possibilities still allowed by the current data. The inputs used in these scenarios are discussed first. Section 4.1 gives the experimental prospects for the relevant CP and branching ratio information of the B → DD decays, while Section 4.2 deals with the future constraints on the additional decay topologies.

Extrapolating from Current Results
The B-factories have pioneered the study of B → DD decays, including the discoveries of numerous B → DD decay modes [71], the measurements of branching fractions [72,73], and the analyses of CP asymmetries [67,68,72]. The LHCb collaboration subsequently continued the study of B → DD decays, notably focusing on the analysis of B 0 s decays [40,61,69], which are abundantly produced at the LHC. Based on the successful performance of LHCb during run I of the LHC, an estimate can be made of its performance with the data samples that are expected to be collected after the upgrade of the LHCb detector. For these extrapolations, an integrated luminosity of 5 fb −1 in run II, from 2015 until 2018, is assumed. In addition, the B production cross section will increase at a centre-of-mass energy of 13 TeV compared to 8 TeV by about 60%. For the upgrade scenario, an integrated luminosity of 50 fb −1 is assumed with increased trigger efficiency, leading to about a three times larger data sample per fb −1 compared to the B yield per fb −1 at run I. Similarly, a prognosis can be made for measurements at Belle II, which is expected to start taking data in 2018. Here we assume that 50 times more data will be collected than currently is available (1 ab −1 ) .
The expectations for the CP asymmetry parameters are listed in Table 2. The extrapolations are done for the currently available measurements only; no attempt is made to forecast the precision on yet-to-be-performed analyses. For example, the LHCb collaboration will also determine the CP asymmetries of the B 0 d → D − d D + d decay, but it remains to be seen what the accuracy will be in comparison with possible Belle II results.
The expectations for the branching fractions are listed in Table 3. The "current measurement" column reflects the best available knowledge at this moment, which in some cases could have been more precise if the ratio of branching fractions were determined directly, rather than dividing the individually measured branching fractions. In the extrapolations it is assumed that the ratios of branching fractions are determined. Moreover, it is assumed that the systematic uncertainties due to f s /f d (4.7%), due to the D-meson branching fractions (3.9%, 2.1% and 1.3% for D + s , D + d and D 0 mesons, respectively) and due to the different B 0 s lifetimes (2.9% (1.5%) for a CP (flavour) eigenstate), remain the same. We assume that the total experimental systematic uncertainty 1.06 +0.14 −0.21 ± 0.08 [68] ±0.08 Not observed Table 3: Experimental prospects for ratios of branching fractions. The second and fourth ratios are obtained from direct determinations of the ratios of branching fractions, whereas the others are calculated from individual branching fractions. The value in brackets indicates the possible uncertainty if this ratio were determined directly. Note that for the calculation of the H observables, additional uncertainties due to |A /A| arise.

Observable
Current Measurement Upgrade Table 4: Experimental prospects for external input. A * indicates an average of the extrapolated measurement with the current PDG average.
will decrease from 5.0% to 4.0%. In some ratios, the uncertainty on the D branching fractions cancels with their contribution to the f s /f d uncertainty. This is taken into account where appropriate. In our upgrade era scenario, systematic uncertainties will be the limiting factor on the ratio of branching fractions. Therefore, we would like to encourage research into f s /f d , B(D + s → K + K − π + )/B(D + d → K − π + π + ) and B lifetime differences. If these three factors could be reduced to a level of about 2%, then that would lead to a systematic uncertainty of (5-6)% for all of these decays, assuming an experimental uncertainty of 4%. Finally, the prospects for improvements on external input parameters are listed in Table 4.

Exchange and Penguin Annihilation Contributions
For the construction of the H observable based on the ratio of hadronic amplitudes in Eq. (147), the contributions from exchange and penguin annihilation topologies, represented by the parametersx andσ , need to be quantified. With future, improved measurements of the B → DD branching ratios it is expected that the picture emerging from the current data, discussed in Section 3.3.2, can be sharpened further. We therefore explore the precision that can be achieved towards the end of the Belle II era and of the LHCb upgrade. Based on the best fit solution obtained from the current data, i.e. Im Re |x |  Eq. (125), and the prospects in Table 3, we start from the following input measurements: corresponding tox = 0.258 +0.017 −0.045 ,σ − ω = (180 ± 15) • .
The associated confidence-level contours are shown in Fig. 12. Note that at first sight these uncertainties do not seem to have improved significantly with respect to the present experimental situation. However, this is merely caused by the shape of the confidence contour in Fig. 12.

Future Scenarios
To achieve the smallest theoretical uncertainty on the H observable, it should be constructed from the semileptonic decay information, see Eq. (45), as explained in Section 2.4. This method is preferred over the direct ratio of hadronic branching fractions in Eq. (25), as it does not rely on form factor information, and is not experimentally limited by f s /f d . However, as the necessary information on dΓ/dq 2 (B 0 s → D − s + ν ) is currently not yet available, we also do not have any estimates for the precision that can be achieved at LHCb or Belle II. For the following discussion, we will thus, like for the fits to the current data, rely on the original definition using the ratio of hadronic branching fractions, Eq. (25).
Using the currently available data, we have illustrated in Section 3.4 that it is possible to simultaneously determine the penguin contributions and the B 0 d -B 0 d mixing phase φ d using the CP and branching ratio information of the B 0 d → D − d D + d decay. However, as the result of φ d in Eq. (171) shows, the precision on φ d is very limited. Also for the Belle II and LHCb upgrade era it will be challenging to reach precisions below the (10-20) • level. A high-precision determination of φ d using the B 0 d → D − d D + d decay is only possible if the direct CP asymmetry and the H observable together are sufficient to unambiguously pin down the penguin parameters a and θ. In such a situation, the phase φ d can then be determined from the mixing-induced CP asymmetry. As Fig. 10 illustrates, a and θ cannot precisely be determined. The situation would arise either for very large values of a, which looks unrealistic, or if the H observable can be determined with a precision of well below the 5% level, which in view of the prospects in Table 3 is unlikely. The determination of φ d from the B 0 d → D − d D + d decay is therefore not competitive with the results from the B 0 d → J/ψK 0 S decay. It is thus advantageous to use external input on φ d for the analysis of the B → DD decays. This extra information breaks the ambiguity that is still present in the confidence-level contours shown in Fig. 10, and can therefore improve the precision on a and θ, and thus also on ∆φ s . For the future benchmark scenarios we therefore only focus on the high precision determination of φ s .
Using external input for φ d in principle makes one of the three observables used by the fit (A dir CP , A mix CP or H) superfluous. As the H observable receives corrections from possible U -spin-breaking effects through |A /A|, which result in large theoretical uncertainties, it is preferred to determine the penguin parameters using information on the CP asymmetries only, omitting H. Such a determination is theoretically clean. This will ultimately lead to the highest precision on a, θ and ∆φ s . In this situation, the branching ratio information can instead be used to gain insight into the hadronic physics We can then follow the opposite path where the fit results for a and θ can be used to determine the H observable with the help of Eq. (29), labelled H (a,θ) below. Since a enters there with the tiny , the U -spin-breaking corrections affecting the U -spin relation in Eq. (179) have a very minor impact. Since we now know the value of H, the relation (25) can be inverted to instead determine the ratio of hadronic amplitudes from the measured ratio of branching ratios. This experimental measurement of the ratio of hadronic amplitudes can be compared with the theoretical result in Eq. (159). This favourable strategy is illustrated by the flow chart in Fig. 13.
However, the ideal scenario described above cannot always be realised. When the value of the mixing-induced CP asymmetry is compatible with 1 (at the 1σ level), its power to constrain the penguin parameters a and θ is limited. This can best be illustrated using the contour plots, like Fig. 10 or Fig. 17 below. In this situation, the annular constraint originating from the mixing-induced CP asymmetry becomes a closed disk, leading to a large overlap region with the direct CP asymmetry constraint. Consequently, it is not possible to conclusively pin down a and θ in such a situation. Additional information is thus needed to improve the picture, and reach our target of matching the Penguin: (a, θ) Quantity is Theoretically Clean Affected by U -Spin-Breaking Effects Figure 13: Flow chart illustrating the favourable strategy to control ∆φ s , which only requires information on the B 0 foreseen experimental precision on φ s with an equally precise determination of ∆φ s . In this situation, the H observable forms an essential ingredient in the fit, and it can therefore not be used to experimentally constrain the ratio of hadronic amplitudes. This less favourable strategy is illustrated by the flow chart in Fig. 14.
Given the current experimental situation, either of the two situations sketched above can still be realised, depending on the future world average for A mix To demonstrate the variety of situations in which we may ultimately end up, we made an overview of six different scenarios, covering both situations. Scenarios 1-3 represent the favourable situation in which the H observable can be omitted from the fit, in which we can determine a and θ in a theoretical clean way, and get experimental access to the ratio of hadronic amplitudes. Scenarios 4-6, on the other hand, fall in the second category and do require information on H to conclusively pin down a and θ. All six scenarios are chosen to be compatible with the current experimental situation, with scenario 5 representing the current best fit point, and have a < R b , which is suggested by Eq. (3). Although it is mathematically possible for a to be larger than R b , it would imply that the penguin topologies are larger than the tree contribution, which seems very unlikely. The condition a = R b thus serves as a naturally upper limit for the size of the penguin contributions. The different scenarios we consider, and the resulting input values of the three observables (A dir CP , A mix CP and H) are listed in Table 5. The choice of input points can be compared with the current fit solution for a and θ in Fig. 15, and with the current measurements of the B 0 d → D − d D + d CP asymmetry parameters in Fig. 16. For each of the six scenarios, the individual constraints coming from A dir CP , A mix CP and H are illustrated in Fig. 17. The three left-most plots represent the favourable situation, Quantity is Theoretically Clean Affected by U -Spin-Breaking Effects   while the three right-most plots fall in the second category. For the three right-most plots the constraint from the mixing-induced CP asymmetry is more disk-like as the central value of A mix CP is closer to one. As a consequence, the overlap with the direct CP asymmetry, which forms a narrow band in all cases, is too large to pin down a and θ. In this respect, scenario 4 should be seen as a limiting case; information of H is not strictly necessary if one is only interested in the 1 sigma results.
For each of the six scenarios we also performed a χ 2 fit, similar to the one described in Section 3.4, but including φ d as a Gaussian constraint. The fit results for a and θ, and the associated values for the shifts ∆φ d and ∆φ s are listed in Table 6. U -spinbreaking effects, parametrised by Eq. (179), have been included in the results for ∆φ s . The associated confidence-level contours are shown in Fig. 17. In all cases we succeed in our goal of matching the foreseen experimental precision on φ s , see Table 2, with an equally precise determination of ∆φ s . This is also the case for φ eff d and ∆φ d , as illustrated in Table 7. For the first three scenarios, which do not include the H observable in the fit, the resulting solution for H (a,θ) and the values for the ratio of hadronic amplitudes are listed in Table 8. The resulting uncertainties are about a factor two smaller that the current theoretical uncertainties derived within the factorisation framework, and of comparable size to the experimental precision that can be obtained on the ratio of hadronic amplitudes describing the B 0 d → J/ψK 0 S and B 0 s → J/ψK 0 S decays [16]. Consequently, the experimental determination of |A /A| is yet another interesting topic for Belle II and the LHCb upgrade. It will provide valuable insights into possible nonfactorisable U -spin-breaking effects and the hadronisation dynamics of the B → DD decays.

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Conclusions
In this paper, we have presented a detailed study of the system of B → DD decays, exploring the picture emerging from the currently available data and proposing new strategies for the era of Belle II and the LHCb upgrade. We find that patterns in the current branching ratio measurements can be accommodated through sizeable contributions from exchange and penguin annihilation topologies, which play a more prominent role than naively expected. This feature suggests that long-distance effects of strong interactions are at work, which cannot be understood within perturbation theory. Using data for the differential semileptonic with the help of the U -spin symmetry of strong interactions. Following these lines, it will be possible to control the penguin effects in the determination of φ s from the CP violation in the B 0 s → D − s D + s channel. We find that the implementation of this strategy depends strongly on the values of the measured CP asymmetries, as we illustrated through a variety of future scenarios which are consistent with the current experimental situation, giving us a guideline for the LHCb upgrade era. We distinguish between two kinds of scenarios: in the first, the direct and mixing-induced CP asymmetries of the B 0 d → D − d D + d channel are sufficient to determine a and θ in a theoretically clean way, allowing us to determine the ratio |A |/A| from the observable H, providing valuable insights into non-factorisable U -spinbreaking effects. In the second -less favourable -class of scenarios, information both from H and the CP asymmetries is needed to determine the penguin parameters. We have demonstrated that the resulting theoretical uncertainty for the penguin shift ∆φ s of the B 0 s → D − s D + s channel will be smaller than the experimental uncertainty for φ eff s (B s → D − s D + s ) in both classes of scenarios. Similar analyses can be performed for B 0 d → D * − d D * + d and B 0 s → D * − s D * + s decays, where time-dependent measurements of the angular distribution of the decay products of the two vector mesons are required [77,78].
Analyses of B → DD decays in the era of Belle II and the LHCb upgrade will offer interesting new insights both into the physics of strong interactions and into CP violation. We look forward to confronting the strategies discussed in this paper with future data!