Exploiting CP-asymmetries in rare charm decays

We analyze patterns from CP-violating new physics (NP) in hadronic and semileptonic rare charm $\vert \Delta c \vert=\vert \Delta u\vert=1$ transitions. Observation of direct CP-violation in hadronic decays, as in $\Delta A_{\text{CP}}$, provides opportunities for $c \to u \,\ell^+ \ell^-$, $\ell=e,\mu$ transitions, and vice versa. For the concrete case of flavorful, anomaly-free $Z^\prime$-models a NP-interpretation of $\Delta A_{\text{CP}}$ suggests measurable CP-asymmetries in semileptonic decays such as $D \to \pi\, \ell^+ \ell^-$ or $D \to \pi \pi\, \ell^+ \ell^-$. Conversely, an observation of CP-violation in $c \to u\, e^+ e^-$ or $c \to u\, \mu^+ \mu^-$ decays supports a NP--interpretation of $\Delta A_{\text{CP}}$. Flavorful $U(1)^\prime$-extensions provide explicit U--spin and isospin breaking which can be probed in patterns of hadronic decays of charm mesons. We work out signatures for CP-asymmetries in $D^0 \to \pi^+ \pi^-$, $D^0 \to K^+ K^-$ and $D^0 \to \pi^0 \pi^0$, $D^+ \to \pi^+ \pi^0$ decays, which can be probed in the future at LHCb and Belle II and provide further informative cross checks.

While this leaves room for NP, due to the sizable uncertainties of hadronic D-decays, Eqs. (1) and (3) provide no clear-cut sign of NP. On the other hand, ∆A CP as large as the permille level is non-trivial to achieve in concrete models of NP. Correlations with other observables in charm and the down-quark sector exist, which are subject to partly very strong flavor constraints. For recent works, see Refs. [3][4][5][6][7][8][9][10][11][12]. Turning this around, the study of patterns using different sectors can hence disfavor or support a particular ∆A CP interpretation, and vice versa. * Electronic address: rigo.bause@tu-dortmund.de † Electronic address: hector.gisbert@tu-dortmund.de ‡ Electronic address: marcel.golz@tu-dortmund.de § Electronic address: ghiller@physik.uni-dortmund.de In this work we pursue a global analysis of CPasymmetries in rare hadronic and semileptonic charm decays.
Our focus is on NP patterns induced by four-fermion operators. Links via dipole operators between hadronic and semileptonic CP-asymmetries in D → π + − decays have been pointed out by Ref. [13]. We work out predictions and correlations for anomalyfree Z -extensions of the SM with generation-dependent U (1) -charges, see Refs. [14][15][16][17][18][19][20] for recent phenomenological works. Flavorful charges can give rise to explicit isospin and U-spin breaking effects. It is our goal to work out corresponding experimental signatures for hadronic charm decays, exploiting yet another SM null test strategy in charm [21]. This paper is organized as follows: In Section II we briefly review CP-violation in hadronic D-decays, D-mixing and semileptonic c → u + − transitions. In Section III we analyze effects of anomaly-free U (1) -extensions with generation-dependent charges in hadronic 2-body D-decays and how D-mixing constraints can be evaded to address ∆A CP . Patterns among CP-asymmetries in D 0 → π + π − , D 0 → K + K − , D 0 → π 0 π 0 and D + → π + π 0 decays are worked out in Section IV. Correlations with CP-asymmetries in rare semileptonic decays are studied in Section V. We conclude in Section VI. Auxiliary information is given in several appendices.
A. Direct CP-violation in D 0 → π + π − , K + K − The single-Cabibbo-suppressed (SCS) D 0 (D 0 ) decay amplitudes A f (A f ) to CP-eigenstates f can be written as where η CP = ±1 is the CP-eigenvalue of f . The dominant SCS "tree" amplitude in the SM is denoted by A T f e ± i φ T f , and r f parametrizes the relative magnitude of all subleading amplitudes. Inserting Eqs. (4) into Eq. (2), in the limit of r f 1, yields requiring both strong (δ f ) and weak (φ f ) relative phases for a non-vanishing direct CP-asymmetry. Beyond the SM the SCS D 0 decay amplitude can be written as where the first term corresponds to the SM contribution with CKM-factors λ q = V * cq V uq made explicit, and the second term accounts for NP. Using CKM unitarity λ d + λ s + λ b = 0 and writing for the final states K + K − and π + π − in the subscripts f = K and f = π, respectively, one finds Here, the first term is the SCS contribution and the second one corresponds to "penguin" contributions with small Wilson coefficients which are strongly CKMsuppressed with respect to the SCS one by λ b /λ s,d . The last term A NP K(π) encodes NP contributions. Using Eqs. (4), (5) and (7), we obtain and and r π,K 1. The strong phases δ π,K are associated with the NP amplitudes. Since we are interested in maximal NP contributions, we employ in our numerical analysis sin δ π, K ∼ 1. Note, there is a priori no information on the sign of ∆r NP as it depends on products of strong and weak phases. The branching ratios of the D → f modes are dominated by their respective SM contributions. We can therefore extract (A s (d) Here we consider constraints from charm meson mixing. The D 0 -D 0 transition amplitude can be written as which can be parametrized in terms of the following physical quantities Here, x 12 and y 12 are CP-conserving, while φ 12 is a phase difference that results in CP-violation in mixing. A global fit from the HFLAV collaboration [2] results in In absence of a sufficiently controlled SM prediction of the mixing parameters, we require the NP contributions to saturate the current world averages (13), C. CP-violation in c → u + − CP-violation in semileptonic rare charm decays arises from complex-valued Wilson coefficients C i , C i in the effective Hamiltonian [18], i=9,10 with the operators Here, α e denotes the fine structure constant, G F is Fermi's constant and L = (1 − γ 5 )/2, R = (1 + γ 5 )/2 are chiral projectors. CP-violation has not been observed in semileptonic |∆c| = |∆u| = 1 decays yet. Available measurements for CP-asymmetries in rare semileptonic charm decays are at the level of few to O(10) % [22], which is close to possible NP effects [13,18,21]. Branching ratio and high-p T data imply the following constraints, barring cancellations [23,24] |C µµ ( ) stronger for muons than for electrons.

III. A FLAVORFUL Z IN CHARM
We work out NP-effects in charm from anomaly-free U (1) -extensions of the SM with fermion charges F ψi that depend on the generation, i = 1, 2, 3. Specifically, SM fermion multiplets plus possibly right-handed neutrinos ψ = Q, u, d, L, e, ν in representations of SU can be characterized, in that order, as Concrete models with F ψi -assignments that fulfill the anomaly-cancellation conditions and induce c → u flavor changing neutral currents (FCNCs) are given in TABLE I. Related models (models 1 to 8) have been studied previously in the context of semileptonic rare charm decays in Ref. [18], to which we refer for further details. The models in TABLE I satisfy   3 i=1 (F Qi − F Li + 2 F ui − F di − F ei ) = 0 and therefore avoid kinetic mixing at one-loop [25]. In Section III A we discuss couplings of the fermions to the Z -boson, which arises from the U (1) -group. We assume the Z to have a mass M Z of the electroweak scale or heavier. We discuss the induced c → u fourquark operators and Wilson coefficients in Section III B.
In Section III C we discuss how to bypass constraints from D 0 -D 0 mixing. We work out predictions for ∆A CP in Section III D.

A. Z -FCNCs
The Z -couplings relevant to charm FCNCs can be written as with = e, µ, τ . The flavor diagonal couplings g d,s L,R and g L,R are given as the U (1) -gauge coupling g 4 times the associated charge F ψ .
The |∆c| = |∆u| = 1 FCNC couplings g uc L,R are generated via rotations from the gauge to the mass basis, and are in general complex-valued. Four different unitary rotations exist in the quark sector, corresponding to the left-handed (LH) and right-handed (RH) ones both for up-and down-type quarks. The product of LH up-and down-type rotations gives the CKM-matrix. In order to evade the severe constraints in the kaon sector, we assume the CKM-matrix to predominantly stem from the LH up-type rotation, implying where λ CKM ∼ 0.2 denotes the Wolfenstein parameter and we used λ CKM 1. In contrast, the RH rotation is a priori unconstrained and induces where θ u is the up-charm mixing angle for the up-quark singlets, ∆F R = F u2 − F u1 and φ R the corresponding CP-phase.

B. Four-fermion operators and matching
Generation-dependent quark-couplings result in additional operators in the effective weak Hamiltonian beyond the ones considered usually, i.e. Ref. [26]. At the scale m b < µ < µ EWK , with the new operators where (V ± A) refers to the Dirac structures γ µ (1 ± γ 5 ), q = u, c, d, s, b and α, β are the color indices. The strength of these operators is given by their respective  I: Sample solutions of an anomaly-free U (1) -extension of the SM+3 νR with FQ 1 = FQ 2 . Models 2, 4 and 5 are taken from Ref. [18]. Models 9 and 10 feature FQ i = 0. In general, the ordering of generations is arbitrary due to permutation invariance. However, our analysis explicitly uses the ordering stated here, that is, the ith entry corresponds to the ith generation. Model 10µ is the same as model 10 with the smallest lepton-coupling to muons.
Wilson coefficients C i , C i which depend on both heavy masses and weak phases responsible for CP-violating phenomena. The Wilson coefficients induced by the Lagrangian (20) read Rare |∆c| = |∆u| = 1 decays are induced in the Zmodels by operators with coefficients proportional to g uc L or g uc R in Eq. (32). These couplings induce at second order D 0 -D 0 mixing (13), and are constrained as with X ∼ 20 for M Z in the TeV range [18]. This constraint on x 12 can be evaded if both g uc L and g uc R are present, for either g uc L ∼ Xg uc R or g uc L ∼ 1/Xg uc R . However, in these cases the CP-phases have to be aligned Arg(g uc L ) ∼ Arg(g uc R ) to fulfill Eq. (33). As kaon constraints force Arg(g uc L ) to be SM-like, CP-violating effects in charm become negligible. We therefore choose g uc L = 0, which can be achieved with ∆F L = 0. The models in TABLE I satisfy for this reason F Q1 = F Q2 . Consequently, we focus on FCNCs in the up-singlet sector (22), that is, g uc R = 0 and complex. If there is a single coupling only, the above mixing con- straint on x 12 becomes The even tighter constraint (14) for CP-violating couplings on x 12 sin φ 12 can be bypassed for Arg(g uc R ) = φ R around π/2 (or 3π/2), as the CP-phase of the mixing amplitude is twice the one of the |∆c| = |∆u| = 1 FCNC [26]. The contributions to ∆A CP become maximal while simultaneously mixing constraints are satisfied. This interplay of φ R versus the coupling g 4 /M Z ( TeV −1 ) for model 2 and fixed θ u = 1 · 10 −4 is illustrated in FIG. 1. The red (hatched) area corresponds to the D 0 -D 0 mixing constraints on the imaginary part x 12 sin φ 12 (absolute value x 12 ). Z -induced values of ∆A CP are shown in green. Indeed the region around φ R ∼ π/2 is viable and can induce ∆A NP CP ∼ 10 −3 .

D. Z -effects for ∆ACP
Taking into account the running from M Z to m c , details of which are given in Appendix B, we find that ∆A CP can be written as with where As explained in the previous Section III C, we analyze models with g uc L = 0 and Im(g uc R ) large. In Eq. (36) we use sin δ π,K sin φ R ∼ 1 and anticipated θ u 1. The parameters c K,π and d K,π depend on the chiral factors χ K,π at the charm scale, the LO QCD running functions r 1,2 (m c , M Z ) and the tree-level contributions a K,π , which are determined experimentally. Further details can be found in Appendices A-D. Numerical values of c K,π and d K,π for different Z masses are displayed in TA-BLE II.
start to be competitive with mixing constraints close to the non-perturbativity region (black region). This is particularly relevant for model 9 and 10, which exhibit large couplings to leptons. To evade the muon constraints and allow for slightly larger values of ∆A CP we also consider model 10µ, which is the same as model 10 with the lepton-charges ordered in such a way that the smallest ones are for muons, stressing the interplay between hadronic and leptonic sectors; model 10 can accommodate ∆A NP CP up to 1.5 · 10 −3 , while model 10µ can reach 1.8 · 10 −3 . FIG. 2 shows the stronger bound for each model, i.e., Eq. (39) for models 2, 5, 9 and 10µ (dashdotted) and Eq. (38) for model 10 (dotted). In FIGS. 1 and 2 we show benchmark points. They pass constraints from D-mixing and semi(-muonic) decays, while giving ∆A NP CP ∼ 10 −3 . The golden star corresponds to model 2 with ∆F R = 12 and The pink diamond corresponds to model 10µ with ∆F R = 19 and We learn that Z -models with charges as in TABLE I can provide concrete NP-interpretations of ∆A CP of the order of 10 −3 . D 0 -D 0 mixing provides upper limits on the achievable ∆A NP CP . To distinguish the different model scenarios we explore correlations of ∆A CP with other sectors, hadronic 2-body D-decays in Section IV and semileptonic c → u + − transitions in Section V.

IV. PATTERNS IN HADRONIC DECAYS
Z -models with non-universal charges F ψ can give rise to large flavor-breaking effects which could explicitly violate relations between hadronic charm decays [28][29][30][31]. We study signatures of Z -induced U-spin and isospin breaking in Section IV A and Section IV B, respectively. A CP in D 0 → π 0 π 0 is studied in Section IV C.
To quantify deviations from this relation we define 2 In the U-spin limit U tot break = 0.
2 For model 10(µ) we use instead 1 + A CP (D 0 →K + K − ) to avoid U tot break > 1. It is tacitly understood that K, Q 2 , d 2 and π, Q 1 , d 1indices in Eq. (44) and following need to be swapped in this case.
Using Eqs. (36), U tot break can be written as In TABLE III we give U tot break for models 2, 4, 5, 9 and 10(µ), for M Z = 6 TeV. The variation of U tot break with M Z in the range shown is within a few percent. Taking advantage of the smallness of the parameters d K,π relative to c K,π , we perform a Taylor expansion in Eq. (44) up to O(d K , d π ) to qualitatively understand how U-spin breaking in our models emerges. This leads to for F Q1 = F Q2 = 0 (models 2, 4 and 5), while for F Q1 = F Q2 = 0 (models 9 and 10(µ)) Eq. (44) simply becomes For models with F Q1 = F Q2 = 0 different sources of U-spin breaking exist. The second term in Eq. (45) accounts for effects originating from interference between the SM-amplitude and the F Q1,2 -charges. This contribution is responsible for 22 % U-spin breaking, which is of the same order of magnitude as the expected U-spin breaking uncertainty of the SM. In contrast, the last two terms in Eq. (45) are pure NP U-spin breaking effects. Eq. (45) can further be simplified with d K ≈ c K cπ d π due to χ π ≈ χ K , which holds numerically at the level of O(0.1 − 1) %. It follows that highlighting that pure NP U-spin breaking effects are 1.
-17 induced by which indicates how both the pion chiral enhancement and r 2 suppress U-spin breaking in these models. Therefore, values of F d2 − F d1 ∼ O(1) such as in model 5, induce U-spin breaking within the range expected within the SM 30 %. In model 4, F di = 0 and U NP break = 0, that is, U-spin breaking is SM-like. On the other hand, for F d2 − F d1 ∼ O(10) as in model 2, large U-spin breaking effects can arise and would be discernible with future sensitivities for A CP (K + K − ) and A CP (π + π − ) shown in TABLE IV. For models with F Q1 = F Q2 = 0 we obtain for the pure NP U-spin breaking from Eq. (46) which, unlike in Eq. (48), is unsuppressed. Models with F Q1 = F Q2 = 0 are therefore prime candidates for sizable NP U-spin breaking effects. Models 9 and 10(µ) have been constructed for this purpose. However, in model 9 F d2 = F d1 and U-spin breaking arises from d K = −d π only, and is SM-like. Note, the strong phases associated with NP are assumed to be similar, sin δ π sin δ K , and order one; violation of Eq. (42) can be suppressed or even further enhanced by U-spin breaking in the strong phases. While this is an uncertainty on the NP interpretation, Z -signals could even be more striking.
In FIGs. 3 and 4 we show the contributions of models 2, 5, 9 and 10(µ) to the individual CP-asymmetries A CP (K + K − ) and A CP (π + π − ) in blue, magenta, yellow and cyan, respectively. The U-spin limit is given by the The orange error ellipses illustrate the NP sensitivity of the projected uncertainties of A CP (K + K − ) and A CP (π + π − ) assuming no correlations. A future databased analysis which takes into account correlations between the individual asymmetries and ∆A CP can be expected to be more powerful. U-spin symmetry within the SM is broken at the level of 30 %. We find that flavorful Z -models can exceed this by far (model 10(µ)), or moderately (model 2), which makes the measurements of A CP (K + K − ) and A CP (π + π − ) smoking guns for NP, within reach of Belle II and LHCb with the projected sensitivities.
B. Isospin breaking patterns in D + → π + π 0 Isospin breaking arises in Z -models if F u1 = F d1 . In charm physics, the hadronic decay D + → π + π 0 represents a formidable candidate to study these effects, because the CP-asymmetry A CP (π + π 0 ), defined by with f ± = π ± π 0 is a clean SM null test [35]. Following the same procedure as in Section III D for ∆A NP CP we obtain, using θ u 1, with Here, a π denotes the tree-level contribution to D + → π + π 0 whose modulus has been fixed experimentally, see Appendix A for details. Numerical values of d π for different values of M Z are given in where Values of β π for M Z = 6 TeV and different Z -models can be seen in TABLE III. Since we have lost information about the signs of the leading SM decay amplitudes with which NP is interfering, we cannot predict the relative sign between the CP-asymmetries in Eq. (54) without relying on assumptions on the strong interaction. Note, unlike for A CP (K + K − ) and A CP (π + π − ), there is no SM flavor symmetry here at work. We find that model 9 and 10(µ) induce values near which for ∆A NP CP ∼ 10 −3 is within the projected sensitivity of Belle II with 50 ab −1 [34], see TABLE IV. Model 2, 4 and 5 induce A NP CP (π + π 0 ) 0.1 · ∆A NP CP ∼ 10 −4 , beyond the reach of current facilities. This behavior can be understood by expanding Eq. (55) in the d i up to O(d i ). For F Q1 = F Q2 = 0 (model 9 and 10(µ)), we find that β π scales with d π /d K ≈ −1.6 times a combination of charges (F d1 − F u1 )/F d2 (1 + ...) ∼ O(1) resulting in O(1) isospin breaking effects. For models with F Q1 = F Q2 = 0 instead a suppression factor d π /(c K − c π ) ≈ 0.03 exists from the chiral enhancement of the (V − A) × (V + A) operators, leading to β π of O(10 −2 − 10 −1 ).
We work out the CP-asymmetry for D 0 → π 0 π 0 decays because of its potential to diagnose patterns of NP [36]. In addition, the experimental prospect at Belle II for A CP (D 0 → π 0 π 0 ) is about a factor of two better than for A CP (D + → π + π 0 ), see TABLE IV. In the Z -models, A CP (D 0 → π 0 π 0 ) is obtained from Eqs. (52) and (53) after replacing subscripts π by π 0 with otherwise identical expressions. Therefore, with β π 0 given in TABLE III, hence with the limit saturated by model 9, and which is within the sensitivity of Belle II with 50 ab −1 [34], see TA-BLE IV. Furthermore, holds universally for all Z -models with F u1 = F d1 . Experimental tests of Eq. (59) can support a Zinterpretation, however, additional uncertainties from large, unknown strong phases exist, which can modify the relation. As discussed after Eq. (54), we cannot predict the relative sign between the CP-asymmetries (57), (59) without relying on input on the strong interaction.

V. SEMILEPTONIC DECAYS VS. ∆ACP
The dominant Wilson coefficients in c → u + − transitions are C ( ) 9/10 , defined in Eq. (15). In flavorful Zmodels [18] where g R = g 4 F ei and g L = g 4 F Li with in general different couplings for muons and electrons. As explained in Section III C, we analyze in this work Z -models with g uc L = 0 and Im(g uc R ) large. CP-asymmetries in the branching ratios are induced by interference of NP, here through g uc R , with C eff 9 , the effective coefficient of O 9 present in the SM, which is leptonuniversal, depends on the dilepton invariant mass and has sizable hadronic contributions and provides sizable strong phases. This interference term is sensitive to C 9 only. Angular analysis offers further opportunities. An interesting recent example for the latter is D 0 → π + π − µ + µ − decays [21,22,37]. Notably, the angular observables I 5,6,7 are GIM-protected in the SM and clean null tests [21]. In the Z -models under consideration, I 5,6 are induced by Re(C 9 · C * 10 ) and Im(C 10 · C eff * 9 ), whereas I 7 is induced by Re (C eff * 9 − C 9 ) · C * 10 ) . CPasymmetries in angular asymmetries, on the other hand, can stem from naïve T -odd observables and do not rely on strong phases (I 7,8,9 ). CP-odd ones (I 5,6,8,9 ) provide CP-asymmetries that can be measured without tagging, see Ref. [21] for details. A complete and detailed analysis of angular asymmetries in Z -models is beyond the scope of this work. What we do want to point out here is that a global analysis of angular and CP-asymmetries can probe both C 9 and C 10 for electrons, = e and muons, = µ separately, and therefore can distinguish different U (1) -charge assignments. Taking the imaginary part of Eq. (61) and employing Eq. (35), we obtain where Values of β 9/10 for = µ, e in (TeV) −2 are given in TA-BLE III. For ∆A NP CP ∼ 10 −3 we find Im(C 9/10 ) ∼ 0.03 (TeV) 2 · β 9/10 , consistent with C 9/10 = O(10 −2 ) for g uc L = 0, g uc R = 0 [18] and for β 9/10 = O(1/TeV 2 ) (models 2, 4 and 5). Models 9 and 10(µ) have sizable couplings to leptons, and in addition F Q1,2 = 0, which bring a factor In the plots to the right the correlation (62) between Im(C 9 ) (solid) and Im(C 10 ) (dashed) and |∆ACP| in the Z -models 2, 9, 10 and 10µ is made explicit. The golden star and pink diamond are benchmark points (40) (model 2) and (41) (model 10µ), respectively. The shaded areas correspond to the upper limits (18). See text for details. of c π,K /d π,K , see Eq. (63), score β 9/10 = O(10/TeV 2 ) and sizable C 9/10 = O(10 −1 ). As values of Im(C 9/10 ) O(10 −2 −10 −1 ) suffice to induce CP-asymmetries beyond the SM in semileptonic D-decays at the few percent level and above [13,18,21,23], all models can simultaneously lead to ∆A NP CP ∼ 10 −3 with NP patterns in c → u + − decays.
In FIG. 5 we show the imaginary part of Wilson coefficients with di-electrons (upper plots) and di-muons (lower plots) for different models as in Eq. (62). Plots to the left show lepton vector couplings versus lepton axial vector couplings, Im(C 9 ) vs. Im(C 10 ), respectively. Also given is Im(C 10 ) = −Im(C 9 ) (thin gray line). The lines corresponding to model 2, 4, and 5 end when the corresponding ∆A NP CP exceeds 2 · 10 −3 . Results are lepton non-universal as anticipated and sensitive to the lepton doublet and singlet charges. In the plots to the right the correlation (62) between Im(C 9 ) (solid) and Im(C 10 ) (dashed) and |∆A NP CP | in the Z -models 2, 9, 10 and 10µ is made explicit. Curves for models 4 and 5 are only in mild excess of those for model 2, or smaller, see TA-BLE III, and are not shown to avoid clutter.
As couplings to electrons and muons differ, lepton nonuniversality in charm [18,21,38] is induced, for example in the ratio of branching ratios of D → π µ + µ − and D → π e + e − using identical kinematic cuts, R D π . To better control SM backgrounds from intermediate resonances R = φ, η ( ) , ρ, . . ., via D → πR(→ + − ), interesting regions are for low (high) dilepton mass, below the η-mass (above the φ-mass), see Ref. [18] for details. We focus on the high mass region as it has fewer sensitivity to unknown strong phases from the resonances.
Using β 9/10 from TABLE III and Eq. (64) we find that all models yield order one deviations from the universality limit R D π = 1. Except for model 10µ, which has smaller couplings to muons by construction, all models can induce significant enhancements or suppressions from the SM. In particular, in the high mass region, for φ R = π/2 and varying strong resonance phases, see Ref. [18] allowing to signal NP.

VI. CONCLUSIONS
Patterns of observables are indispensable for pinning down an underlying NP-dynamics. We looked globally into hadronic and semileptonic charm decays and their respective CP-asymmetries. We find that there is strong benefit in doing so.
Most important, all flavorful, anomaly-free Z -models in TABLE I can simultaneously accommodate ∆A NP CP ∼ 10 −3 and induce measurable CP-asymmetries in the semileptonic c → u + − modes for = e or = µ above the SM. An observation of CP-violation in, for instance, D → π + − or D → ππ + − decays supports a NPinterpretation of ∆A CP , Eqs. (1) and (3) Additional cross checks are provided by CP-asymmetries in D 0 → π + π − , D 0 → K + K − , which probe for U-spin breaking NP, see FIGs. 3 and 4 for present data and future sensitivities, respectively. In addition, isospin violating NP can be observed with projected sensitivities at Belle II in D 0 → π 0 π 0 , D + → π + π 0 decays, whose CP-asymmetries can exceed ∆A CP , Eqs. (54) and (57). In the Z -models lepton non-universality is generic, and observable in the ratio of branching fractions of D → π µ + µ − and D → π e + e − decays, as briefly discussed in Section V. The Z -model 9 with order one enhancements over the universality limit, R D π 1, also induces A NP CP (π + π 0 ) ∼ A NP CP (π 0 π 0 ) 2 · ∆A NP CP . Zmodel 10µ with order one suppression of the universality limit, R D π < 1 exhibits sizable NP U-spin breaking A NP CP (π + π − ) A NP CP (K + K − ) ∼ ∆A CP . Checking correlations pins down models. Improved data and sensitivities from LHCb and Belle II are important in this program. We encourage and look forward to further CP-studies of rare semileptonic and hadronic charm decays.

Appendix B: Evolution of Wilson coefficients
The Wilson coefficients C ( ) 7,8,9,10 at the Z mass scale (32) are evolved to the charm mass scale at LO in α s . The requisite anomalous dimension matrix for the operators Q 7,8,9,10 can be inferred from Ref. [41]. We obtain where N C = 3 is the number of colors. Since QCD conserves parity, γ 0 F is identical for Q i and Q i . Using