Dispersive and Absorptive CP Violation in $D^0- \overline{D^0}$ Mixing

In the precision era, CP violation in $D-\bar D$ mixing is ideally described in terms of the dispersive and absorptive phases $\phi_f^M$ and $\phi_f^\Gamma$, parametrizing CP violation (CPV) in the interference of $D^0$ decays with and without dispersive (absorptive) mixing. These are distinct and separately measurable effects. This formalism is applied to (i) Cabibbo favored/doubly Cabibbo suppressed (CF/DCS) decays $D^0 \to K^\pm X$; (ii) CF/DCS decays $D^0 \to K_{S,L} X$, including the impact of $\epsilon_K$, and (iii) singly Cabibbo suppressed (SCS) decays. Expressions for the time-dependent CP asymmetries simplify: Indirect CPV only depends on $\phi_f^M$ (dispersive CPV), whereas $\phi_f^\Gamma$ (absorptive CPV) can only be probed with non-CP eigenstate final states. Measurements of the final state dependent phases $\phi_f^M$, $\phi_f^\Gamma$ determine the phases $\phi_2^M$ and $\phi_2^\Gamma$, which are the arguments of the dispersive and absorptive mixing amplitudes $M_{12}$ and $\Gamma_{12}$, relative to their dominant ($\Delta U=2$) $U$-spin components. $\phi_2^M$ and $\phi_2^\Gamma$ are experimentally accessible due to approximate universality: in the SM, $\phi_f^M-\phi_2^M$ and $\phi_f^\Gamma-\phi_2^\Gamma$ are negligible in case (i) above; and below $10\% $ in (ii), up to precisely known $O(\epsilon_K )$ corrections. In case (iii), the pollution enters at $O(\epsilon)$ in $U$-spin breaking and can be significant, but is $O(\epsilon^2)$ in the average over $f=K^+K^-$, $\pi^+\pi^-$. U-spin based estimates yield $\phi_2^M, \phi_2^\Gamma = O(0.2\%)$ in the SM. The current fit to the data thus implies an $O(10)$ window for new physics at $2\sigma$. A fit based on naively extrapolated experimental precision at the LHCb Phase II upgrade suggests that sensitivity to $\phi_2^{M,\Gamma}$ in the SM may be achievable in the precision era.

In the precision era, CP violation in D 0 − D 0 mixing is ideally described in terms of the dispersive and absorptive "weak phases" φ M f and φ Γ f , parametrizing CP violation (CPV) originating from the interference of D 0 decays with and without dispersive mixing, and with and without absorptive mixing, respectively, for CP conjugate hadronic final states f , f .These are distinct and separately measurable effects.This formalism is applied to the three relevant classes of decays: (i) Cabibbo favored/doubly Cabibbo suppressed (CF/DCS) decays D 0 → K ± X; (ii) CF/DCS decays D 0 → KS,LX, including the impact of K at LHCb and Belle-II, and (iii) singly Cabibbo suppressed (SCS) decays.Expressions for the time-dependent CP asymmetries simplify, compared to the more familiar parametrization, yielding a physically transparent strong phase dependence.Thus, we learn that for CP eigenstate final states, indirect CPV only depends on φ M f (dispersive CPV), whereas φ Γ f (absorptive CPV) can only be probed with non-CP eigenstate final states.Measurements of the final state dependent phases φ M f , φ Γ f determine the intrinsic dispersive and absorptive mixing phases, φ M 2 and φ Γ 2 , respectively.They are the arguments of the total dispersive and absorptive mixing amplitudes M12 and Γ12, respectively, relative to their dominant (∆U = 2) U -spin components.The latter are ∝ (VcsV * us − V cd V * ud ) 2 , and account for the D 0 mass and width differences.The intrinsic phases are experimentally accessible due to approximate universality: in the SM, and in extensions with negligible new CPV phases in CF/DCS decays, the relative difference (final state pollution) between φ M 2 , φ Γ 2 and φ M f , φ Γ f is negligible in case (i) above; and below 10% in (ii), up to precisely known O( K ) corrections.In case (iii), in the SM and in extensions with CP-odd QCD penguins of same order, the pollution enters at O( ) in U -spin breaking and can be significant, but is O( 2 ) in the average over f = K + K − , π + π − .U -spin based estimates yield φ M 2 , φ Γ 2 = O(0.2%) in the SM.The current fit to the data thus implies an O (10) window for new physics at 2σ.A fit based on naively extrapolated experimental precision at the LHCb Phase II upgrade suggests that sensitivity to φ M,Γ 2 in the SM may be achievable in the precision era.

I. INTRODUCTION
In the Standard Model (SM), CP violation (CPV) enters D 0 − D 0 mixing and D decays at O(V cb V ub /V cs V us ) ∼ 10 −3 , due to the weak phase γ.Consequently, all three types of CPV [1] are realized: (i) direct CPV, (ii) CPV in pure mixing (CPVMIX), which is due to interference of the dispersive and absorptive mixing amplitudes, and (iii) CPV due to the interference of decay amplitudes with and without mixing (CPVINT).In this work, we are particularly interested in the latter two, which result from D 0 − D 0 mixing, and which we collectively refer to as "indirect CPV".We would like to answer the following questions: How large are the indirect CPV asymmetries in the SM?What is the minimal parametrization appropriate for the LHCb/Belle-II precision era?How large is the current window for new physics (NP)?Can this window be closed by LHCb and Belle-II?
In order to address these questions we first develop the description of indirect CPV in terms of the CP violating (CP-odd) and final state dependent dispersive and ab-sorptive "weak phases".These phases, which we denote as φ M f and φ Γ f , respectively, for CP conjugate final states f and f , parametrize CPVINT contributions originating from the interference of D 0 decays with and without dispersive mixing, and with and without absorptive mixing, respectively.These are distinct measurable effects, as we will see below.Their difference equals the CPVMIX weak phase, i.e. φ 12 ≡ arg(M 12 /Γ 12 ) = φ M f − φ Γ f .An immediate consequence of our approach is that it yields simplified expressions for the indirect CP asymmetries, which have a transparent physical interpretation (unlike the more familiar description in terms of the mixing parameter |q/p|, and the weak phase φ λ f ).In particular, the requirement that the underlying interfering amplitudes possess non-trivial CP-even "strong-phase" differences is manifest, and accounts for the differences between the φ M f and φ Γ f dependence of the CP asymmetries.For example, we will see that the time-dependent CPVINT asymmetries in decays to CP eigenstate final states are purely dispersive, i.e. they only depend on φ M f (apart from subleading direct CPV effects).
In the SM, the dispersive and absorptive D 0 − D 0 mixing amplitudes are due to the long distance exchanges of all off-shell and on-shell intermediate states, respectively (short distance dispersive mixing is negligible).The CPVINT asymmetries are due to the CP-odd contributions of the subleading O(V cb V ub /V cs V us ) ∆C = 1 transitions to the mixing amplitudes (via intermediate states) and the decay amplitudes (via final states).The combined effects of these two CPV contributions can be expressed in terms of the underlying final state dependent phases φ M,Γ f , as noted above.Unfortunately, due to their non-perturbative nature, these phases can not currently be calculated from first principles QCD.However, we will be able to make meaningful statements using SU (3) F flavor symmetry arguments.
In order to estimate the magnitudes and final state dependence of φ M,Γ f in the different classes of decays, we compare them to an intrinsic or theoretical pair of dispersive and absorptive phases, which follow from the Uspin decomposition of the mixing amplitudes.They are, in general, defined as the arguments of the total dispersive (M 12 ) and absorptive (Γ 12 ) amplitudes, respectively, relative to a basis choice for the real axis in the complex mixing plane, given by the common direction of the dominant ∆U = 2 mixing amplitudes.Hence, we denote them as φ M 2 and φ Γ 2 , respectively.(The ∆U = 2 mixing amplitudes are proportional to (V cs V * us − V cd V * ud ) 2 , and are responsible for the observed D 0 meson mass and width differences.)Note that these phases are quark (or meson) phase convention independent and physical, like the phases φ M,Γ f directly measured in the decays.U -spin based estimates yield φ M 2 , φ Γ 2 = O(0.2%) in the SM.In principle, they could be measured on the lattice in the future.
In the SM, and for the Cabibbo favored and doubly Cabibbo suppressed decays (CF/DCS), the differences between φ M f and φ M 2 , or φ Γ f and φ Γ 2 are essentially known, thus allowing for precise determinations of the theoretical phases, and comparison with the U -spin based estimates and future lattice measurements.For example, for the CF/DCS decays D 0 → K ± X, e.g.D 0 → K ± π ∓ , the differences between φ M,Γ f and φ M,Γ 2 are given by a negligible and precisely known final state independent term of O(|λ b /λ s | 2 ) = O(10 −6 ), where λ i = V ci V * ui .For the CF/DCS decays D 0 → K S,L X, e.g.D 0 → K S π + π − , K S ω, K S π 0 , the differences between φ M f and φ M 2 , or φ Γ f and φ Γ 2 are dominated by two precisely known contributions.The smaller one is the CKM related quantity, |λ b /λ s | sin γ.The larger one, due to CPV in K 0 −K 0 mixing, is given by 2 Im( K ), and is of the same order as our estimates for φ M,Γ 2 .Thus, CPV in K 0 − K 0 mixing should be accounted for in order to achieve sensitivity to φ M,Γ 2 in the SM.Two additional contributions, associated with / and the DCS amplitudes, lie an order of magnitude below φ M,Γ 2 and can be neglected.Finally, for the singly Cabibbo suppressed (SCS) decays, e.g.D 0 → K + K − , π + π − , the final state dependence of φ M f ,φ Γ f in the SM originates from currently incalculable QCD penguin operator effects, and is of the same order as the corresponding direct CP asymmetries.These effects generally enter at first order in U -spin breaking, i.e. at O( ) (unless the leading "tree" decay amplitude is also subleading).However, the final state dependence could turn out to be O (1), due to the actual sizes of the QCD penguin amplitudes in certain decay modes.For example, the recent LHCb measurement of ∆A CP [2], suggests a nominal effect of O(0.4 φ M,Γ

2
) for D 0 → π + π − , K + K − .Nevertheless, our order of magnitude estimates for φ M,Γ 2 would still apply to φ M,Γ f .Fortunately, for the averages of φ M,Γ f over f = K + K − and π + π − , the deviations from φ M,Γ 2 are of O( 2 ).We conclude that a single pair of intrinsic dispersive and absorptive mixing phases suffices to parametrize all indirect CPV effects in CF/DCS decays, whereas for SCS decays this could cease to be the case as SM sensitivity is approached.We refer to this fortunate state of affairs as approximate universality.In particular, the approximate universality phases are identified with the intrinsic mixing phases, φ M 2 and φ Γ 2 .Once non-universality is hinted at in the SCS phases, the SCS observables could be dropped from the global fits.Instead, one could combine the CF/DCS based fit results for φ M,Γ 2 with measurements of φ M,Γ f and direct CPV in the SCS decays, to learn about the anatomy of the (subleading) SCS QCD penguin amplitudes.For example, in the SM one could separately determine their relative magnitudes, and strong phases.
One can also introduce a "phenomenological" intrinsic mixing phase φ 2 (again defined relative to the direction of the dominant ∆U = 2 mixing amplitudes), corresponding to the familiar phenomenological final state dependent phases φ λ f .In fact, to very good approximation, the two are weighted averages over φ M 2 and φ Γ 2 , and over φ M f and φ Γ f , respectively, where the weights in both cases are the dispersive and absorptive contributions to the CP averaged mixing probability.Moreover, the familiar CPV mixing parameter 1 − |q/p| (which determines the semileptonic CP asymmetries) is proportional to sin φ 12 = sin(φ M 2,f − φ Γ 2,f ).Thus, the approximate universality fit to the "theoretical" intrinsic CPV phases φ M 2 and φ Γ 2 is equivalent to a fit to the "phenomenological" intrinsic CPV parameters 1 − |q/p| and φ 2 .Whereas the former isolate the physically distinct phenomena of dispersive and absorptive CPV in the mixing amplitudes, the latter parametrize phenomenologically motivated combinations of the two.
Approximate universality generalizes beyond the SM under the following conservative assumptions regarding potential subleading decay amplitudes containing new weak phases: (i) they can be neglected in Cabibbo favored and doubly Cabibbo suppressed (CF/DCS) decays, given that an exotic NP flavor structure would otherwise be required in order to evade the K constraint [3]; (ii) in singly Cabibbo suppressed (SCS) decays, their magnitudes are similar to, or smaller than the SM QCD penguin amplitudes, as already hinted at by current bounds on direct CPV in D 0 → K + K − , π + π − decays.These assumptions can ultimately be tested by future direct CPV measurements at LHCb and Belle-II.
The most stringent experimental bounds on indirect CPV phases have been obtained in the superweak limit [4][5][6], in which the SM weak phase γ and potential NP weak phases in the decay amplitudes are set to zero in the indirect CPV observables.In this limit, the dispersive and absorptive mixing phases satisfy φ M f = φ M 2 and φ Γ f = φ Γ 2 = 0. Thus, φ 12 is identified with φ M 2 , and is entirely due to short-distance NP.The superweak fits are highly constrained, given that only one CPV phase, φ 12 , controls all indirect CPV (CPVMIX and CPVINT are therefore related).Comparison of superweak fit results with our estimate, φ M 2 , φ Γ 2 = O(0.2%)suggests that there is currently an O(10) window for NP in indirect CPV.
Moving forward, the increased precision at LHCb and Belle-II will require fits to the indirect CPV data to be carried out for both φ M 2 and φ Γ 2 , in the approximate universality framework.The addition of φ Γ 2 yields a less constrained fit.However, this should ultimately be overcome by a large increase in statistics.
Throughout this work we develop, in parallel, the description of indirect CPV for the three relevant classes of decays: (i) SCS (both CP eigenstate and non-CP eigenstate final states), (ii) CF/DCS decays to K ± X, and (iii) CF/DCS decays to K 0 X, K 0 X.The last one requires special care due to the intervention of CPV in K 0 − K 0 mixing.In Section II, the formalism for mixing and indirect CPV is presented, based on the final state dependent dispersive and absorptive CPVINT observables.A translation between the dispersive and absorptive CPV phases, φ M f , φ Γ f , and more widely used CPV parameters is also provided.In Section III, we apply this formalism to the derivation of general expressions for the time dependent decay widths and indirect CP asymmetries in terms of φ M f , φ Γ f .In CF/DCS decays to K 0 X, K 0 X, the widths depend on two elapsed time intervals: the time at which the D decays, and the time at which the K decays, following their respective production.Approximate universality is discussed in Section IV.We begin with the Uspin decomposition of the mixing amplitudes in the SM, introduce the intrinsic mixing phases φ M 2 , φ Γ 2 , estimate their magnitudes, and derive their deviations from the final state dependent phases.In Section V we explain how to convert the expressions for the time dependent decay widths and indirect CP asymmetries, collected in Section III, to the approximate universality framework.In the case of CF/DCS decays to K 0 X, K 0 X, the effects of K on the K decay time scales of relevance for LHCb and Belle-II are compared.Superweak and approximate universality fits to the current data are presented in Section VI, together with future projections.We conclude with a summary of our results in Section VII.Appendix A contains expressions for a selection of time-integrated CP asymmetries, demonstrating that they can also be used to separately measure φ M 2 and φ Γ 2 .

A. Mixing and time evolution
The time evolution of an arbitrary linear combination of the neutral D 0 and D 0 mesons, follows from the time-dependent Schrödinger equation (see e.g.[1]), The 2 × 2 matrices M and Γ are Hermitian, where the former is referred to as the mass matrix, and the latter yields exponential decays of the neutral mesons.CP T invariance implies H 11 = H 22 .The transition amplitudes for D 0 − D 0 mixing are given by the off-diagonal entries ( M 12 is the dispersive mixing amplitude.In the SM it is dominated by the long-distance contributions of off-shell intermediate states.A significant short distance effect would be due to new physics (NP).Γ 12 is the absorptive mixing amplitude, and is due to the long distance contributions of on-shell intermediate states, i.e. decays.The D meson mass eigenstates are where The differences between the masses and widths of the mass eigenstates, ∆M D = m 2 − m 1 and ∆Γ D = Γ 2 − Γ 1 , are expressed in terms of the observables where the averaged D 0 lifetime and mass are denoted by Γ D and M D .We can define three "theoretical" physical mixing parameters: two CP conserving ones, and a CP violating pure mixing (CPVMIX) phase The CP-odd phases are separately meson and quark phase convention dependent and unphysical.The CP conserving parameters in ( 6) and ( 7) are related as yielding up to negligible corrections quadratic in sin φ 12 .Two other useful relations are (12) Measurements of the D 0 meson mass and lifetime differences and CPV asymmetries imply that x 12 , y 12 ∼ 0.5%, while sin φ 12 0.1, cf.Section V.One is free to identify D 2 or D 1 with either the short-lived meson, or the heavier meson, by redefining q → −q.This is equivalent to choosing a sign-convention for y, which in turn fixes the sign of x, or vice-versa, via the imaginary part of (10).In the HFLAV [7] convention, D 2 is identified with the would be CP-even state in the limit of no CPV.Given that the short-lived meson is approximately CP-even, this is equivalent to the choice y > 0.
The time-evolved mesons D 0 (t) and D 0 (t) denote the mesons which start out as a D 0 and D 0 at t = 0, respectively.Solving (2) for their time-dependent components yields, with D 0 |D 0 (t) obtained from D 0 |D 0 (t) by substituting M * 12 → M 12 and Γ * 12 → Γ 12 .The phase π/2 in the first relation of (13) originates from the time derivative in (2), and is a dispersive CP-even "strong phase".We will keep track of its role in the derivation of the indirect CP asymmetries in Section III.For the time intervals relevant to experiment, i.e. t 1/Γ D , (13) reduces to up to negligible corrections entering at O(t 3 ) and beyond, and where use has been made of (10) in the last relation.

B. The decay amplitudes
The amplitudes for D 0 and D 0 decays to CP conjugate final states f and f are denoted as where H is the |∆C| = 1 weak interaction effective Hamiltonian.The tree-level dominated decay amplitudes can, in general, be written as where A 0 f and A 0 f are the magnitudes of the dominant SM contributions, the ratios r f and r f are the relative magnitudes of the subleading amplitudes (which are CKM suppressed in the SM, and potentially contain NP contributions), φ 0 f , φ 0 f , φ f , and φ f are CP-odd weak phases and ∆ 0 f , δ f , and δ f are CP-even strong phases.With the exception of the weak phases, the quantities entering ( 16) are understood to be phase space dependent for 3-body and higher multiplicity decays.Note that φ 0 f and φ 0 f are quark and meson phase convention dependent.However, this dependence cancels in physical observables.
In the case of decays to CP eigenstates, ∆ 0 f = 0 (π) for CP even (odd) final states.Eq. ( 16) therefore reduces to where η CP f = +(−) for CP even (odd) final states.For SCS decays, the choice of the dominant and subleading SM amplitudes in (16) and ( 17) is convention dependent.For example, using CKM unitarity, the leading SCS D 0 decay amplitudes could be chosen to be proportional to The last choice is a particularly convenient one that is motivated by U -spin flavor symmetry, cf.Section IV A. In all cases, the subleading SM amplitudes are ∝ V * cb V ub , and are included in the second term on the RHS of each relation in (16), (17).However, the physical observables must be convention independent.
We divide the CF/DCS decays into two categories: (i) decays to K ± X, where indirect CP requires interference between a CF and a DCS decay chain, e.g.D 0 → K − π + and D 0 → D0 → K − π + , respectively; (ii) decays to K 0 X, K 0 X, where indirect CPV is dominated by interference between two CF decay chains, e.g.D 0 → K 0 π + π − and D 0 → D0 → K 0 π + π − , with subsequent decays K 0 /K 0 → π + π − .In the SM, the CF and DCS D 0 decay amplitudes are proportional to V * cs V ud and V * cd V us , respectively.Thus, only the first terms in (16) are present.We choose the CF and DCS amplitudes to be A f , Ā f and A f , Āf , respectively.For the computation of the indirect CP asymmetries in case (i), all four amplitudes in (16) must be included, whereas in case (ii) we will see that the contributions of the two DCS amplitudes can be neglected to good approximation.

C. The CPVINT observables
We are now ready to define the CPV phases φ M f and φ Γ f , responsible for dispersive and absorptive CPVINT, respectively. 1

SCS decays to CP eigenstates
For SCS decays to CP eigenstate final states, φ M f and φ Γ f are the arguments of the CPVINT observables They are given by, to first order in r f , cf. ( 9), (17).We will see that φ M f , φ Γ f ≈ 0 (rather than π), given the sign of the CP conserving observable y f CP , f = π + π − ,K + K − , cf. ( 62), (64).

SCS decays to non-CP eigenstates
For SCS decays to non-CP eigenstate final states, e.g.
1 In [8] it was noted that a non-zero value for arg[M equivalent to 2φ M f and 2φ Γ f , respectively, cf.(18), (20), (21), implies CP violation.However, the phenomenology of these phases was not discussed.and The dispersive and absorptive CPV phases now satisfy, cf. ( 9), ( 16), while the overall strong phase difference in the decay amplitude ratios is given by to first order in r f and r f .
3. CF/DCS decays to K ± X For CF/DCS decays to K ± X, e.g.D 0 → K ± π ∓ , the definitions in (20), ( 21) apply (recall that A f is the CF amplitude), however we introduce overall minus signs in the equalities, i.e.
Thus, the dispersive and absorptive CPV phases satisfy and the expression for the strong phase in ( 23) is not modified.The sign convention in (24) yields φ M f , φ Γ f ≈ 0 (rather than π), as in SCS decays.In the SM and, more generally, in models with negligible new weak phases in CF/DCS decays, the second line in (25) is absent, and the dispersive and absorptive phases are separately equal for all decays in this class.Moreover, the absence of direct CPV yields the relation 4. CF/DCS decays to K 0 X, K 0 X Next, we define the CPVINT observables for D 0 /D 0 decays to final states f = [π + π − ]X, where the square brackets indicate that the pion pair originates from decays of a K S or K L , i.e. two step transitions of the form In order to achieve SM sensitivity to CPVINT, the contributions of CPV in the K system must be taken into account.The neutral K mass eigenkets are written as, The corresponding eigenbras are given in the "reciprocal basis" [8,9], where CPT invariance has been assumed.To excellent approximation (see, e.g.[1]), The experimental values of the real and imaginary parts of the kaon CPV parameter K are [10], We have obtained them from the quoted measurements of η 00 and η +− , ignoring correlations in their errors.
In general, due to the presence of the two intermediate states K S X and K L X, there are four pairs of CPVINT observables, where the first and second lines correspond to the CP conjugate final states f = [π + π − ]X and f = [π + π − ]X, respectively.Note that for the important case of X = π + π − , f corresponds to interchange of the Dalitz plot variables (p K + p π + ) 2 ↔ (p K + p π − ) 2 in f .We can express the CPVINT observables (30) in the form where the overall plus and minus signs refer to the K S and K L , respectively.The four CPVINT phases and two strong phases in (31) respectively.
The D decay amplitudes in (30) satisfy, where we have used the reciprocal basis (27), and the first and second terms on the RHS in each relation are the dominant CF and subleading DCS contributions, respectively.
In the SM and, more generally, in models with negligible new CPV phases in CF/DCS decays, the DCS decay amplitudes introduce relative corrections of O(θ 2 C ) to the weak phases, strong phases, and magnitudes of λ M,Γ K S/L X , λ Γ K S/L X , making it a good approximation to neglect them.(We assess the impact of the DCS amplitudes on approximate universality in Section IV C 3.) In this limit, (30) reduces to, Thus, in the limit of negligible new CPV phases in CF/DCS decays, it is a good approximation to consider a single pair of CPVINT observables for final state f = [π + π − ]X, and a single pair for f = [π + π − ]X, which we have denoted in (33) as λ M f , λ Γ f and λ M f , λ Γ f , respectively.They can be expressed in terms of dispersive and absorptive CPVINT phases as, where the amplitude relations, valid in the limit of vanishing direct CPV, have been employed in the second relation.Note that the weak phases φ M ,Γ [K S/L X] and strong phases ∆[K S/L X], defined in general in (31), reduce to φ M,Γ f and ∆ f , respectively.The strong phase difference ∆ f (between A K 0 X and A K 0 X ) is generally non-vanishing and phase space dependent for multi-body intermediate states, e.g.X = π + π − .The weak phases satisfy where φ 0 K 0 X is the weak phase of the CF amplitudes 16), while arg(p K /q K ) introduces a dependence on CPV in the K system, cf.Section IV C 3. Note that φ M f and φ Γ f are separately equal for all final states in this class.
In the case of two-body (and quasi two-body) intermediate states, the CPVINT observables in (34) reduce to, where L is the orbital angular momentum of the intermediate states K S/L X, and CP [X] = + (−) for CP even (odd) Finally, we point out that in all three classes of D 0 decays discussed in this Section, the quark (CKM) phase convention dependence cancels in φ M f and φ Γ f , i.e. between the first two terms on the RHS of ( 19), the first three terms on the RHS of (22), and between all three terms in (36), cf.Section IV C.Moreover, they are always related to the pure mixing phase φ 12 as, i.e. the final state dependent effects are common to the dispersive and absorptive phases.

Relation to other parametrizations of CPVINT
It is instructive to relate the parametrization of indirect CPV effects in terms of absorptive and dispersive phases to the more familiar one currently in use.The latter consists of the CPVMIX parameter, and the final state dependent phenomenological CPVINT phases φ λ f , which appear in the arguments of the observables λ f , see e.g.[1].We begin with the definitions of the λ f , corresponding to the absorptive and dispersive observables λ M,Γ f , in the different classes of decays.For SCS decays to CP eigenstate final states, they correspond to the observables in (18), and are given by For SCS decays to non-CP eigenstate final states, and CF/DCS decays to K ± X, the λ f corresponding to the observables in ( 20), (21), and ( 24) are given by, where the ∓ sign conventions in the right-most relations apply to the SCS and CF/DCS cases, respectively.Finally, for CF/DCS decays to K 0 X, K 0 X (given negligible new CPV phases in the decay amplitudes, and neglecting the DCS contributions) the λ f correspond to the absorptive and dispersive observables in (33), (34), and are given by In the case of two-body or quasi two-body intermediate states, corresponding to the observables in ( 37), these expressions reduce to, The sign conventions in the right-most relations of ( 41)-( 44) yield all φ λ f ≈ 0 (HFLAV convention for D 2 ), or all ≈ π, for the three classes of decays.
The CPV parameters |q/p| − 1 and φ λ f are expressed in terms of the absorptive and dispersive CPV phases as, where Eq. ( 46) is obtained by multiplying both sides of ( 5) by ( Āf /A f ) 2 and ( Āf Ā f /A f A f ) for CP eigenstate and non-CP eigenstate final states, respectively, and holds for all classes of decays.To lowest order in the CPV phases, it equates the phenomenological CPVINT phase φ λ f to a sum over the dispersive and absorptive CPVINT phases, φ M f and φ Γ f , weighted by the ratios x 2  12 /(x 2 12 + y 2 12 ) and y 2 12 /(x 2 12 + y 2 12 ), respectively.The weights are, respectively, the leading dispersive and absorptive contributions to the CP averaged mixing probability, (14).
Finally, we remark on the CPV observables ∆x f [11] and ∆y f , which have been measured in tandem by the LHCb collaboration [12] in D 0 → K S π + π − decays.They are defined in terms of φ λ f and |q/p| as2 The observable −∆y f is equivalent to the familiar CPVINT asymmetry ∆Y f for SCS decays to CP eigenstate final states, cf.(61).Translating to the dispersive/absorptive parametrization via ( 45), ( 46), we obtain to leading order in sin φ M,Γ f .Thus, the use of the parameters ∆x f and ∆y f is equivalent to the CPVINT parametrization in terms of φ M f and φ Γ f , respectively, modulo the corresponding dispersive and absorptive mixing factors.(It is amusing that interchange of the ∆x and ∆y labels turns out to be appropriate).Interestingly, we will see that experimental sensitivity to φ Γ f (or ∆x f ) requires a non-trivial strong phase difference between decay amplitudes, i.e. non-CP eigenstate final states, e.g.

III. THE INDIRECT CP ASYMMETRIES
We can now derive expressions for the time-dependent decay widths and CP asymmetries in terms of the absorptive and dispersive CPV phases.(A discussion of CPV in certain time-integrated decays is deferred to Appendix A.)

A. Semileptonic decays
We begin with the CPVMIX "wrong sign" semileptonic CP asymmetry, In the second line the semileptonic decay amplitude factors have been cancelled, given negligible direct CPV in these decays, i.e.
In turn, the expressions for the mixed amplitudes in ( 13) or ( 14) yield the semileptonic asymmetry, Note that the CP-even phase difference between the interfering dispersive and absorptive mixing amplitudes, required to obtain CPVMIX, is provided by the dispersive mixing phase π/2 in the first line of (13).

B. Hadronic decays
The hadronic decay amplitudes sum over contributions with and without mixing, The corresponding time-dependent decay rates are identified with their magnitudes squared.They are expressed in terms of the CPVINT observables λ M f, 18), ( 20), (21), as (τ ≡ Γ D t), with the expressions for Γ(D 0 (t) → f ) and Γ(D 0 (t) → f ) obtained via the substitutions f → f in (52).Note that throughout this work appropriate normalization factors are implicit in all decay width formulae, including (52).The expressions in (52) are applied to the following cases: SCS decays to CP eigenstates, SCS decays to non-CP eigenstates, and CF/DCS decays to K ± X.The description of CF/DCS decays to K 0 X, K 0 X requires a separate treatment, cf Section III C.

SCS decays to CP eigenstates
This category includes, for example, the decays D 0 → K + K − /π + π − .(We comment on the decay D 0 → K 0 K 0 at the end of Section IV C 1 ).The time-dependent decay widths D 0 (t) → f and 19), and the direct CP asymmetry, cf. (17), are given by where the coefficients Terms involving a d f have been expanded to first order in CPV quantities, and the semileptonic CP asymmetry, expressed in terms of φ 12 , is given in (50).
The O(τ 2 ) terms in the SCS widths are usually neglected, due to an O(x 12 , y 12 ) suppression relative to the O(τ ) term.Thus, it has been traditional to express the SCS widths in the approximate exponential forms, where the decay rate parameters satisfy, cf. (55).As the goal of SM sensitivity comes into view, i.e. φ M f , φ Γ f = O(few) × 10 −2 , this will not necessarily be a good approximation, as can be seen by comparing the CP-odd terms in c ± f , and the CP-even term in c ± f .However, the CP-odd terms in c ± f are further suppressed by CPV parameters, and can be neglected.Thus, to good approximation, Measurements of the time-dependent decay rates at linear order in τ yield the known CP conserving observables, and the CPVINT asymmetries, The average of ∆Y f over f = K + K − , π + π − is denoted by A Γ .In the exponential approximation, the corresponding definitions are, Applying (55), and neglecting contributions quadratic in CPV, we obtain The experimental average over f to excellent approximation.Furthermore, fits to the data [7,13] yield xy > 0 at 3σ, or φ 12 ≈ 0 (rather than π), cf.(10).Thus, we learn that both At first order in CPV, (55) yields the relation (already noted in (48) for the CPVINT part), The direct CPV contribution in ( 65) is formally subleading, cf.Section IV C 1.In general, it can be disentangled experimentally from the dispersive CPV contribution with the help of time integrated CPV measurements, in which a d f enters without mixing suppression, cf.Appendix A.
It is noteworthy that ∆Y f depends on φ M f , but not on φ Γ f .This is because CP asymmetries require a nontrivial CP-even phase difference δ between the interfering amplitudes, i.e., they are proportional to sin δ.In general, for CP eigenstate final states there is a CP-even phase difference between decays with and without dispersive mixing, namely the π/2 dispersive phase in (13).However, there is none between decays with and without absorptive mixing (the strong phase between A f and A f is trivial).Therefore, in general, φ Γ f can only be measured in decays to non-CP eigenstate final states, where the requisite CP-even phase is provided by the strong phase difference ∆ f between A f and A f , as we will see explicitly below.Finally, in the case of CP averaged decay rates, interference terms are in general proportional to cos δ, rather than sin δ.Therefore, in the CP averaged time dependent decay rates for CP eigenstate final states, the interference between decays with and without dispersive mixing will vanish at leading order in the mixing, i.e.O(τ ), only leaving a dependence on y 12 .This is borne out by the expression for y f CP in (62).

SCS decays to non-CP eigenstates
This category includes, for example, the decays D 0 → ρπ, K * + K − .The time dependent decay widths are of the form, for final state f , and for final state f , where In general, the ratios satisfy R f , R f = O(1) for SCS decays.The coefficients c ± f and c ± f in ( 66), (67), expressed in terms of φ M f , φ Γ f , and ∆ f , cf. ( 20)-( 23), are given by The coefficients in the O(τ 2 ) terms satisfy, As in the prior case of decays to CP eigenstates, the CPeven terms in c ± f, f should be kept, with future sensitivity at the level of SM indirect CPV in mind.However, the CP-odd terms (∝ a SL ) can be neglected.
The time dependent measurements yield pairs of CPVINT asymmetries (normalized rate differences for D 0 (t) → f vs. D 0 (t) → f , and To first order in CPV parameters, (69) yields the expres- where the direct CP asymmetries, cf. ( 16), enter via the deviation of R f R f from unity.In (72), replacing the numerator and denominator in the ratio R f , cf. ( 68), with their CP averaged counterparts would introduce a negligible higher order correction in the CPV parameters.Note that the CP-even phase differences for dispersive and absorptive CPVINT are given by ∆ f − π/2 and ∆ f , respectively, where π/2 is the "dispersive" phase in the first line of ( 13), thus accounting for the factors cos ∆ f and sin ∆ f in the first two terms of ∆Y f and ∆Y f in (72).In particular, Eq. ( 72) confirms that sensitivity to the absorptive phase φ Γ f requires a strong phase difference between decay amplitudes, i.e. non-CP eigenstate final states, as argued at the end of Section III B 1.

CF/DCS decays to K ± X
This category consists of the CF/DCS decays D 0 → K ± X, with a single K in the final state.As noted previously, we choose the DCS decay amplitudes in ( 16), (20), (21), and (24), to be A f and Āf , e.g.f = K + π − .Thus, we denote the time dependent CF/DCS decays to "wrong-sign" (WS) final states as D 0 (t) → f and D 0 (t) → f .The O(τ 2 ) terms in (52) and its CP conjugate can not be neglected, given that the decay amplitude ratios entering λ M,Γ f, ).The WS decay widths following from ( 52) and ( 64) can be expressed as, where R ± f are the DCS to CF ratios and the coefficients c ± f , c ± f , to first order in CPV parameters, are given by The (CF) direct CP asymmetry, a d f , appearing in (76) is given by and vanishes in the SM.The last four terms in c ± f , two CP-even, and two CP-odd, yield contributions to the time-dependent decay widths which are suppressed in the SM by O( 14 ), respectively, relative to the O(τ ) CP-odd terms, i.e. by more than an order of magnitude in both cases, and can therefore be neglected.In particular, the O(τ 2 ) coefficients are well approximated as, The prefactors in (74) are, to excellent approximation, equal to the right sign (RS) time dependent decay widths, where the subleading DCS contributions have been neglected.
A fit to the time-dependence in (74), (79) yields measurements of R ± f , c ± f , c ± f , and the indirect CP asymmetries, Note that the last terms in (80) for δc f and δc f are absent in the SM and, more generally, in models with negligible CP violating NP in CF/DCS decays.As in (72), the cos ∆ f and sin ∆ f dependence in the first two terms of δc f originates from the total CP-even phase differences ∆ f − π/2 and ∆ f , between decays with and without dispersive mixing and decays with and without absorptive mixing, respectively.This again confirms that strong phase differences are required in order to measure the absorptive CPV phases, φ Γ f .
C. CF/DCS decays to K 0 X , K 0 X We derive expressions for the time-dependent D 0 and D 0 decay rates for two step CF/DCS decays of the form to final states f = [π + π − ]X.These decays depend on two elapsed time intervals, t and t , at which the D and K decay following their respective production.The D 0 (t) and D 0 (t) decay amplitudes now sum over contributions with and without D 0 −D 0 mixing, and with and without K 0 − K 0 mixing.The kaon time evolution is conveniently described in the mass basis, where M S,L , Γ S,L , and τ S,L are the corresponding masses, widths, and lifetimes.The time-dependent amplitudes for the decay of an initial D 0 to final state f = [π + π − ]X, and for the CP conjugate decay of an initial D 0 to final state f = [π + π − ]X, are given by where expressions for the D decay amplitudes A KaX , etc. appear in (32).The K S,L → ππ decay amplitudes satisfy, with The amplitudes A f (t, t ) and A f (t, t ) are obtained by substituting |D 0 (t) → |D 0 (t) and vice versa in the first and second relations of (83), respectively.Expressing the amplitudes in terms of the CPVINT observables in (30) yields the general expressions, where A f (t, t ) is obtained by substituting A KaX → A KaX and λ KaX in the first relation, and KaX in the second relation.The time-dependent decay rates are obtained by squaring the magnitudes of the amplitudes in (86), e.g.Γ f (t, t ) = |A f (t, t )| 2 etc., and assuming that CP violating NP is negligible in CF/DCS decays.Therefore, as in the SM, we assume vanishing direct CPV in the CF decays, neglect the DCS amplitudes (their impact is discussed in Section IV C 3), and employ the expressions for the CPVINT observables given in (34).We work to first order in CPV quantities, and also employ the relations (see e.g.[1]), In particular, the last relation in (87) implies that we can neglect the purely K L contributions to the widths.The expressions for the time-dependent decay rates are then of the form, for final state f , and for final state f , where  34)- (36), and K .For the purely K S X contributions (e −Γ S t dependence), they are given by CP-odd contributions to the coefficients c ± f , c ± f are of O[(x 2  12 , y 2 12 ) × ( K , φ 12 )] and have been neglected, i.e. they are O(x 12 , y 12 ) suppressed relative to the CP-odd terms arising at O(τ ).Interference between the amplitudes containing intermediate K S X and K L X (e −Γ K t dependence) yields, We have neglected interference contributions of O(x 2 12 K , y 2 12 K ) arising at O(τ 2 ) in (88), (89).Again, they are O(x 12 , y 12 ) suppressed relative to the CP-odd terms arising at O(τ ).
The indirect CP asymmetries are obtained by taking normalized rate differences between Γ f and Γ f , and between Γ f and Γ f .To first order in CPV quantities, the phases φ M f , φ Γ f only enter the CP asymmetries of the purely K S contributions, while the CP asymmetries induced by K S − K L interference only probe K .The first set of CP asymmetries, between the coefficients in (91), are given by (δc is negligible), Again, ∆ f = 0, π is required in order to measure φ Γ f , due to the lack of a non-trivial CP-even phase in the absorptive mixing ampltiude.The six CP asymmetries in the second set of coefficients, cf.(92), are In principle, each of the CP asymmetries in (93) , (94) can be measured by fitting to the dependence of the decay rates on t and t .
In Section IV B we will see that in the SM, φ M f and φ Γ f are expected to be of same order as K , implying that the CPVINT asymmetries in (93) and (94) are also of same order.Thus, the impact of K , particularly at linear order in τ , on the asymmetry measurements needs to be considered.We will address this point in Section V, taking into account the typical decay times t for the intermediate K 0 's detected at LHCb and Belle-II.
In the case of two body (and quasi two body) intermediate states, e.g.X = π 0 , ω, f 0 , expressions for the time dependent decay rates and CP asymmetries are obtained by setting 94), where η CP f is defined in (37).The resulting decay widths are, with coefficients, The corresponding CP asymmetries, as defined in ( 93), (94), are given by Note that δc f is purely dispersive, similarly to ∆Y f for SCS decays to CP eigenstates, cf.(65) (again, the only CP even phase available for charm CPVINT is the dispersive mixing phase π/2).Finally, the CP conserving observable, y f CP , for SCS decays to CP eigenstates, cf. ( 59), (61), can be carried over to the case of two body and quasi two body intermediate states discussed above.It is analogously defined as However, the K S decay time dependence, e −Γ S t , in (95),(96), must be accounted for in order to avoid additional systematic errors in its extraction.Employing (97) yields up to negligible corrections quadratic in CPV parameters.For example, we expect y f CP = −y 12 for X = ω, π 0 (opposite in sign to y f CP for K + K − , π + π − ), and y f CP = +y 12 for X = f 0 .

IV. APPROXIMATE UNIVERSALITY
In the previous section, all indirect CPV effects were parametrized in full generality, in terms of final state dependent pairs of dispersive and absorptive weak phases (φ M f , φ Γ f ).In order to understand how best to parametrize indirect CPV effects in the upcoming precision era, we need to estimate the final state dependence.We accomplish this via a U -spin flavor symmetry decomposition of the SM D 0 −D 0 mixing amplitudes.Crucially, this also yields estimates of indirect CPV effects in the SM.

A. U-spin decomposition
The SM D 0 − D 0 mixing amplitudes Γ 12 and M 12 have flavor transitions ∆C = −∆U = 2 and ∆S = ∆D = 0. We can write them as where λ i ≡ V ci V * ui .At the quark level, the transition amplitudes Γ ij and M ij are identified with box diagrams containing, respectively, on-shell and off-shell internal i and j quarks.Thus, they possess the flavor structures (Dirac structure is unimportant for our discussion) and similarly for the M ij .Employing CKM unitarity (λ d + λ s + λ b = 0), the U -spin decomposition of Γ SM 12 is given by where the U -spin amplitudes Γ 2,1,0 are the ∆U 3 = 0 elements of the ∆U = 2, 1, 0 multiplets, respectively.This can be seen from their quark flavor structures, The orders in the U -spin breaking parameter at which they enter are also included, corresponding to the power of the U -spin breaking spurion M ∼ (ss − dd) required to construct each Γ i .The U -spin decomposition of M 12 is analogous to (103), with the exception of additional contributions to M 1 and M 0 , given by (M sb − M db ) and (M sb +M db +M bb ), respectively, and corresponding to box diagrams with internal b quarks at the quark level.The small value of λ b implies that we can neglect the ∆U = 1, 0 contributions to the mass and width differences, even though the ∆U = 2 piece is of higher order in .Thus, x 12 and y 12 are due to Γ 2 and M 2 , respectively, and arise at O( 2 ) [14 -16].Similarly, CPV in mixing arises at O( ) due to Γ 1 and M 1 , while the contributions of Γ 0 and M 0 are negligible.The U -spin amplitudes Γ i , M i are of the form, The exponential factors originate from the choice of meson phase convention, and trivially cancel in physical observables.However, the η i in (105) are physical, can a priori be of either sign, and can be determined from experiment.For example, since φ 12 ≈ 0, we already know that or that η M 2 = η Γ 2 .Moreover, as we shall see shortly, cf.(120), existing measurements also imply that The inclusive [17][18][19][20][21][22][23][24] and exclusive [14-16, 25, 26] approaches to estimating ∆Γ D yield several observations of relevance to our discussion of CPV below.In the inclusive OPE based approach, the flavor amplitudes satisfy Γ ij ∼ Γ D .This is reflected in the ability of this approach to accommodate the charm meson lifetimes [24,27].The individual Γ ij contributions to y 12 are, therefore, about five times larger than the experimental value [28], suggesting that U -spin violation is large, e.g.O( 2 ) ∼ 20% in Γ 2 , cf. (104). 4The exclusive approach addresses its origin by estimating sums over exclusive decay modes.Unfortunately, the charm quark mass is not sufficiently light for D 0 meson decays to be dominated by a few final states.Moreover, the strong phase differences entering y 12 , and the off-shell decay amplitudes in x 12 are not calculable from first principles.However, there is consensus in the literature that accounting for y 12 near 1% requires significant contributions from high multiplicity final states (n ≥ 4), due to the large SU (3) F breaking near threshold.This observation is consistent with the large U -spin breaking required (from duality violations) in the OPE/HQE approach.

B. Intrinsic CPV mixing phases
We introduce three intrinsic CPV mixing phases, defined with respect to the direction of the dominant ∆U = 2 dispersive and absorptive mixing amplitudes in the complex plane, 4 Inclusive OPE based GIM-cancelations between the Γ ij yield y four orders of magnitude below experiment.Evidently, mc and (ms − m d )/Λ QCD are not sufficiently large and small, respectively, for this approach to properly account for U -spin breaking in y 12 .
where Γ 12 , M 12 , and q/p can contain NP contributions.They can be viewed as the pure mixing analogs of the final state dependent phases φ M f , φ Γ f , and φ λ f , respectively.Note that they are quark and meson phase convention independent, like the final state dependent ones, as required for physical phases.For later use we give the expressions for the (phase convention dependent) arguments of M 12 and Γ 12 in terms of φ M 2 and φ Γ 2 , respectively, cf.(105), Employing ( 106), the theoretical or intrinsic mixing phases are seen to satisfy the relations and the analog of ( 46), Together with (45), the above relations allow translation between φ 2 and |q/p|, and any two out of the three phases φ M 2 , φ Γ 2 , and φ 12 .We estimate the magnitudes of the theoretical phases in the SM (Γ 12 = Γ SM 12 , M 12 = M SM 12 ), as well as their deviations from the corresponding final state dependent phases φ Γ f , φ M f , and φ λ f , using U -spin based arguments and experimental input.The U -spin breaking hierarchy and similarly for φ M 2 .In terms of the U -spin breaking parameter , and with the most recent CKM fits [29,30], we obtain the rough SM estimates The third phase, φ 2 , is seen to be of same order, barring large cancelations, cf. ( 111).
An alternative expression for φ Γ 2 in the SM follows from (113), via the relation where in the second and third relations we have, respectively, taken |y| ≈ 0.066% [7], and Γ 1 ∼ Γ D (recall that the inclusive approach yields Γ ij ∼ Γ D ).The estimates for φ Γ 2 in (114) and (115) are consistent (for illustrative purposes, if we identify their respective factors, the two estimates would coincide for ≈ 0.36).However, the dependence in the latter has been shifted to the numerator: |y| = O( 2 ), while Γ 1 = O( ).In principle, Γ 1 can be estimated via the exclusive approach, as more data on SCS D 0 decay branching ratios and direct CP asymmetries become available.It relies on the U -spin decomposition of exclusive contributions to Γ 1 .Details can be found in [31].Unfortunately, the potentially large contributions from high multiplicity final states would complicate this program, as in the case of ∆Γ D .

C. Final state dependence
The misalignments between the final state dependent phases φ M f , φ Γ f , φ λ f , and their theoretical counterparts are equal in magnitude, satisfying Below, we discuss the size of δφ f in the SM for (i) SCS decays, (ii) CF/DCS decays to K ± X, and (iii) CF/DCS decays to K 0 X, K 0 X.

SCS decays
The amplitudes for the SCS decay modes D 0 → f and D 0 → f in the SM can be written as, see e.g.[32], with substitutions f → f for the CP conjugate modes.The first and second terms in each relation are the ∆U = 1 and ∆U = 0 transition amplitudes, respectively, where the former is due to the current-current operators The amplitudes for decays to CP eigenstates are generally of the form given in (17).In the case of SCS decays, comparison with (117) yields the weak phase, ) where the sum of the first two terms on the RHS is identified with 2φ 0 f (the second term originates from the choice of meson phase convention), and in the SM, Combining ( 109) and ( 118) yields the following expressions for the CPVINT phases φ M f , φ Γ f , cf. ( 18), ( 19), 64), we learn that the first term on the RHS must vanish in both relations of (120), i.e. η M 2 = η Γ 2 = +, as claimed in (107).In turn, the misalignment in (116) for a CP eigenstate final state, is given by where the direct CP asymmetry, a d f , has been defined in (53).
It is instructive to rewrite the CPVINT asymmetry ∆Y f , cf. (65), in terms of φ M 2 , and the subleading decay amplitude parameters r f , φ f , and δ f , (122) Previously, we saw that the leading amplitude contribution is purely dispersive for CP eigenstate final states, because the requisite CP-even phase difference is only present in the dispersive mixing amplitude (δ = π/2).Similarly, it is now clear that the strong phase dependence of the dispersive and absorptive contributions entering at first order in the subleading amplitudes, cf.(122), can be attributed to the strong phase differences π/2+δ f and δ f between their respective interfering decay chains.
In the case of SCS decays to non-CP eigenstates, the misalignments of the CPVINT phases, cf. ( 20)-( 22), generalize as, where r f , δ f are defined as in (119); r f , δ f correspond to the substitutions f → f therein; and The direct CP asymmetries have been defined in (73).
The misalignments (121), (123) for SCS decays are non-perturbative, and incalculable at present, like the direct CP asymmetries.However, the strong phases are expected to satisfy δ f, f = O(1), due to large rescattering at the charm mass scale, yielding the order of magnitude estimates δφ f = O(λ b sin γ/θ C ).In particular, the misalignments, like the direct CP asymmetries a d f are O(1) in SU (3) F breaking.Thus, they are parametrically suppressed relative to the theoretical phases in the SM, cf.(113), For example, the recent LHCb discovery [2] of a nonvanishing difference between the D 0 → K + K − and D 0 → π + π − direct CP asymmetries yields the world average [7], In the U -spin symmetric limit, a d π + π − = −a d K + K − [33], implying the rough estimate δφ f ∼ 0.08% for these decays.Dividing by the SM estimates for φ M 2 and φ Γ 2 in (114) or (115) yields significant misalignments, consistent with the parametric suppression in (124) for sizable ∼ 0.4.Fortunately, the K + K − and π + π − misalignments, like the direct CP asymmetries [33], are equal and opposite in the U -spin limit, i.e. (δφ Thus, the average of φ M,Γ f and the average of the time dependent CP asymmetries in (65) satisfies, where we have used the relations x 12 ∼ y 12 and δφ f ∼ a d f .As has already been noted, large U -spin violation is likely to play an important role in mixing.Moreover, the δφ f for SCS decays are inherently non-perturbative.Therefore, while (124) implies that the order of magnitude estimates (114), (115) for φ M,Γ 2 apply equally well to the measured phases φ M,Γ f in the SM, O(1) variations can not be ruled out.The latter possibility would correspond to the weakest form of approximate universality.Ultimately, precision measurements of the indirect and direct CP asymmetries in a host of SCS decays will clarify the situation.
We point out that in the presence of NP in SCS decays, the expressions for the misalignments, δφ f , in the second relations of (121), (123) remain valid.In particular, the direct CP asymmetries a d f, f and the strong phases δ f, f now depend on the total subleading amplitudes, i.e the sums of the QCD penguin and NP amplitudes.The δφ f would be of same order as in the SM, provided that the CP-odd NP amplitudes are similar in size, or smaller than the SM QCD penguin amplitudes, as already hinted at by the current bounds on direct CPV in D 0 → K + K − , π + π − decays.
Finally, we mention two SCS decay modes, D 0 → K 0 K 0 and D 0 → K * 0 K 0 , which violate the O( ) counting in (124).For D 0 → K 0 K 0 , the first term in (117) is suppressed by O( ) (as reflected in the rate), yielding O(1/ ) enhancements of δφ f , the direct CP asymmetry [34], [35], and the misalignment, i.e. δφ f /φ M,Γ 2 = O(1) in the SM.For D 0 → K * 0 K 0 , the first term in (117) is not formally suppressed by O( ).However, a large accidental cancelation between contributions related by K * 0 ↔ K 0 interchange (again reflected in the measured decay rate), once more enhances δφ f , and the direct CP asymmetry [36].Thus, in effect, the misalignment could be O(1), as for K 0 K 0 .

CF/DCS decays to K ± X
The CPVINT observables in this class are given in (20), (21), with the modified sign convention of (24).The CKM factors enter the CF/DCS amplitudes as A f ∝ V * cs V ud (CF) and Āf ∝ V cd V * us (DCS).Thus, in the SM and, more generally, in models with negligible new weak phases in CF/DCS decays, Eqs. ( 25) and (109) yield the absorptive and dispersive phases, Employing CKM unitarity, the misalignments, given by the second term on the RHS, are seen to satisfy To summarize, for CF/DCS decays to K ± X, the misalignments vanish up to a negligible (and precisely known) final-state independent correction of O(10 −6 ).This represents the strongest form of approximate universality, i.e. the universal limit.Thus, in these decays, CPVINT measurements directly determine the theoretical phases.
3. CF/DCS decays to K 0 X, K 0 X We begin with a discussion of the misalignments in this class of decays in the limit that the DCS decays are neglected.Expressions for the CPVINT observables and time-dependent decay widths in this approximation are given in ( 33)- (36) and Section III C, respectively.The misalignments follow from (36).One ingredient is the phase of q K /p K .To excellent approximation [1], this ratio satisfies the relation where A 0 ,2 denote the K 0 → (ππ) I=0 ,2 amplitudes, respectively, i.e. they are ∆I = 1/2 , 3/2 transitions.Keeping track of the CKM factors, these amplitudes can be written as, A second ingredient is the CP -odd phase in the ratio of CF amplitudes, Finally, combining (109),(133), and (134) yields the final state independent absorptive and dispersive phases, The last term in (135) is non-perturbative in origin.However, it enters the kaon CPV observable, K / K , as 6   Re where ω ≡ (A 2 /A 0 ) ≈ 1/22.Equating the measured value of Re[ K / K ] with the first term on the RHS of the second relation in (136), i.e. assuming modest cancelation with A 2 [37], yields the estimate Similarly, the dominant chirally enhanced penguin operator (Q 6 ) contribution to A 0 yields [37], where the matrix element parameter B (1/2) 6 = 1 in the large N C limit.(A recent study [38] claiming that the SM prediction for / could be significantly smaller than the measured value obtains Im[r 0 ] < 10 −4 ).
Thus, in the limit that the DCS amplitudes are neglected, the misalignments satisfy up to a small CP-odd ratio of K → ππ amplitudes, given by −2Im[r 0 ] = O(10 −4 ).The latter lies an order of magnitude below our SM estimates for the theoretical phases φ M 2 , φ Γ 2 in (114), (115) and can be neglected.Finally, we address the impact of the DCS amplitudes.Expanding the CPVINT observables in (30) to first order in the DCS amplitudes, the weak and strong phases in λ M,Γ K S/L X are seen to be related to those in λ M,Γ f (cf.(31) and (34), respectively), as where δφ f is given in (139).We recall that φ M,Γ f are the CPV phases in the absence of the DCS amplitudes, r f and r f are the magnitudes of DCS to CF amplitude ratios, and δ f , δ f are the strong phase differences of the corresponding amplitude ratios.Finally, their magnitudes are related as, and similarly for M → Γ. Expressions for the time dependent decay widths, including the DCS amplitudes, are obtained via insertion of the CPVINT observables (31) and the full expressions for the decay amplitudes (32) into the general formulae (86) for the time-dependent amplitudes.The result can be brought into the same general form as (88), (89).Effectively, the prefactors in Eqs. ( 88), (89), the ratios R f , and the expressions (91), (92) for the coefficients are modified at O(r f , r f ), i.e.O(θ 2 C ).For example, the coefficients contain new CP-even terms of O(r f, f ), and new CP-odd terms of O( K r f, f ).These corrections produce relative shifts in the CP averaged decay rates, as well as the indirect CP asymmetries listed in (93), (94), (98), of O(λ 2 C ).Our primary focus here is on the absorptive and dispersive CPVINT phases.As previously noted, they only reside in the pure K S contributions to the time dependent widths (to first order in CPV).In particular, φ M,Γ f are replaced by φ M,Γ [K S X] in the coefficients c ± f , c ± f , cf.
(140), (91).Consequently, the misalignments (139) are modified as, Thus, while the DCS corrections to the CPVINT phases are final state dependent, they are of O(2θ ) in the SM.This represents a more generic form of approximate universality than what we found in the previous two classes of decays, i.e. an O(10%) variation among the φ M f and φ Γ f , corresponding to a similar variation in the CPVINT asymmetries.The shifts in the asymmetries remain at this order when taking all of the DCS corrections to the widths into account.We therefore conclude that their inclusion in (88), (89) is not warranted for the interpretation of CPVINT data at SM sensitivity.

V. IMPLEMENTATION OF APPROXIMATE UNIVERSALITY
In this section, we discuss how to convert the general expressions for the time dependent decay widths and indirect CP asymmetries obtained in Section III B to the approximate universality parametrization, in the three classes of decays.For CF/DCS decays to K 0 X, K 0 X, we pay special attention to K induced effects at LHCb and Belle-II.

A. SCS decays
For SCS decays, the theoretical absorptive and dispersive CPV phases replace the final state dependent ones via the substitutions, in the expressions for the time dependent decay widths and CP asymmetries.For decays to CP eigenstates, they enter the expressions for the decay widths (54) (via Eq. ( 55) for c ± f ) and the CP asymmetry ∆Y f (65).For decays to non-CP eigenstates, they enter the expressions for the decay widths (66), (67) (via Eq. ( 69) for c ± f ) and the indirect CP asymmetries ∆Y f , ∆Y f (72).Note that the misalignments δφ f are dropped on the RHS of (144), as they are not calculable from first principles QCD.Moreover, while formally of O( ) in U -spin breaking relative to φ M,Γ 2 , they could, in principle, yield O(1) variations in φ M f and φ Γ f in the SM.In Section VI B we discuss a strategy for fits carried out once SM sensitivity is achieved, and final state dependent effects in φ M f , φ Γ f become accessible to experiment.The direct CPV (a d f ) and misalignment (δφ f ) contributions to the CPVINT asymmetries in (65), (72) are of same order, cf.(121).Therefore, consistency requires us to drop the a d f , a d f terms in the CPVINT asymmetries, if we neglect δφ f in (144).For example, for CP eigenstate final states, and in the approximate universality parametrization, (65) reduces to, and similarly for the non-CP eigenstates (the first line of each asymmetry in (72 ).However, we recall that in the average of ∆Y f over f = K + K − , π + π − , i.e.A Γ , the error incurred by dropping δφ f and a d f is of O( 2), cf.(127) (128).
B. CF/DCS decays to K ± X For CF/DCS decays to K ± X, substitute in the expressions for the decay widths (74) (via Eq. ( 76) for the coefficients c ± ), and the indirect CP asymmetries δc f (80).However, in contrast to the SCS decays, the misalignments are entirely negligible, cf.(130).
C. CF/DCS decays to K 0 X, K 0 X In CF/DCS decays to K 0 X, K 0 X, the final state dependent phases for f = π + π − X are replaced by the theoretical phases via the substitutions, in the widths (88), (89) (via Eq. ( 91) for the coefficients c ± f , c ± f ), and in the indirect CP asymmetries δc f , δc f (93).The sum of the last two terms in (147) equals the misalignment δφ f (139), up to negligible corrections lying an order of magnitude below our SM estimates of φ M,Γ 2 , cf. (137), (138),(143).
At LHCb, the bulk of observed K 0 /K 0 → π + π − decays take place within a time interval7 t τ S /3, while at Belle-II they can be detected over far longer time intervals8 , e.g.t O(10 τ S ).This has important consequences for the impact of K on the CP asymmetries, e.g. in D 0 → K S π + π − decays, which we discuss below.
The function F 0 is associated with direct CPV via integration over τ , and agrees with the expression obtained in [39].The functions F 1 and e −Γ S t are associated with the contributions of K and φ M,Γ 2 to the CPVINT asymmetries, respectively.In Fig. 1, we plot the three functions over a short time interval of relevance to LHCb, and a longer time interval of relevance to Belle-II.Over the entire time scale for observed K 0 's at LHCb, e.g.t 0.5τ S , the function F 1 undergoes a remarkable cancelation down to the few percent level, while e −Γ S t = O(1).Thus, at LHCb, the contributions of K to the CPVINT asymmetries are highly suppressed compared to those of φ M,Γ 2 (recall that φ M,Γ 2 ∼ I,R in the SM).The cancelation in F 1 at short times takes place between the contributions to CPVINT from K L − K S interference [δb f, f , δd f, f in (94)], and from the I term in φ M,Γ f (139) [via δc f, f in (93)].Thus, for simplicity, analyses of CPVINT in D 0 → K S,L π + π − decays at LHCb could omit a fit to the interference terms [∝ e −Γ K t τ in (88), (89)], if they substitute rather than (147).In contrast, over the longer K 0 decay time scales that can be explored at Belle-II, the cancelation in F 1 subsides, and K ultimately dominates the CPVINT asymmetries in the SM, cf.Fig. 1 (right).Thus, Belle-II CPVINT analyses must fit for K L − K S interference and employ the substitutions in (147), in order to extract φ M,Γ

2
. Finally, the function F 0 undergoes some cancelation at small time intervals, e.g.t τ S /3, leading to moderate suppression of direct CPV at LHCb.

VI. CURRENT STATUS AND PROJECTIONS
We perform two global analyses of the current experimental data, collected in Table I, in order to assess the current sensitivity to the phases φ M 2 and φ Γ 2 .(The x CP , y CP , ∆x, ∆y entries in Tables I, III correspond to K S π + π − ).We also report on future projections.

A. Superweak limit
Until recently, fits to measurements of indirect CPV were sensitive to values of φ 12 down to the 100 mrad level.This level of precision probed for large short-distance NP effects.In particular, the effects of weak phases in the subleading decay amplitudes could be safely neglected in the indirect CPV observables.In this limit, referred to as the superweak limit, a non-vanishing φ 12 would be entirely due to short-distance NP in M 12 , with the CPVINT phases satisfying For example, the expression for the SCS time dependent CP asymmetry in (65) would reduce to9 Thus, the phase φ M 2 (or φ 12 ) would be the only source of indirect CPV.Consequently, CPVMIX and CPVINT would be related as [4][5][6], or, equivalently, as where (155) is the superweak limit of (46).Superweak fits to the data are highly constrained, given that there is only one CPV parameter controlling all of indirect CPV.The second column in Table II contains the results of our fit to the mixing parameters with current data in the superweak framework.We see that sensitivity to φ M 2 is ≈ 22 mrad at 1σ, and ≈ 54 mrad at 95% probability, while sensitivity to φ 2 is ≈ 5 mrad at 1σ, and ≈ 11 mrad at 95% probability. 10Some superweak correlation plots are also shown in the first row of Fig. 2. The Heavy Flavor Averaging Group (HFLAV) [7] has obtained similar results, φ M 2 = −0.004± 0.016 (1σ), φ 2 = 0.001 ± 0.005 (1σ) .
Comparison with the SM ranges (114) implies that an order of magnitude window for NP remains, at 95% probability, in the CPVINT phases.

B. Approximate universality fits
It is encouraging that the 1σ error on φ 2 in the superweak fit (5 mrad), and the U -spin based SM estimates for φ M,Γ 2 , φ 12 in (114), ( 115) are only about a factor of two apart.However, this means that the approximate universality parametrization is advisable moving forward.Inspection of the relations between φ 2 and φ M,Γ 2 in (111), (112), reinforces this conclusion.Approximate universality fits are less constrained, given that they employ two CPV parameters rather than a single one to describe indirect CPV.Hopefully, this will be overcome in the high statistics LHCb and Belle-II precision era, and SM sensitivity in φ M,Γ 2 will be achieved.This possibility is assessed below.
We remark that an approximate universality fit for any two of the phases φ M 2 , φ Γ 2 , and φ 12 is equivalent to a (traditional) two-parameter fit for φ 2 and |q/p|, with translations provided by (45), (110)-(112).General formulae for the decay widths, given in terms of φ λ f and |q/p|, can be converted to approximate universality formulae which depend on φ 2 and |q/p|, via the substitutions These are analogous to the substitutions for φ M,Γ f in (144), ( 146), (147) , and (152), respectively.
We begin with a fit to the current data, cf.Table I, for the phases φ M 2 and φ Γ 2 .We implement the substitutions for φ M,Γ f given in (144), ( 146), (152), and employ the  expression for ∆Y f in (145).The K L − K S interference terms in the D → K S,L π + π − decay widths (88), (89) are ignored, as in the experimental analyses.As explained in Section V C, this does not affect the determination of φ M,Γ 2 at LHCb, provided that the substitution in (152) is employed.For the Belle D 0 → K S,L π + π − analysis [54], omission of K L − K S interference is not an issue, given its experimental precision.
The results of the approximate universality fit appear in the third column of Table II, and in the second row of correlation plots in Fig. 2. It is interesting to notice that the error on φ M 2 is about a factor of three smaller than the error on φ Γ 2 , and is similar to the corresponding superweak error.This can be traced, in part, to the observable A Γ = −∆Y f , for f = π + π − , K + K − .It has a relatively small experimental error, and it only depends on the product x 12 sin φ M 2 in the fit [compare (145), (154)].However, both φ 2 and |q/p| − 1 are determined with order of magnitude larger uncertainties in the approximate universality framework, due to the presence of a second CPV parameter, φ Γ 2 , in the fit.In the future, as SM sensitivity in CPVINT is approached, a modified strategy will be appropriate.As discussed in Section IV C 1, significant and non-universal Table II: Results of fits to the current and future D mixing data within the superweak and approximate universality frameworks, where the phases are defined in Eq. (108).
5.96 5.98 6 6.02 6.04 6.06 6.08 6.1 6.12 6.14 6.16It is interesting to point out that simultaneous knowledge of φ M,Γ 2 from CF/DCS decays, and of the direct CP asymmetries in the SCS decays could be used to determine the relative magnitudes and strong phases of the corresponding subleading SCS decay amplitudes in the SM, i.e. r f and δ f .This can be seen for CP eigenstate final states via (53) with φ f = γ, (65) with φ M f = φ M 2 + δφ f , and (121), and similarly for non-CP eigenstate final states.Thus, important information on the QCD anatomy of these decays could be obtained.
To illustrate the potential for probing the SM in the precision era, we use the (naïvely) estimated experimental sensitivities reported in Table III for the LHCb Phase II Upgrade era, for three decay modes: D 0 → K S,L π + π − , K + π − , and K + π − π + π − .We caution that scaling the errors on the individual measurements purely based on the expected statistics may be optimistic.The results of the fit are presented in the rightmost columns in Table II and in Figure 3 (including the SCS observable A Γ leads to marginal improvement in the sensitivity to φ M 2 in Phase II).They suggest that SM sensitivity to φ M,Γ 2 may be achievable, particularly if these phases lie on the high end of our U -spin based estimates.Moreover, additional input from Belle-II indirect CPV measurements at 50 ab −1 [68], e.g. for the decays D 0 → K S,L π + π − , K + π − , K + π − π 0 , and A Γ , may improve the sensitivity.

VII. DISCUSSION
In this paper we have developed the description of CP violation in D 0 −D 0 mixing in terms of the final state dependent dispersive and absorptive weak phases φ M f and φ Γ f .They govern CP violation in the interference between decays with and without dispersive mixing, and with and without absorptive mixing, respectively.The expressions for the time dependent decay widths and CP asymmetries undergo extensive simplifications compared to the familiar parametrization in terms of |q/p| and φ λ f (translations are provided), and become physically transparent.For instance, their dependences on the strong phases in the decay amplitudes, as well as the CP-even dispersive mixing phase π/2, are easily understood.This understanding extends to the strong phases of the subleading decay amplitudes, e.g.those responsible for direct CP violation in D 0 → K + K − , π + π − .An important consequence is that the time dependent CP asymmetries for decays to CP eigenstate final states, e.g.f = K + K − , π + π − , depend on φ M f (dispersive CP violation), but not on φ Γ f (absorptive CP violation).Conversely, the φ Γ f can only be probed in decays to non-CP eigenstate final states, e.g. the CF/DCS final states We have applied the dispersive/absorptive formalism to the three classes of decays which contribute to D 0 −D 0 mixing, (i) CF/DCS decays to K ± X, (ii) CF/DCS decays to K 0 X, K 0 X, and (iii) SCS decays (both CP eigenstate and non-CP eigenstate final states).Derivations and expressions have been provided for the time dependent decay widths and asymmetries in all three cases.Appendix A contains expressions for a selection of timeintegrated CP asymmetries, demonstrating that they can also be used to measure φ M,Γ f .The CF/DCS decays to K 0 X, K 0 X require special care due to the effects of CPV in K 0 − K 0 mixing.Moreover, their widths depend on two elapsed time intervals, the D and K decay times, following their respective production.
Measurements of the final state dependent phases φ M f and φ Γ f ultimately determine a pair of intrinsic mixing phases φ M 2 and φ Γ 2 , respectively, cf.(108).The latter are the arguments, in the complex mixing plane, of the total dispersive and absorptive mixing amplitudes M 12 and Γ 12 , relative to their dominant ∆U = 2 (U -spin) components, which are responsible for the neutral D meson mass and width differences.The intrinsic mixing analog of the final state dependent phenomenological phases, φ λ f , is similarly defined as the argument of q/p relative to the ∆U = 2 mixing amplitude.The U -spin decomposition of the dispersive and absorptive mixing amplitudes yields the SM estimates φ M 2 , φ Γ 2 = O(0.2%),cf. ( 113)-(115), with φ 2 of same order.
The intrinsic mixing phases are experimentally accessible due to approximate universality.In particular, we have shown that there is minimal uncontrolled final-state dependent pollution from the decay amplitudes in the measured phases φ M f , φ Γ f : • For the CF/DCS K ± X final states, e.g.K + π − , and in the SM and extensions with negligible new weak phases in these decays, the difference δφ f between φ M,Γ • For the SCS decays, e.g.f = K + K − , π + π − , there is uncontrolled final state dependent QCD penguin pollution.In the SM, and for extensions with CP-odd QCD penguins of same order, the misalignments satisfy δφ f /φ M,Γ Expressions for the time dependent decay widths in the approximate universality parametrization, i.e. in terms of φ M 2 , φ Γ 2 , have been discussed in detail for the three classes of decays, cf.Section V. Our results for the K 0 X final states are particularly noteworthy.On the time scale of sequential K 0 decays at LHCb (t 0.5 τ S ), the effect of kaon CP violation on the time dependent CP asymmetries (due to K L X − K S X interference, and an Im[ K ] component in φ M,Γ f ) undergoes a cancelation at the few percent level.Thus, to very good approximation, LHCb analyses of these modes can neglect the effects of kaon CP violation in measurements of φ M,Γ 2 from the time dependent CP asymmetries.In contrast, over the longer K 0 decay time scales that can be explored at Belle-II, the cancelation subsides, and K ultimately dominates the time dependent CP asymmetries.Thus, Belle-II analyses must fit for K L −K S interference effects, and account for Im[ K ] in the extraction of φ M,Γ 2 .
In the future, the values of φ M,Γ 2 obtained from the CF/DCS decays will allow a determination of the misalignments, δφ f , in the SCS decays.In combination with measurements of the SCS direct CP asymmetries, a d f , it will be possible to determine the anatomy of the QCD penguins in the SM, e.g. for f = K + K − , π + π − .In particular, taking the SM value γ for their weak phases, it will be possible to separately measure their relative magnitudes, and strong phases, thus potentially providing an important test of QCD dynamics, if lattice measurements of these quantities become available in the future.
Past fits to the mixing data were sensitive to values of φ 12 = arg[M 12 /Γ 12 ] = φ M 2 − φ Γ 2 down to the 100 mrad level.This level of precision probed for large short-distance new physics contributions.Thus, the effects of weak phases in the subleading decay amplitudes could be safely neglected in the indirect CPV observables.In this limit, referred to as superweak, the mixing phases satisfy φ 12 = φ M 2 , and φ Γ 2 = 0. We have carried out a fit to the current data set in this limit, yielding φ M 2 = (−0.5 ± 2.2)% at 1σ, consistent with the HFLAV fit result, and corresponding to an O(10) window for New Physics at 2σ.
The approximate universality fit is less constrained, given the description of indirect CP violation in terms of two phases, φ M 2 and φ Γ 2 , rather than just one.Interestingly, in this case, our errors for φ M 2 (≈ 29 mrad) are similar to the superweak fit result, and about a factor of three smaller than the errors for φ Γ 2 (≈ 99 mrad).This is due, in part, to the observable A Γ = −∆Y f (f = π + π − , K + K − ), which depends on φ M 2 but not on φ Γ 2 , and has a relatively small experimental error.The phenomenologically motivated phase φ 2 is a weighted sum over φ M 2 and φ Γ 2 , where the weights are equal to the leading CP averaged dispersive (∝ x 2  12 ) and absorptive (∝ y 2 12 ) mixing probabilities, respectively, cf.(111).The latter is nearly three times larger, according to current fits to the data, thus explaining why the error on φ 2 (≈ 72 mrad) is similar to the error on φ Γ 2 .The U -spin based estimates of φ M 2 and φ Γ 2 imply that probing the SM will require a precision of a few mrad or better for both phases.Given the large theoretical uncertainties, a null result as this sensitivity is approached would effectively close the window for new physics in charm indirect CP violation.Alternatively, the most likely origin for a significantly enhanced signal would be CP violating short distance new physics, yielding φ M 2 φ Γ 2 , with the latter given by its SM value.A second possibility, light CP violating new physics, would enter both the dispersive and absorptive mixing amplitudes via new D 0 decay modes, likely enhancing both φ M 2 and φ Γ 2 .This appears unlikely, given the upper bounds on exotic D 0 decay rates.For instance, for invisible D 0 decays, the upper bound on the branching ratio, Br inv < 9.4×10 −5 (90% CL) [10], constrains the invisible contribution to φ Γ 2 as δφ Γ 2 Br inv /θ 2 C ∼ 0.2%, i.e. the the upper bound lies at the SM level (before taking into account additional suppression due to the relative magnitudes of the interfering invisible decay amplitudes, and their weak and strong phase differences).Moreover, the upper bound on contributions from D 0 → K 0 + invisibles is about a factor of 30 smaller. 11 Finally, based on available LHCb Phase II projections for the decays D 0 → K S,L π + π − , K + π − , K + π − π + π − , and A Γ , we have estimated the precision that could be reached for φ M,Γ 2 in the upcoming high statistics charm era, using an approximate universality fit.Note that our results are intended to be illustrative, given that the LHCb phase II projections do not include systematic errors.The resulting 1σ errors for φ M 2 (≈ 1.2 mrad) and φ Γ 2 (≈ 1.7 mrad) suggest that sensitivity to φ M,Γ SM may be achievable, particularly if these phases lie on the high end of the U -spin based estimates.Measurements of φ M,Γ 2 could one day become available on the lattice.Comparison with their measured values would provide the ultimate precision test for the SM origin of CP violation in charm mixing.
We end with the time integrated CP asymmetries for the SCS final states f = π + π − , K + K − : dt(Γ D 0 (t)→f + Γ D0 (t)→ f ) , (A3) for which we obtain the expression where t is the average (acceptance dependent) decay time of the D 0 mesons in the experimental sample.The ratio t /τ D is very close to one at the B factories, and exceeds one by about 5% − 10% for both final states at LHCb [2].Recall that in the SM, for SCS decays, whereas the average of φ M f over f = K + K − , π + π − differs from φ M 2 by O( 2 ) in U -spin breaking, cf. ( 121), (124), (127).
The time integrated CP asymmetry difference ∆A CP = A CP,K + K − − A CP,π + π − [2] can be expressed in terms of φ M 2 and the direct CP asymmetries as, where δ K,π are the strong phase differences between the leading and subleading K + K − and π + π − decay amplitudes, respectively, a d K,π are the two direct CP asymmetries, and t π,K are the two average decay times.
and the latter is dominated by their QCD penguin contractions.Generically, both amplitudes are O(1) in SU (3) F breaking, and the ∆U = 0 amplitude is parametrically suppressed by O(λ b /θ C ). (Two exceptions are mentioned below).

Figure 2 :
Figure 2: P.d.f.'s for mixing parameters in the superweak (first row) and approximate universality scenarios, see text.Darker (lighter) regions correspond to 68% (95%) probability.Notice the order-of-magnitude difference in the scale of the rightmost plots.

Table III :
Estimated uncertainties on mixing parameters from CF/DCS decays in the LHCb Phase II Upgrade.Correlations from current results have been used where available.