Null tests from angular distributions in $D \to P_1 P_2 l^+l^-$, $l=e,\mu$ decays on and off peak

We systematically analyze the full angular distribution in $D \to P_1 P_2 l^+ l^-$ decays, where $P_{1,2}=\pi,K$, $l=e,\mu$. We identify several null tests of the standard model (SM). Notably, the angular coefficients $I_{5,6,7}$, driven by the leptons' axial-vector coupling $C_{10}^{(\prime)}$, vanish by means of a superior GIM-cancellation and are protected by parity invariance below the weak scale. CP-odd observables related to the angular coefficients $I_{5,6,8,9}$ allow to measure CP-asymmetries without $D$-tagging. The corresponding observables $A_{5,6,8,9}$ constitute null tests of the SM. Lepton universality in $|\Delta c| =|\Delta u|=1$ transitions can be tested by comparing $D \to P_1 P_2 \mu^+ \mu^-$ to $D \to P_1 P_2 e^+ e^-$ decays. Data for $P_1 P_2=\pi^+ \pi^-$ and $K^+ K^-$ on muon modes are available from LHCb and on electron modes from BESIII. Corresponding ratios of dimuon to dielectron branching fractions are at least about an order of magnitude away from probing the SM. In the future electron and muon measurements should be made available for the same cuts as corresponding ratios $R_{P_1 P_2}^D$ provide null tests of $e$-$\mu$-universality. We work out beyond-SM signals model-independently and in SM extensions with leptoquarks.


I. INTRODUCTION
Rare charm decays are notoriously challenging theoretically, yet offer singular insights into flavor in the up-quark sector [1].With standard model (SM) branching ratios of |∆c| = |∆u| = 1 modes in the 10 −7 −10 −6 (semileptonic) and 10 −6 −10 −4 (radiative) range, precision studies are feasible at the experiments LHCb [2], Belle II [3] and BESIII [4].In view of the substantial hadronic uncertainties there are three main avenues to probe for beyond the standard model (BSM) physics in charm: i) a measurement in an obvious excess of the SM such as the D → πµ + µ − branching ratio at high dilepton mass [5] -a window that can be closing soon [6], ii) extract the SM contribution from a SM-dominated mode and use SU (3) F , e.g., recently demonstrated for D → V γ, V = ρ, K * , φ and of BSM couplings, however, will again be subject to hadronic uncertainties.This paper is organized as follows: In section II we review the weak Lagrangian, SM values and constraints on |∆c| = |∆u| = 1 couplings.The D → P 1 P 2 l + l − angular distribution is given in section III.Phenomenological resonance contributions are discussed in section IV.BSM signals are worked out in section V, where we also discuss LNU-sensitive observables, probing BSM interactions which distinguish between electrons and muons.In section VI we conclude.Auxiliary information on D → P 1 P 2 l + l − matrix elements is given in the appendix.

II. WEAK LAGRANGIAN
We consider BSM effects in the semileptonic operators, Q 10 = (ūγ µ P L c) lγ µ γ 5 l , Q 10 = (ūγ µ P R c) lγ µ γ 5 l , in the effective Lagrangian i + i=9,10,S,P where G F is the Fermi constant, α e denotes the fine structure constant and V ij are CKM matrix elements.P L , P R denote left-and right-chiral projectors, respectively.
Here, the coefficient of the dipole operator Q 7 = mc e (ūσ µν P R c)F µν , where F µν denotes the electromagnetic field strength tensor, is also given for completeness.The effective coefficients C eff 7,9 equal C 7,9 up to matrix elements of 4-quark operators which relax the GIM-cancellation, thus being the dominant contribution [6,22] and inducing a q 2 -dependence, see [23].
In addition, all primed coefficients C i are negligible in the SM.Experimental constraints, available from the upper limit on the D + → π + µ + µ − branching ratio, and D 0 → ρ 0 γ are presently very weak, at least about two orders of magnitude away from the SM [6,24] |C ( ) see [6] for correlated constraints.Corresponding constraints on c → ue + e − processes are about a factor 2-4 (5 times for C T,T 5 ) weaker than the ones in (9) on dimuons.Constraints on LFV processes c → ue ± µ ∓ are 6-7 times (4 times for C ( ) S,P ) weaker than the dimuon constraints.To discuss LNU or LFV, Wilson coefficients and operators become lepton-flavor dependent.To avoid clutter, we refrain from showing lepton flavor superscripts throughout this paper.

III. FULL ANGULAR DISTRIBUTION
In section III A we discuss the full angular distribution for D → P 1 P 2 l + l − decays and identify SM null tests that exist thanks to the extreme GIM-suppression in charm.In section III B we give the angular distribution in the low hadronic recoil OPE, which defines a factorization-type framework at leading order in 1/m c .To estimate possible BSM signals, which involve SM-BSM interference, we need to estimate SM contributions to decay amplitudes as well.The phenomenological description of the dominant resonance-induced contributions is detailed in section IV.

A. General case
The D → P 1 P 2 l + l − angular distribution, with the angles θ l , θ P 1 , φ defined as in [25] taking into account footnote 2 of Ref. [26], can be written as where q 2 , p 2 denotes the invariant mass-squared of the dileptons, (P 1 P 2 )-subsystem, respectively, and θ l denotes the angle between the l − -momentum and the D-momentum in the dilepton center-ofmass system (cms), θ P 1 is the angle between the P 1 -momentum and the negative direction of flight of the D-meson in the (P 1 P 2 )-cms, and φ is the angle between the normals of the (P 1 P 2 )-plane and the (ll)-plane in the D rest frame.The angles are within the ranges P 1 is the meson that contains the quark emitted from the semileptonic weak ūcll vertex.For instance, P 1 = π + and P 1 = K + in the D 0 , D + -decays in (1).
The angular coefficients I i ≡ I i (q 2 , p 2 , cos θ P 1 ) are given in terms of transversity amplitudes 1 as The subscript 0, and ⊥ stands for longitudinal, parallel and perpendicular polarization, respectively.Here, L, R denotes the handedness of the lepton current.In the SM electromagneticallyinduced contributions dominate c → ul + l − transitions due to the GIM-mechanism (8).Hence, by inspecting the relative signs between the left-handed and the right-handed contributions in (13), it follows that I 5,6,7 constitute null tests, as they require axial-vector contributions to be non-vanishing.
One may wonder about backgrounds to I SM 5,6,7 = 0. Intermediate pseudo-scalar resonances D → P 1 P 2 η * → P 1 P 2 l + l − induce a contribution to pseudo-scalar operators Q P not included in (13).The impact can be read off from the D → V (→ P 1 P 2 )l + l − angular distribution [26]: Contributions from C P to I 5,6,7 require the presence of tensor operators.Similarly, lepton mass effects pose no challenge 1 No tensor and no (pseudo)-scalar operators included, and for vanishing lepton mass.
to the null tests, as finite m l contributions require the presence of scalar or tensor operators which are both negligible in the SM (8).Finite SM contributions to axial-vector couplings are expected to arise from higher order electromagnetic effects.For instance, a 2-loop diagram with an insertion of Q (q) 1,2 with two photons induces a contribution at the relative order α e /(4π), about permille level.We estimate contributions from electromagnetic operator mixing as C 10 < 0.01 C 9 [23,27,28], which is small, at most 10 −4 in the SM (8).As will be shown in section V A, order one BSM contributions are needed to generate finite angular coefficients up to few percent.Therefore, higher order effects are of no concern to the null tests I 5,6,7 within the accuracy that can be achieved in the foreseeable future, 3%(1%) at Run II (upgrade) on D 0 → π + π − µ + µ − asymmetries at LHCb [29].
We learn that angular analysis in charm is simpler than in B-decays because charm is dominated by resonances.
Integrating (10) over φ, cos θ l and both, respectively, yields the decay distributions The forward-backward asymmetry in the leptons, A FB ∝ I 6 can be obtained from asymmetric cos θ l integration The observables I 7 and I 5 can be obtained, for instance, as follows Methods to get angular coefficients for P-wave contributions are given in [25].
At the kinematic end point of zero hadronic recoil the following exact relations hold [13] The corresponding observables of the CP-conjugated D decays are given by I 1,2,3,4,7 → Ī1,2,3,4,7 and I 5,6,8,9 → − Ī5,6,8,9 , where Ī equals I with the weak phases flipped.In D-decays, θ l is the angle between the l − -momentum and D-momentum in the dilepton cms, θ P 1 is the angle between the P 1 -momentum and the negative D-momentum in the (P 1 P 2 )-cms, and φ the angle between the (P 1 P 2 )-and (ll)-planes.We keep the definition of P 1 from D decays for D decays.
The observables I 7,8,9 are (naive) T-odd and corresponding CP asymmetries are not suppressed by small strong phases.The observables I 5,6,8,9 are odd under the CP-transformation.Therefore, if distributions from (untagged) D 0 and D0 decays are averaged one measures a CP-asymmetry, Due to the smallness of V * cb V ub /(V * cs V us ) these constitute null tests of the SM.Note that time-dependent effects in angular observables [25] are suppressed by the small D 0 − D0 width difference [30].

B. OPE and factorization
At leading order low recoil OPE, long-and short-distance physics factorizes as follows [13] where the short-distance coefficients read As we are anticipating BSM contributions to semileptonic operators (2), (3) only2 we dropped the contributions from dipole operators, which enter as ∝ (m c m D /q 2 )C eff 7 for clarity.Full formulae can be seen in [13].We explicitly checked that contributions from dipole operators are negligible for the purpose of our analysis.
The transversity form factors F i , i = 0, ⊥, can be written as Here, λ = λ(m 2 D , q 2 , p 2 ) and λ p = λ(p 2 , m 2 P 1 , m 2 P 2 ), where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc).The D → P 1 P 2 transition form factors are defined as where the right-hand sides have to be multiplied by an isospin factor of 1/ √ 2 for every neutral pion in the final state and we tacitly suppressed the dependence on q 2 , p 2 and cos θ P 1 in the form factors. Here, Since the dipole operators in the SM are negligible and we do not consider BSM tensor operators, the dipole form factors (25) are not needed for our analysis.r ( ) does not contribute to D → P 1 P 2 l + l − decays for m l = 0 3 .The relevant non-resonant D → P 1 P 2 form factors w ± , h are available from HHχPT [20].Numerical input is given in the appendix.Note that HHχPT applies if the participating light mesons are sufficiently soft.We find that , where E π denotes the energy of any of the pions in the D-cms.The region above m 2 φ is kinematically closed for D → K + K − l + l − decays.In these decays E K − m K in the D-cms does not exceed 0.3 GeV for all q 2 , where E K denotes the energy of any of the kaons in the D-cms.Although formally they are limited to low hadronic recoil we use the HHχPT form factors in the full phase space also for D → π + π − l + l − in absence of other estimates.
Figure 1: Phase space and dominant resonances in q 2 and p 2 for D 0 → π + π − µ + µ − decays (left) and The OPE formally applies for q 2 = O(m 2 c ).This is approximately the region above the φ-peak in D → π + π − l + l − decays, and nowhere in D → K + K − l + l − .QCDF at least formally works for p 2 = O(Λ 2 ) and p 2 ∼ q 2 , that is, when the (P 1 P 2 )-system is light and energetic in the D-cms, see also [32].While QCDF therefore can be used in D 0 → π + π − l + l − for low q 2 , this region is mostly occupied by resonances.In D → K + K − l + l − with p 2 min ≈ 1 GeV 2 there is little room left.For p 2 = O(m 2 c ) the dilepton system is soft in the D-cms.A related discussion of phase space has been given in [33] for B → ππlν decays.Due to the lower value of the heavy quark mass the phase space in charm is much more compressed than in b-decays.

IV. RESONANCE CONTRIBUTIONS
Several resonances contribute to D → P 1 P 2 l + l − decays.First we consider resonances in the (P 1 P 2 )-subsystem, that is, in p 2 .Depending on the spin j = 0, 1, ... of the resonance, such contri-butions are termed S,P,...-wave, respectively.Due to the lower mass of the D-mesons relative to the B-ones, there are fewer resonances and ones with lower spin contributing in charm.Lowest lying resonances with sizable branching ratios into ππ are the ρ and scalars σ = f 0 (500) and f 0 (980).At spin 2 there is the f 2 (1270).For K + K − , it is essentially the φ, and for Kπ there is the K * (892), the scalars κ and K * 0 (1430) and the spin 2 resonance K * 2 (1430).We model the resonance structure in p 2 for D 0 → π + π − l + l − decays by the ρ-contribution, which is dominant at least in the wider vicinity of p 2 ≈ m 2 ρ .D → ρ form factors are taken from [34], see appendix.D-waves and higher are phase space suppressed relative to the ρ and contribute to small q 2 0.4 GeV 2 only.Further study including scalar contributions, which are rather wide and less known, is beyond the scope of this work, which aims at identifying null tests and illustrating the sensitivity to BSM physics.We stress, however, that since there is no S-wave contribution to I 3,6,9 [13] these angular coefficients are unaffected by scalars.In addition, the S-P interference terms in I 4,5,7,8 can be separated from the P-wave contribution by angular analysis, therefore scalars can be experimentally subtracted in these coefficients.
We model these contributions with a phenomenological Breit-Wigner shape for C 9 → C R 9 for vector and C P → C R P for pseudoscalar mesons [6, 8] where m M , Γ M denotes the mass and total width, respectively, of the resonance M = η ( ) , ρ 0 , ω, φ, and we used isospin to relate the ρ 0 to the ω.Corresponding transversity form factors are given in the appendix, eqs.( 45)- (48).LHCb [17] has provided branching ratios in q 2 -bins around the resonances ρ/ω and φ, where we added uncertainties in quadrature and neglected correlations.The resonance parameters in C R 9,P are in general p 2 -dependent.We assume that the dominant p 2 -dependence is taken care of by the ρ-lineshape specified in the appendix such that the a M are fixed by ( 27), (28) at p 2 ≈ m 2 ρ : ) and take the right-hand side from data [35] together with B(η → µ + µ − ) ∼ O(10 −7 ) [35,36].We obtain To implement the pseudo-scalar contributions we employed the D → ρl + l − distributions that can be inferred from [26].We note that fitting M = ρ 0 , ω, φ in the zero-width approximation [35] and using 2 × B(D 0 → ρ 0 ρ 0 ) × B(ρ 0 → µ + µ − ) for the ρ 0 , one obtains parameters consistent with (30), with a ω somewhat below the isospin prediction a ρ /3 as already noticed for D + → π + µ + µ − [6].The strong phases δ M remain undetermined by this and introduce theoretical uncertainties.
The situation in the D 0 → K + K − l + l − channel is different as the obvious resonance, the φ, is not produced through a significant form-factor type contribution in D 0 -decays.The small uū admixture in the φ should give approximately few percent of the corresponding ρ → ππ amplitude.Similarly, lowest lying mesons with larger uū content, f 2 (1270), a 2 (1320), decay with about 5% branching ratio to K K, which again is a correction.The dominant contribution is expected to originate from annihilation topologies D 0 → φ(→ K + K − )γ * , recently discussed in [10] for D → ρl + l − decays within QCDF.Here we continue following a phenomenological approach, as in [12], based on factorization and vector meson dominance, and use where V = ρ 0 , ω, and we neglect differences between the D → ρ 0 and D → ω form factors.The corresponding amplitude in D → π + π − l + l − decays, that is, when the ρ 0 which decays to π + π − is created at the weak vertex rather than through a form factor, is effectively included in our prescription with resonance parameters fixed by data -allowing for the extra amplitude would merely result in re-fitting a ππ φ and a ππ ρ4 .Specifically, for D 0 → K + K − l + l − decays we use C R 9 as in (26) with a φ = 0, and the transversity form factors F iφ given in the appendix.The φ-lineshape is parameterized by a Breit-Wigner distribution.To include the contribution from η → l + l − we use Note, the η is kinematically forbidden.We then obtain from ( 29) and the zero-width approximation branching ratio Table I: Branching ratios for D 0 → π + π − l + l − and D 0 → K + K − l + l − from data, LHCb [17] (l = µ) and BESIII [18] (l = e), our evaluation, resonant and non-resonant, and [12].Upper limits are at 90% CL. † Statistical and systematic uncertainties are added in quadrature.
In table I branching ratio data on D 0 → π + π − l + l − and D 0 → K + K − l + l − decays from LHCb [17] and BESIII [18] are shown, together with our evaluation for resonant and non-resonant branching ratios, and the predictions from [12] 5 .In figure 2 we show the differential branching ratio dB/dq 2 for δ ρ − δ φ = π (red solid curve) and δ ρ − δ φ = 0 (red dotted curve).The ρ/ω-φ interference matters in the regions around the resonances.The η ( ) contributions are subleading.The purely non-resonant -neither q 2 nor p 2 resonances are included -SM contribution (blue band) is much smaller than the resonance-induced distributions except for very low q 2 .This remains true with BSM couplings (long-dashed purple curve) as illustrated for a maximal scenario C BSM 9 = 1 (9).We learn that, unlike presently in D → πµ + µ − decays, in the branching ratio of D → ππµ + µ − decays and with non-resonant form factors (44) there is no room left to probe BSM physics in the high q 2 region above the φ.In figure 2 we also show the prediction by [12] (green dashed curve).The rise of the branching ratio at very low q 2 in [12] is due to the onset of bremsstrahlung, computed using an extrapolation of Low's theorem [37], an effect which will be more pronounced for electrons as lower values of q 2 can be accessed.We recall that the soft photon approximation holds for photon energies up to m 2 P /E P [38], P = π, K, which limits its controlled use to q 2 0.1 GeV 2 in D 0 → K + K − l + l − and to q 2 0.001 GeV 2 in D 0 → π + π − e + e − decays.As it is a small effect on the D 0 → P 1 P 2 µ + µ − branching ratios and, except for the difference in phase space a lepton universal one, we refrain from including this effect in our numerics.We comment on bremsstrahlung in the discussion of LNU in section V C. = 1 on the non-resonant distribution.The resonance curves are our evaluation for δ ρ − δ φ = π (red solid), as from SU (3) F , and δ ρ − δ φ = 0 (red dotted) to illustrate uncertainties related to strong phases, compared to the model [12] (green, dashed).The latter employs fixed δ ρ − δ φ = π and the relative sign between the ρ and the ω is as in (26), see footnote 5.

So far we discussed
the only one from (1) with a proper distribution at high q 2 above the φ.The latter decay is special as it is the only mode from (1) which only proceeds through the annihilation-type topology.On the other hand, the decays D + → K + K0 l + l − are expected to have a more pronounced non-resonant contribution in p 2 as the presumably leading resonance in K + K0 is a 2 (1320), with only a small branching ratio to K K.The rare, semileptonic 4-body D s decays are somewhere between the two D 0 -decays, with contributions from both topologies, however, with color-enhanced annihilation at q 2 m 2 φ and m 2 η ( ) .We stress that we employ such a phenomenological description only to obtain BSM signatures, worked out in the next section.The SM predictions, that is, specific observables being null tests, are independent of the resonance model.

V. BSM SIGNATURES
In this section we work out BSM signatures of SM null tests model-independently and in BSM scenarios with leptoquarks.For null tests related to the angular observables I 5−9 largest effects are expected from SM-BSM interference near the resonances ρ/ω and φ.The dependence on the semileptonic |∆c| = |∆u| = 1 coefficients can be taken from ( 21), (22).
In section V A and V B we study the angular null tests I 5,6,7 and CP asymmetries, respectively.
In section V C we discuss ratios of dimuon to dielectron branching ratios as a probe of LNU.LFV can be obtained by flipping the signs of the I 5,6,7 , see (21), (22).The latter also explain why I 5 and I 6 have similar BSM-sensitivity and why I 7 is different.As anticipated, the effects are largest where the SM contribution peaks, around the ρ/ω and the φ resonances.The shape between the resonances depends on their relative strong phase, shown here for δ ρ − δ φ = π.The effect of δ ρ − δ φ = 0 is a reflection of the φ-peak at the x-axes.Our findings for the magnitude of I 6 are consistent with [12] 6 .

B. CP asymmetries without tagging
The CP asymmetries corresponding to the CP-odd angular coefficients I k , k = 5, 6, 8, 9 are defined as [25] A k = 2 where Γ ave corresponds to the CP-averaged decay rate.The observables I 8 and I 9 can be obtained from the angular distribution (10), for instance, as follows 6 Note, θ l is defined in [12] with respect to the positively charged lepton, whereas we use the negatively charged one.
We define the integrated angular coefficients I 8 analogous to I 5,7 , (36), and I 9 analogous to I 6 , (35).From here we obtain the integrated CP asymmetries A k = ( I k − Īk )/Γ ave .
Numerical values for high q 2 , q 2 min = (1.1 GeV) 2 in BSM-benchmarks are given in table II.To obtain the ranges given we varied strong phases and explicitly verified that the sign of C 9 in the first and C 9 in the second case does not matter, in agreement with (22).In the analysis of the CP asymmetries in (26) we effectively take into account the CKM factors V * cd V ud and V * cs V us for the ρ/ω and φ, respectively.The SM predictions for A 8,9 SM at high q 2 are below the permille level, and zero for A 5,6 SM due to the GIM-mechanism, C SM 10 = 0. CP-asymmetries integrated over the full q 2 region are at most permille level in BSM models, and smaller in the SM.Table II: Ranges for the high q 2 , q 2 min = (1.1 GeV) 2 , integrated CP asymmetries A i for D 0 → π + π − µ + µ − decays for different BSM benchmarks, varying strong phases.

C. Testing lepton universality
LNU-ratios in semileptonic decays [13,39,40] with the same cuts in the dielectron and dimuon measurement provide yet another null test of the SM in charm as R D P 1 P 2 | SM 1. Phase space corrections of the order m 2 µ /m 2 c amount to percent level effects.Electromagnetic effects are another source of non-universality, and expected at order α em /(4π) × logarithms, parametrically suppressed [25,28,41].A detailed calculation is beyond the scope of this work.Within the SM we obtain for q 2 min = 4m 2 µ and q 2 max = (m D − m Beyond the SM, R D ππ can be modified significantly.Varying strong phases and Wilson coefficients C ( ) 9,10 one at a time within allowed ranges (9), we obtain R D ππ | BSM ∈ [0.85, 0.99] and R D KK | BSM ∈ [0.94, 0.97].The latter is barely distinguishable from (41), as well as R D ππ and R D KK in leptoquark models, e.g., [6,40].It is advantageous to consider the LNU-ratios in bins with a smaller SM contribution to increase the BSM sensitivity.For ππ, this is, for instance, the high q 2 region above the φ, q 2 min = (1.1 GeV) 2 , as in [17], and with the SM prediction (41) intact.Here, in this high q 2 bin, leptoquark effects are within R D ππ | high q 2 LQ ∈ [0.7, 4.4], consistent with related sizable SM deviations in D → πl + l − decays at high q 2 [40].Such sizable deviations from universality are possible for the scalar and vector SU (2) L -singlet and doublet representations S 1,2 , Ṽ1,2 , respectively, which escape kaon bounds because there is no coupling to quark doublets [6].The other leptoquark representations give SM-like values for R D P 1 P 2 .For D 0 → K + K − l + l − decays we investigate possibilities to enhance the BSM sensitivity by lowering q 2 max .This increases the sensitivity to lepton mass effects such that (41) does not hold anymore.We find, even when simultaneously increasing q 2 min , that leptoquark-induced LNU cannot be unambiguously distinguished from the SM in the KK mode.The long-distance dominance of the branching ratio even with BSM contributions is also manifest from figure 2. For instance, below the η, for q 2 max = (0.525GeV) 2 [17], we find that R D KK in leptoquark models is within the ballpark of the SM prediction, R D SM KK = 0.83 ± O(%).On the other hand, model-independently R D KK can be suppressed relative to the SM, R D KK | <η BSM ∈ [0.60, 0.87].While data on muons [17] and electrons [18] exist for D 0 → π + π − l + l − and D 0 → K + K − l + l − decays, see table I, unfortunately, this does not permit to compute the respective clean LNU-ratios (40) due to incompatible q 2 -cuts employed by the two experiments.In particular, BESIII included q 2 -regions not accessible with dimuons and vetoed the φ → e + e − region.We recommend to give dielectron results for q 2 values above the dimuon threshold to allow for a measurement of R D P 1 P 2 (40).Naive ratios of the branching ratio measurements [17,18] given in table I result in lower limits, whose respective SM predictions are, due to the different q 2 -cuts, subject to sizable hadronic uncertainties.Using the same cuts as in the BESIII analysis -none on the dielectron invariant mass squared except for excluding the region [0.935, 1.053] GeV [18] -we find in the model of [12] RD SM π + π − 0.9 and RD SM K + K − 0.1, about an order of magnitude away from the data.The smallness of the ratio RD SM K + K − follows from the bremsstrahlung enhancement for electrons.A similar effect is present in the ππ mode, however, here it is lifted by the contribution of the φ in the dimuon mode.
The main difference between our resonance model and [12] is, besides the use of on-peak data ( 27)- (29), the inclusion of bremsstrahlung effects at very low q 2 , subject to systematic uncertainties as briefly discussed in section IV.Lepton mass corrections in our model are small such that the main difference between electrons and muons is due to the vetoed φ in the denominator, RD SM π + π − ∼ 2, and RD SM K + K − ∼ 1.A measurement with identical cuts (40) would avoid this model-dependence.

VI. CONCLUSIONS
The SM angular distribution in semileptonic 4-body D-decays is considerably simpler than in B-decays because of long-distance dominance in charm.The latter implies P-conservation and equal chirality of the lepton currents.As a result, the angular coefficients I 5,6,7 are null tests of the SM.BSM-contributions to the axial-vector coupling, C ( ) 10 , can, on the other hand, induce rates at few percent level, see figure 3.
Rare semileptonic D 0 -decays are not self-tagging and benefit from the CP-asymmetries related to I 5,6,8,9 , which are CP-odd and do not require D-tagging.Due to the smallness of V * cb V ub /(V * cs V us ) corresponding CP-asymmetries A 5,6,8,9 constitute null tests of the SM.BSM-induced integrated asymmetries can reach few percent, see table II.
Ratios of branching fractions into muons and electrons (40) probe lepton universality in the up-sector and complement studies with B-decays.LNU-tests in charm are presently not very constraining as only upper limits on branching ratios of D → P 1 P 2 e + e − decays exist.We strongly encourage experimenters to provide in the future data based on the same kinematic cuts for muons and electrons, enabling more powerful SM tests.Leptonic P-invariance and suppression of SM CP violation holds in the whole (p 2 , q 2 )-phase space on and off resonance peaks.Therefore, there is no particular need for cutting on ππ around or outside the ρ, or ll around φ or ρ/ω and one can collect events from the whole phase space.Yet, experimental information on the otherwise SM-dominated branching ratios with on-resonance cuts assists tuning the hadronic model parameters.Note, near-resonance BSM signals in the angular observables I 5−9 are larger due to enhanced interference with the SM, as exploited in [12,42] and evident in figure 3. On the other hand, deviations from lepton universality in the ratios (40) are enhanced in regions where the SM-contribution is smaller, such as in the high q 2 region above the φ in D 0 → π + π − l + l − decays, where order one BSM effects are possible.LFV branching ratios B(D 0 → π + π − e ± µ ∓ ) and B(D 0 → K + K − e ± µ ∓ ) can reach 10 −7 and 10 −9 , respectively.
The bands correspond to (mass ± width)2 .The very wide scalar resonances f 0 (500) and f 0 (980) would fill everything below the f 2 in the ππ plot and are not shown.We use this prescription, factorization plus HHχPT form factors, for the BSM short-distance contributions from 4-fermion operators (2), (3) to estimate BSM signals in the whole phase space for both ππ and KK modes.In figure 1 the q 2 , p 2 -phase space for D 0 → π + π − l + l − (plot to the left) and D 0 → K + K − l + l − decays (plot to the right) is shown with dominant resonances.