Two is better than one: The U -spin-CP anomaly in charm

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I. INTRODUCTION
The LHCb collaboration measured the CP-asymmetry in D 0 → K + K − decays [1] A CP (K + K − ) = (6.8± 5.4 ± 1.6) where the first and second errors are statistical and systematic, respectively.Together with the previous LHCb measurement [2] ∆A CP = A CP (K = (−15.4± 2.9) • 10 −4 , LHCb performed a fit determining both direct CPasymmetries [1] a d K − K + = (7.7 ± 5.7) with a correlation ρ(a d K − K + , a d π − π + ) = 0.88, and leading to 3.8 σ evidence of CP-violation in D 0 → π + π − decays.This is puzzling for two reasons: First, the CPasymmetry a d π − π + is larger than |∆A CP |.Therefore, a standard model (SM) interpretation of the former needs even more dynamical enhancement of higher-order contributions h over the tree-level amplitude t to compensate the Cabibbo-Kobayashi-Maskawa (CKM) suppression , with data (3) pointing to h/t ∼ 2.Here an order one strong phase is assumed, and the enhancement is even bigger if the latter is suppressed, see App.A for details.Secondly, the new result implies a violation of U -spin symmetry, that is, violation of a d K − K + = −a d π − π + , at the level of 2.7σ [1].Two approximate symmetries of the SM are thus being challenged.While it is too early to draw firm conclusions given the significant hadronic uncertainties in D-decays, the recent data make new physics (NP) searches with rare charm decays just more interestingthis could be a hint for flavorful physics beyond the SM.This interplay of the CP-asymmetries is illustrated in Fig. 1.The small value of A CP (K + K − ), combined with ∆A CP (green-shaded area), implies a sizable CPasymmetry in π + π − , together with substantial U -spin breaking, which has also been pointed out in [4].Predictions in the U-spin limit (red dashed line) and 30% SM-like breaking (red-shaded cones) are indicated.The LHCb-fit (orange-shaded area) is two sigmas outside of this cone.The U-spin splitting in the D → π + π − and D → K + K − branching ratios is very well-known [5], and can be explained within the SM with 30% breaking, for instance, [6][7][8].Roughly speaking, because (1+1/3) 2 (1−1/3) 2 = 4 an assumed 33 % contribution to both decays of opposite sign is more than enough to explain the enhancement of B(D → K + K − )/B(D → π + π − ) 2.8, with or without considering the different phase space or factorizable flavor breaking from, e.g., decay constants and form factors [9].The splitting in the leading SM decay amplitudes W / e u / c x X 1 3 x 8 p s j + A P v 8 w e n R p c r < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a n x U t Q e B I D t + p X w U R + 2 l 0 l s F g q M = " W / e u / c x X 1 3 x 8 p s j + A P v 8 w e n R p c r < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a n x U t Q e B I D t + p X w U R + 2 l 0 l s F g q M = " W / e u / c x X 1 3 x 8 p s j + A P v 8 w e n R p c r < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a n x U t Q e B I D t + p X w U R + 2 l 0 l s F g q M = " W / e u / c x X 1 3 x 8 p s j + A P v 8 w e n R p c r < / l a t e x i t >

BM II
< l a t e x i t s h a 1 _ b a s e 6 4 = " J a j i 2   3) are shown with correlation at 68 % and 95 % CL [1] (orange-shaded).Also shown is the U-spin limit (red dashed line) together with 30% SM-like breaking (red cones), and the modified U-spin relation (4) (red dotted line).Thick straight lines relate to the new benchmark models of this work: BM I (magenta), BM II (teal), BM III (for G = 0) and BM IV (both brown), see Table I.
suggests a modified U-spin relation, see App.A, also indicated in Fig. 1 (dotted red line).Even though this effect slightly alleviates the anomaly, it still leaves the bulk of it unaltered, and the quest for models to explain it remains open.Enhanced chromomagnetic dipole operators such as from supersymmetric loops are flavor singlets, feature therefore SM-like symmetries, and are not able to account for the significant U-spin breaking.Models that generically break flavor beyond the SM are Z models with generation-dependent charges.Their impact on CPasymmetries in charm has been studied in [10].
In this work, we analyze the new data (3) within flavorful U (1) extensions of the SM.Interestingly, due to empirical constraints it turns out that the Z mass has to be below the weak scale to induce a permille level CP-asymmetry in charm.This is an important finding and we plan to derive it step-by-step in the paper: Z models and constraints from charm processes are discussed in Sec.II, pointing to a low mass Z of O(10) GeV.
In Sec.III we work out constraints applicable to this mass range from searches in dilepton and dijet signatures and quarkonium decays.In Sec.IV we analyze the high energy behavior, including Landau poles.In Sec.V we conclude.SM decay amplitudes and observables are parametrized in App. A. Details on the estimation of hadronic parameters are given in App.B. Charges for leptophobic, anomaly-free models with vanishing one-loop kinetic mixing are derived in App.C, and their higher order kinetic mixing is studied in App.D.

II. FLAVORFUL Z MODELS
We consider the SM extended by an abelian gauge group, with generation-dependent charges to fundamental fermions.We present the model set up in Sec.II A and work out constraints from charm in Sec.II B. In Sec.II C predictions for CP-asymmetries in D → π 0 π 0 and D + → π + π 0 are given.

A. Z model set-up
We denote the U (1) -charges of the SM fermions ψ = Q, U, D, L, E and and possibly also right-handed neutrinos ν R as F ψi , where i = 1, 2, 3 corresponds to the generation label.The charges are subject to anomaly cancellation conditions (C1)-(C6).The SM Higgs is uncharged under the U (1) to avoid mixing with the electroweak sector.The theory has a rescaling invariance with a constant k as F ψ → k F ψ , g 4 → g 4 /k, where g 4 denotes the U (1) -gauge coupling.It is therefore useful to consider rescaling invariant quantities such as F ψ g 4 , F ψ /F ψ or dg 4 /g 4 .Here we choose to show integer charges for notational convenience.
The Z induces D 0 → π + π − and D 0 → K + K − at tree level, as illustrated in Fig. 2. Contributions to the CP-asymmetries in (3) can be parametrized as [10] 1 where c π,K , d π,K are hadronic parameters (see App. B) and stands for the right-handed c → u FCNC coupling.As apparent, it requires non-universal charges F u2 = F u1 , as well as mixing between first and second generation righthanded up-quarks, described by the angle θ u .We treat θ u as a free parameter and adjust it accordingly.We assume that the corresponding angle in the down sector is sufficiently small to avoid kaon constraints.For the same reason, we consider only models with F Q1,2 = 0. 2 Explaining a sizable a d π − π + and near or even SM-like a d poses a challenge to BSM model building.To estimate the maximal reach we assumed in (5), which arises from interference between the SM and the Z amplitudes, that the relative strong and CP-phases are maximal.Having the latter near π/2 also evades constraints from CPviolation in D-mixing, see [10] for details.
The efficiency of the BSM model in explaining data (3) is determined by the hadronic parameters d K,π , c K,π [10], of which here only d π , d K matter.They include the leading order renormalization group (RG) running of Wilson coefficients in the weak effective theory from M Z to the charm mass scale, as well as the hadronic matrix elements.The latter are subject to sizable hadronic uncertainties [10][11][12][13], see App.B for details.The resulting CP-asymmetries serve rather as an indication of what is achievable in Z models.
To construct models, that is, identify suitable charge assignments, which account for the new LHCb results on CP-violation in charm (3), we follow similar lines as [10]: Our starting point is the cancellation of gauge anomalies, decoupling from kaons F Q1,2 = 0, inducing an c → u FCNC F u2 = F u1 and explicit U -spin breaking F d2 = F d1 .Absence of one-loop induced Z-Z -mixing is 1 The Z induces also annihilation-type contributions to the D 0 → π + π − and D 0 → K + K − amplitudes.Annihilation contributions are power-suppressed and require gluon exchange; however, the actual size of suppressions in D-decays is within wider ranges [11].We, therefore, refrain from including them in the numerical analysis, as we are focusing on the reach of models addressing data (3).In addition, note that contributions induced by Fu 1 do not break U-spin. 2 This is also the reason why we do not consider scalar singlet mediators contributing predominantly to D 0 → π + π − decays: They would couple to left-handed (and right-handed) down quarks, and after CKM-mixing induce ds-FCNCs, which are severely constrained.
preferred.In the following the models are further narrowed down.We discuss the theoretical and experimental constraints that arise and the corresponding selection criteria for charge patterns, which lead to the benchmark models Tab.I. Let us also ask about the mass scale one would generically expect to address (3) from Z -tree-level exchange.Very roughly, assuming order one couplings, g 2 4 ∆ FR F d1 ∼ 1, this gives a Z mass around where RG-effects in d π reduce this to the few TeV-range, see (5), App.B and [10] for details.In the next section, we learn that the constraints from D-mixing require suppressed couplings, and a significantly lighter Z than (7).

Charm CP-asymmetries
Using Eqs. ( 3) and ( 5) with F Q1,2 = 0, the ratio between F d2 and F d1 is fixed, resulting in a large hierarchy |F d2 | |F d1 |.The uncertainty in Eq. ( 8) is computed from the χ 2 function χ 2 (a d π − π + , a d K − K + ) with correlations included.This function can be expressed in terms of a d π − π + (or a d K − K + ) and the ratio a d K − K + /a d π − π + .We extract the uncertainty imposing ∆χ 2 = 1 and scanning a d π − π + (or a d K − K + ) within its 1σ range.The non-parabolic behavior results in asymmetric uncertainties.Similar results were obtained in Ref. [4].Note that renormalization group equation (RGE) effects cancel in the ratio d π /d K = −a K /a π −1.27±0.10[10], therefore Eq. ( 8) is independent of the Z mass, and only a parametric dependence with the quantities a π,K extracted from measured D 0 → π + π − and D 0 → K + K − branching ratios survive [10].Given the order of magnitude of a d K − K + , within the ballpark of SM estimations, we also consider models with F d2 = 0.

D-meson mixing constrains right-handed up-quark couplings as
Model   (34) by Z → ee, µµ searches for a light Z .While, in general, the ordering of generations is arbitrary due to permutation invariance, we use the ordering as stated here (the ith entry corresponds to the ith generation).Note, BM III with only the charges in the right-handed up sector swapped, Fu 1 = −F, Fu 2 = G, is equally viable; we refer to it as BM III-s.where the right-hand side of this equation depends mildly on the Z mass from RGE effects (it is a few percent for M Z ∈ [10, 10 4 ] GeV.)The limit (9) takes into account the recent update from HFLAV [14] where the new D-mixing experimental data from LHCb have been included [15].The bound for heavy Z masses is somewhat stronger than the previous one, 8 The available parameter space is presented in Fig. 3 in a way that is independent of the U (1) charge normalization.Shown are curves in g 4 F d1 /M Z versus ∆ F R /F d1 that explain a d π − π + , with uncertainties from data (3) which have been increased with an additional 30% of uncertainty to account for hadronic effects.Roughly, In addition, the excluded 95 % C.L. region by Dmixing ( 9) is shown in red.Thus, ∆ F R /F d1 1 via small mixing θ u is instrumental to generate sizable CP-asymmetries while simultaneously avoiding D-mixing constraints.We recall that ∆ F R (6) contains the mixing angle θ u which can be freely adjusted.We provide other (central) values of a d π − π + as black dashed lines.We learn that the minimal value of g 4 F d1 /M Z with current data is around ∼ 30 TeV −1 , suggesting a low, sub-electroweak Z -mass.

Charm dilepton and invisibles data
Concerning charming dilepton processes, the constraints from branching ratios of (semi-)muonic D-decays read [16,17] Here, we employed the recent LHCb measurement [16], which is a factor two stronger than the previous one.For = e, τ Drell-Yan constraints [18] are stronger than those from rare decays Using the relation (10) imposed by a d π − π + , displayed in Fig. 3, the dilepton bounds are satisfied if that is, couplings to the leptons should not be excessive compared to the ones to the quarks.We also work out limits from data on c → u plus missing energy.Missing energy can stem from right-handed neutrinos ν and/or vector-like dark BSM fermions χ charged under the U (1) only, with mass not exceeding m D /2 ≈ 0.9 GeV.We start with D 0 → π 0 + invisibles, whose branching ratio is constrained by BES III data [19] B(D 0 → π 0 inv.) < 2.1 • 10 −4 (90 % C.L.) .
Neglecting finite m χ corrections, the branching ratio can be written as [20,21] where A + = 9 • 10 −9 [20,21], and the sum over all flavors of the ν and the χ is understood.Following the previous analysis for the charged lepton constraints, we obtain Note the bound can be stronger if more than one kind or flavor contributes.The upper limit on D 0 → invisibles by Belle [22] B(D 0 → inv.) < 9.4 is in principal beneficial for massive invisibles (respecting m inv < m D /2), however does not constrain decays to fermions with purely vectorial coupling to the Z , such as g 4 F χ χγ µ χ.

Synopsis charm constraints and benchmarks
Charm constraints imply further selection criteria on the model charges: U -spin breaking and hierarchy F d1 F d2 (8), on lepton couplings ( 16)-( 18), and on invisible and neutrino couplings (21).All benchmarks BM I -IV given in Tab.I pass these constraints.Note that the BSM benchmarks from [10] are disfavored by the new data.BMs I and II are obtained by scanning integers.BMs III and IV are targeted towards more minimal models, with BM IV designed to have no one-loop kinetic mixing.BMs III and IV pass the additional constraints that arise from light Z searches discussed in the next Sec.III, while BMs I and II fail to do so.BM III-s, a variant of BM III with the charges between first and second generation up-type quark singlets swapped, F u1 = −F , F u2 = G is equally viable.It has a different phenomenology than the other BMs, as it does not couple necessarily directly to charm quarks.Indeed the main impact from D-mixing is that the mass of the Z is light, below the weak scale.Using Eq. ( 5) with F Q1 = 0, and the D-mixing bound we obtain a useful relation indicating a low NP mass scale, significantly lower than the naive estimate (7) due to the severe constraints from Eq. ( 9).The ratio of coupling over mass required to explain ∆A CP alone [10] is approximately a factor of a few smaller than the one from a d π − π + (3) due to the smaller value of the CP-asymmetry, and cooperating contributions from both KK and ππ asymmetries at least for modest U-spin breaking.The contribution of the flavorful Z to four-quark operators ū c q q, q = d, s is about 2−3 orders of magnitude smaller than the one induced in the SM by W -exchange.Therefore, the Z -contribution is irrelevant for the D → π + π − and D → K + K − branching ratios.We observe that the anomaly-free models feature U-spin breaking and also isospin breaking, see Table I.This implies signal in other 2-body charm CP-asymmetries, such as π + π 0 , and π 0 π 0 , see [10], and recently [23].We work out predictions in Sec.II C.
Flavorful Z models for ∆A CP also induce CPasymmetries in the D → π 0 π 0 and D → π + π 0 decays [10].They are of similar size and we recall that A CP (π + π 0 ) requires isospin violation to be finite.Using F d2 F d1 , we find that all CP-asymmetries involving pions are generically correlated as Since all d π 's are roughly of the same size, and noting that viable benchmarks obey |F u1 | < |F d1 | (see Tab.I and Sec.III), the Z -induced CP-asymmetries are at the level of ∆A CP , which is a permille.We also note the opposite sign of ∆A CP with respect to the others, hence A CP (π 0 π 0 ) and A CP (π + π 0 ) are positive in our models.Since F u1 /F d1 can have either sign the relative factor (1− F u1 /F d1 ) can be bigger or smaller than one.Concretely, it is 1 for BM III (with G = 0), 2 for the twisted BM III-s (with G = 0) and within 1 ∓ 1/ √ 2 for BM IV, depending on charges.For the BM IV solution given by Eq. (C30), we obtain a factor 1.7.

III. A FLAVORFUL Z OF THE ORDER 10
GEV?
Due to their strong impact on the viable mass range of the Z , we begin analyzing constraints from couplings to quarks in Sec.III A. The scale required to explain charm data (23) points to a light Z .Searches for U (1) extensions, including dark photons, B − L and B-models in dileptons provide severe constraints in the 1-100 GeV range, in particular in couplings to electrons and muons [24].Consequently, couplings to electrons and muons, or leptons altogether should be suppressed, much stronger than in ( 16)- (17).As such, BM I and II become excluded, and will not be considered any further.We quantify this and constraints in Sec.III B, also working out couplings of the leptons that are induced by kinetic mixing.This effect is larger in BM III as kinetic mixing arises here already at one loop.We analyse this and its impact in Sec.III C. In Sec.III D we work out branching ratios of the Z .

A. Mass constraints from q q
Constraints arise from dijets.For 10 GeV M Z 50 GeV, the strongest constraints are from CMS [25], and their dijet plus initial state radiation (ISR) search [26].Using their results, approximately g 4 F d1 0.5, together with the constraint from charm (3), (23) we arrive at the allowed mass range 10 GeV M Z 20 GeV .
Around and below 10 GeV, constraints depend on the benchmark models.Strong constraints from Υ → jj decays exist [27] around 10 GeV.They apply to BM IV due to its U obtain the allowed regions for BM IV with (C30) from Υ(1s)-decays respecting CP-data (3), (23) as On the other hand, BM III and BM III-s have no Z coupling to b's, and hence evade the Υ-limits.Charmonium decays provide additional constraints below 10 GeV on BM III and BM IV, but not on the "swapped" model BM III-s, as it does not couple to charm (for G = 0).BM III-s with mass below ( 26) can be probed in low energy hadronic processes involving first generation quarks, and invisibles.Due to (23) the Z below a GeV interacts feebly.A detailed assesment of constraints and opportunities for forward facilities [29] is below the scope of this work.
We work out the constraints on BM III and BM IV from , with contributions illustrated in Fig. 4. Following [30], we obtain for the branching ratios normalized to the ones into electrons, which depend on the ratio of the Z -induced amplitude A Z to the SM-photon one A γ , and the pion form factor F π .The left-hand side of Eq. ( 28) is defined in such a way that by switching off the NP-amplitude it equals one.We employ the values of the pion form factor |F π (m J/ψ )| = 0.056, |F π (m ψ )| = 0.04 from [31], which uses data on e + e − → π + π − and pion-electron scattering as input.As our models are electro-phob, we can safely assume that these data are not affected by the Z .The Z -width Γ(Z ) is obtained from (44).The constraints on BM III are shown in Fig. 5 (top), and for BM IV with (C30) (bottom).The grey line denotes the SM prediction corresponding to the ratios in (28) being equal to one.One notices that the NP-contribution decouples very slowly for larger Z masses, the reason is the growth of coupling with mass by means of (23).We also include the experimental uncertainties from the CPdata (3).The main difference between the models BM III and BM IV is stemming from the γ −Z -interference term which has opposite sign but similar size.We observe for BM III that Z -masses around [2.3, 2.8] GeV and [3.2, 3.5]  28) from 1 sigma ranges of J/ψ-data with |Fπ(m J/ψ )| = 0.056 (ψ decays with |Fπ(m ψ )| = 0.04.).Values of the pion form factor are from [31].Curves correspond to the predictions (righthand side of ( 28)) in BM III with F G (top) and BM IV with (C30) (bottom) using (23) including experimental uncertainties from (3).The SM prediction via photon-exchange is shown by the grey line.
GeV are consistent with both the J/Ψ-data (red horizontal bands) and the charm anomalies (red curves).The ranges obtained using ψ -data (blue horizontal bands), which have larger uncertainties, are [1.9, 3.3] GeV or [3.9, 4.5] GeV.BM IV can explain charm CP-data and J/ψ → ππ decays for masses within [2.8, 3.0] GeV, or starting from 3.6 GeV until the Υ-limit (27) kicks in, at about 7 GeV.The corresponding ranges from the ψ -data read [3.1, 3.5] GeV, or [4.6, 7] GeV.Assuming compatibility with both charmonia [32], Υ and charm CP data determines the Z -mass, depending on the model, as With these parameters our models provide an opportunity to resolve the longstanding tension between F π extracted from J/ψ-decays assuming the leading photonexchange contribution and QCD, e.g.[33].Note that the explanation of charmonia in BM III is a resonant effect, while in BM IV it is in the tails.
We recall that our results are subject to sizable uncertainties: the weak effective theory is challenged since a Z as light as a few GeV is close to the charm scale, in addition to the uncertainties from hadronic matrix elements.We also neglect G-parity violating contributions in the SM to the charmonia decays, e.g.[34], noting that the tension is significant, 7σ for the J/π and 1.8σ for the ψ .As such, we consider the phenomenology and formal constraints also in wider viable regions of M Z .For BM III, additional constraints from charmonia to taus or invisibles exist.Using a similar computation as in (28), we find that B(ψ → τ + τ − ) [32] gives the allowed mass ranges M Z 2.2 GeV or within 4.0 − 4.8 GeV, very close to the windows implied by the pion form factor (30).In view of the large uncertainties further analyses are promising and desirable.Furthermore, B(ψ → nothing) < 7 • 10 −4 [32] requires either M Z 0.7 GeV, which is in conflict with (30), or the BSM neutrino which couples to the Z to be heavier than half the ψ mass to forbid the decay kinematically.This suggests that this benchmark solution to the charm CP-data can be probed in charmonium decays.

B. Z → ee and µµ bounds
We work out constraints from Z → e + e − , µ + µ − decay searches.First, we study models where the Z couples directly to electrons or muons [35,36] such as BM I, II, III (with G = 0), and IV, The experimental search limits are given in terms of the mixing parameter ε, defined as L ε = −ε e J µ Z µ where J µ is the electromagnetic current of SM fermions.For the range of interest (40), the current experimental limit on ε both for electrons and muons is [24,37] Combining Eq. ( 31) with (32), one gets Inspecting Fig. 3, we observe that g 4 F d1 0.3 for M Z 10 GeV, which in combination with Eq. ( 33) leads to a strong suppression of electron and muon couplings over the down-type quark one, significantly stronger then the rare D-decay and Drell-Yan constraints ( 16) and (17).Note that Eq. ( 34 with G = 0, one can still induce a small coupling ε to L ε from Z − γ gauge-kinetic mixing which yields Note that gauge-kinetic mixing is in general not technically natural and cannot be switched off at more than one scale.It is also related to the gauge-kinetic mixing between the Z and the hypercharge field B before electroweak symmetry breaking via η = η B−Z cos θ W .This implies Z − Z mass mixing generating a correction δM Z to the unmixed tree-level mass of the Z boson, which affects the ρ parameter which for light Z 's is negative, but vanishes quadratically with ε.Here, ρ SM denotes the ρ parameters SM value, which is close to one.The global fit of electroweak precision parameters [32] suggests a relative NP contribution as Thus, the light Z contributes with the opposite sign.However, the correction is within 2σ of (38) if for a Z mass above 3 GeV (15 GeV).This has to be compatible with the constraint (32) at the M Z scale.As the running of ε is in general not technically natural, avoiding both constraints may results in highly non-trivial conditions of all U (1) charges for a light Z .In the following section, we discuss benchmark models that are feasible in this regard.

C. Viable scenarios
We consider the benchmarks BM III and BM IV, see Tab.I, which allow for sizable couplings to quarks but not leptons (34).Note that in these models the top quark has no direct coupling to the U (1) .This is also beneficial in suppressing contributions from kinetic mixing.
A) BM IV follows the constructions of App. As see Fig. 3, we roughly find a running between the Z and the electroweak scale, where in the last line we used M Z 3 GeV.Thus, the running can accommodate both ρ parameter and Z → constraints ( 32) and (39) if |ε(M Z )| ∼ O(10 −2 ).Using the four-loop running [38,39], we verified that the approximation (40) holds well for the lower end of (42).Larger values of g 4 (M Z )F may result in (43) increasing in the order of magnitude, eventually spoiling compatibility with the kinetic mixing constraints.
To summarize, bounds from ee and µµ can always be evaded: In BM IV, the hierarchy between quark and lepton charges can simply be chosen larger.In BM III, while kinetic mixing induces couplings to SM fermions that are uncharged before going to the gauge boson mass basis, the impact of this can be avoided by tuning with the contribution at the matching scale (40), at the level of 0.1.If experimental dilepton constraints on ε improve in the future, this can be accommodated with an increase of tuning at similar level.Of course, in a UV model this choice is not possible, therefore, models of type BM III can be just around the corner and show up in the next round of dilepton searches.

D. Z decay
In this section, we work out branching ratios of the light Z boson.The partial decay width of the Z to fermions ψ with mass m ψ < M Z /2 is given as [36] Γ  and Fν = 0. Branching ratios in all BMs differ perceptibly between the low and high M Z windows, ( 26) and (30), as the decays Z → b b, cc, τ + τ − are kinematically forbidden or suppressed in the few GeV range.Corrections to branching ratios from kinetic mixing are generically 10 −7 .
of the Z (26) decay channels into the SM electroweak gauge bosons or the Higgs are kinematically forbidden.For low M Z few × GeV, also the fermionic decays Z → b b, cc, τ + τ − can be either kinematically forbidden or severely phase space suppressed.In the limit m 2 ψ /M 2 Z 1 on the other hand the branching ratios are simply given by Numerical results for branching ratios in BM III, IIIs and IV are shown in Tab.II for different M Z .In BM III and III-s results are given in the limit G F where dimuon bounds are avoided.For M Z = 20, 10 and 3 GeV using (23) in BM III we obtain the width Γ(Z ) = i Γ(Z → ψ i ψi )θ(M Z − 2m ψi ) = 1.8, 0.2 and 4 • 10 −3 GeV, respectively.Very similar values for Γ(Z ) are found in BM III-s and IV.We note that for the lower masses Eq. ( 44) is not accurate as hadronic final states should rather be taken into account.In BM IV, results depend on the charge assignments F u,d,e,ν .However, dilepton bounds suggest suppression of lepton couplings (34).When setting |F e |, |F ν | |F u,d | we asymptotically approach a leptophobic model, with decays only to b, c and jets, i.e., light quarks.We also provide branching ratios for the concrete scenario (C30) in Tab.II.All benchmark models lead to a promptly decaying Z .Another possibility to suppress branching ratios to quarks and charged leptons is In this case the Z boson decays mostly invisibly to right-handed neutrinos.The same effect can also be achieved in all BMs by adding a light and dark vector-like BSM fermion χ with U (1) charge F χ .Due to its vector-like nature χ does not contribute to any gauge anomalies and it can have a simple Dirac mass term.Assuming m χ < 2M Z as well as |F χ | |F ψ | for ψ = Q, u, d, L, e, ν the Z will decay predominantly invisibly as Z → χ χ, see (45).For a heavy Z this possibility has already been explored in the context of the B-anomalies [36].The Z can be radiated off quarks, which in the above scenario leads to characteristic signatures such as hadrons in association with invisibles, i.e. missing energy, see Fig. 6.Unfortunately, to our knowledge in the mass range of our interest there is no experimental analysis for this process available.However, an invisibly decaying Z radiated off final state hadrons would be the smoking gun signature of this scenario at e + e − machines potentially giving rise to tight bounds on F ν,χ .This is in contrast to bounds from existing searches for e + e − → γ ISR + (Z → χ χ) [40][41][42] where cross sections are suppressed by tiny factors ε 2 or F 2 e even if B(Z → χ χ) 100%.However, note that F ν,χ cannot be arbitrarily large, as by adding additional U (1) -charged matter we always increase the RG growth of g 4 which finally might give rise to a low energy Landau pole, see Sec.IV for details.There are also constraints (21) on the charges from rare decays for fermions lighter than m D /2.A q q-pair is produced via an s-channel photon and radiates of a Z invisibly decaying to χ χ or ν ν, leading to a final state containing hadrons in association with missing energy.
We comment on corrections to the Z decay rate to SM fermions from kinetic mixing.As discussed in III B the Z thus couples to the electromagnetic current J µ via L ε = −εeJ µ Z µ , where the kinetic mixing parameter ε is defined in (35), (36).The corresponding partial decay width can be obtained from (45) by replacing g 4 → ε e and the U (1) charges with electric ones, F ψ L,R → q ψ .Experimentally, kinetic mixing is constrained by (32) to

V. CONCLUSIONS
The recent data on charm CP-violation (1), taken at face value, together with the measurement of ∆A CP (2), require to accept a huge amount of U -spin breaking, or NP below the weak scale.Explaining Eq. ( 3) indeed poses a challenge to model building, given the low NP scale and the severe constraints from rare decays, mixing, and searches for BSM bosons in dilepton and dijets channels.We obtain viable explanations of data (3) from a light Z -boson, M Z < few × 10 GeV, with novel, characteristic patterns in couplings to fermions: successful model benchmarks, see Tab.I, accidentally couple only to righthanded fermions and are leptophobic (BM IV) or do not have U (1) -charges to electrons or muons (BM III).The latter is subject to kinetic mixing at one loop, which requires a non-excessive tuning of the mixing parameter at the level of ten percent.The former is a novel benchmark using a Diophantine construction to maintain anomaly freedom and the absence of one-loop kinetic mixing, since it couples mostly to quarks without introducing new matter fields, derived in App. C. Models do not couple to top quarks allowing for the top Yukawa to be written down directly.BM III allows also the bottom Yukawa at tree level.We stress that we are not addressing the origin of flavor, which is beyond the scope of this work.Models can also include a dark sector, which can significantly speed up the running such that low scale Landau poles arise which point to a UVcompletion as low as a TeV, see Fig. 7.However, in view of the "the lighter, the safer"-rule a sufficiently light Z , with details depending on the dark sector, can avoid this.Model frameworks BM III and BM IV are unique as they are minimal models passing the very many theoretical and experimental constraints.Together with viable variants obtained by swapping charges within one species, such as BM III-s, they are indeed the only, minimal options, inducing CP-violation in the cR u R current and an enhanced dR d R current, but it cannot be ruled out that further, highly tuned scenarios with cross talk between species may be constructed.Another intriguing feature of the models is that they can simultaneously explain the charm CP-data (3) and the J/ψ → π + π − , ψ → π + π − branching ratios for a Z around ∼ 3 GeV (BM III) or ∼ (5 − 7) GeV (BM IV), see Sec.III A, providing a NPexplanation to the long-standing pion form factor puzzle.Models can be searched for in low mass dijets along the lines of [25] or Υ and J/ψ or ψ -decays and related dark photon searches.The dominant Z branching ratios are given in Tab.II.Signatures include enhanced production in π + π − , or DD, and τ τ in BM III.If dark fermions are also present signatures as in Fig. 6 with hadrons and invisibles are promising smoking guns.We stress that BM III-s, a variant of BM III with the charges between first and second generation up-type quark singlets flipped, F u1 = −F , F u2 = G is equally viable.It has a different phenomenology than the other benchmark models, as it does couple essentially to first generation quarks, the τ , and neutrinos, hence evades charmonium limits, and could be as light as O(GeV) or possibly below.All viable models further predict isospin violation and pattern in the CP-asymmetries in hadronic charm decays, see (25), in addition to A CP (D → K S K S ), which also requires U-spin breaking [11].Models are tightly constrained by D-mixing and Z -searches into electrons and muons.They can hence signal NP in the next round of data.
In the U -spin limit the phase and reduced amplitudes are universal, such as t = t s = t and so on.The observed U -spin breaking in the branching ratios B(D → K + K − )/B(D → π + π − ) 2.8 can be explained with a flavor-dependent correction δt, as t s = t+δt, t = t−δt at the nominal level of flavor breaking, δt/ t 30% [7].This U -spin breaking in the branching ratios and correspondingly between the leading contributors to the amplitudes t, t s implies a shift in the CP-asymmetries e.g.(A3) and leads to the modified U -spin relation (4).= 1, that is the naïve factorization approach.Here, dπ crosses zero around 40 GeV.Blue lines account for a deviation of 30% from the naïve factorization limit, that is B π + π − 9,10 = 1 ± 0.3.The green line represents the large-NC limit.
for m b ≤ M Z ≤ m t .For the computation of the D 0 → K + K − and D 0 → π + π − hadronic matrix elements we employ factorization of currents, (B9) = P + (q 1 Γ 1 q 2 ) |0 P − (q 3 Γ 2 q 4 ) D 0 B P + P − i , with P = π, K, Q i = (q 1 Γ 1 q 2 ) (q 3 Γ 2 q 4 ) is a 4-quark operator and Γ 1,2 represent Dirac and color structures while q j denote quarks.The factor B P + P − i parametrizes the deviation of the true hadronic matrix element from its naïve approximation, B P + P − i | naïve = 1.Including these effects the CP-asymmetries become Setting F Q1,2,3 = 0 we avoid kaon constraints and for the BMs III and IV couplings to the top, and arrive at the simple condition and therefore from Eq. (C8) to Let us work on the remaining ACCs, Eqs.(C5) and (C6).The non-linear behavior of these equations makes it challenging to solve them.We have already some information about the trace of these matrices, they are zero (C9).To solve the problem mathematical relations between F A , F 2 A and F 3 A would be helpful.Noting that for 3 × 3 matrices holds and det(F A ) = F A1 F A2 F A3 it follows from F A = 0 that F 3 A vanishes if one charge vanishes, for instance, F A3 = 0; then Eq. (C6) is fulfilled.In general, the charge matrices F A can then be written as where F A are integers.Note that the order of (+1, −1, 0) can be changed for each species A independently.It remains to solve Eq. (C5), which now simply reduces to We can start exploring this equation by setting F e = F L .This solution is motivated from the phenomenological point of view because we need small values of lepton charges as well as being disconnected from quark charges.However in this case the system has only the trivial solution F u = F d = 0 because F 2 d = 2 F 2 u can not be fulfilled for integers.Let us simplify (C12) by setting F L = 0 which is motivated by the strong constraints on lepton couplings.We obtain (b) Any solution (F d , F e , F u ) to Eq. (C13) can be parametrised by integers (p, q, r) F e = p 2 − 2 r 2 − q 2 − 2 p q + 4 q r, (C15) The first non-trivial solutions (F d , F e , F u ) are found to be (F d , F e , F u ) = (1, 1, 1), (7,1,5), (17,7,13), and so on, and integer multiples thereof.Note that each term can also have either sign, and that permutations between F d and F e are permitted.
Although Eq. (C13) gets fully solved by Eqs.(C14)-(C16), in the following, we illustrate a more practical approach that allows us to run directly into those solutions that accommodate the constraint given by Eq. ( 34).Let us rewrite (C13) as with |F = (F u , F d ) and For illustration, we start with the trivial solution (F d , F e , F u ) = (1, 1, 1).Using Eq. (C18), we obtain with Now, the question we should ask is: are there more solutions such as F e F u , F d ?One possibility is if a transformation by a 2 × 2 matrix with integer entries U , leads invariant Eq. (C20) so that we can generate recursively solutions which could get enlarged while keeping F e fixed to 1 for this particular case.This matrix needs to satisfy We learn that in order to avoid electron and muon constraints of Eq. ( 34), we need solutions |F i with i ≥ 4.
Note that we could have also used a different setup for η 11 , ... to get different solutions.In addition, we can also find more solutions by choosing another initial integer triple (F d , F u , F e ).

Appendix D: Kinetic mixing
In this appendix we discuss the naturalness of the gaugekinetic mixing, occurring due to the parameter η as in (35) between the Z and the photon, or equivalently the Z and hypercharge gauge boson before electroweak symmetry breaking.Ideally, one would like the RG evolution of this parameter to be technically natural, which means dη d ln µ ∝ η . (D1) This would imply that kinetic mixing can naturally be switched off at all scales, or made to remain arbitrarily small.
For theories with the charge configuration (C11), as well as the Higgs not carrying a U (1) charge, kinetic mixing is natural at one-loop order.If only gauge contributions are taken into account, the naturalness remains intact even at higher loops, which is the result of the symmetry implied by (C11).We have verified that this is the case until four loops, using the results of [38,39].
Starting at two-loop order, Yukawa interactions violate the naturalness (D1), unless they retain certain flavor textures.If the top quark does not carry U (1) charge as in BM IV, the naturalness-violating terms do not feature the top Yukawa coupling at two loops, and are either suppressed by the other, much smaller Yukawas and/or loop factors.Thus, these terms are negligible for the running of η, which becomes effectively natural.
However, the naturalness is broken at one loop below the scale where the first field carrying U (1) charge is integrated out, i.e. the bottom quark in BM IV.
limit < l a t e x i t s h a 1 _ b a s e 6 4 = " t m 2 g j Y W r c S 4 h 1 r C E 6 c t b 5 T H s x 1 8 = " > A A A C D H i c b V D L S s N A F J 3 U V 6 2 v a J d u B o v g x p K I Y J c F N y 4 r m F Z o Q 5 l M p + 3 Q m U m Y u R F D 6 C / 4 C 2 5 1 7 0 7 c + g 9 u / R K n b R b a e u D C 4 Z x 7 O Z c T J Y I b 8 L w v p 7 S 2 v r G 5 V d 6 u 7 O z u 7 R + 4 h 0 d t E 6 e a s o D G I t b 3 E TF M c M U C 4 C D Y f a I Z k Z F g n W h y P f M 7 D 0 w b H q s 7 y B I W S j J S f M g p A S v 1 3 W r Q A / Y I + b l J u M K C S w 7 T v l v z 6 t 4 c e J X 4 B a m h A q 2 + + 9 0 b x D S V T A E V x J i u 7 y U Q 5 k Q D p 4 J N K 7 3 U s I T Q C R m x r q W K S G b C f P 7 8 F J 9 a Z Y C H s b a j A M / V 3 x c 5 k c Z k M r K b k s D Y L H s z 8 V 8 v k k v J M G y E O V d J C k z R R f A w F R h i P G s G D 7 h m F E R m C a G a 2 9 8 x H R N N K N j + K r Y U f 7 m C V d K + q P t e 3 b + 9 r D U b R T 1 l d I x O 0 B n y 0 R V q o h v U Q g G i K E P P 6 A W 9 O k / O m / P u f C x W S 0 5 x U 0 V / 4 H z + A K w Y m 5 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t m 2 g j Y W r c S 4 h 1 r C E 6 c t b 5 T H s x 1 8 = " > A A A C D H i c b V D L S s N A F J 3 U V 6 2 v a J d u B o v g x p K I Y J c F N y 4 r m F Z o Q 5 l M p + 3 Q m U m Y u R F D 6 C / 4 C 2 5 1 7 0 7 c + g 9 u / R K n b R b a e u D C 4 Z x 7 O Z c T J Y I b 8 L w v p 7 S 2 v r G 5 V d 6 u 7 O z u 7 R + 4 h 0 d t E 6 e a s o D G I t b 3 E T F M c M U C 4 C D Y f a I Z k Z F g n W h y P f M 7 D 0 w b H q s 7 y B I W S j J S f M g p A S v 1 3 W r Q A / Y I + b l J u M K C S w 7 T v l v z 6 t 4 c e J X 4 B a m h A q 2 + + 9 0 b x D S V T A E V x J i u 7 y U Q 5 k Q D p 4 J N K 7 3 U s I T Q C R m x r q W K S G b C f P 7 8 F J 9 a Z Y C H s b a j A M / V 3 x c 5 k c Z k M r K b k s D Y L H s z 8 V 8 v k k v J M G y E O V d J C k z R R f A w F R h i P G s G D 7 h m F E R m C a G a 2 9 8 x H R N N K N j + K r Y U f 7 m C V d K + q P t e 3 b + 9 r D U b R T 1 l d I x O 0 B n y 0 R V q o h v U Q g G iK E P P 6 A W 9 O k / O m / P u f C x W S 0 5 x U 0 V / 4 H z + A K w Y m 5 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t m 2 g j Y W r c S 4 h 1 r C E 6 c t b 5 T H s x 1 8 = " > A A A C D H i c b V D L S s N A F J 3 U V 6 2 v a J d u B o v g x p K I Y J c F N y 4 r m F Z o Q 5 l M p + 3 Q m U m Y u R F D 6 C / 4 C 2 5 1 7 0 7 c + g 9 u / R K n b R b a e u D C 4 Z x 7 O Z c T J Y I b 8 L w v p 7 S 2 v r G 5 V d 6 u 7 O z u 7 R + 4 h 0 d t E 6 e a s o D G I t b 3 E T F M c M U C 4 C D Y f a I Z k Z F g n W h y P f M 7 D 0 w b H q s 7 y B I W S j J S f M g p A S v 1 3 W r Q A / Y I + b l J u M K C S w 7 T v l v z 6 t 4 c e J X 4 B a m h A q 2 + + 9 0 b x DS V T A E V x J i u 7 y U Q 5 k Q D p 4 J N K 7 3 U s I T Q C R m x r q W K S G b C f P 7 8 F J 9 a Z Y C H s b a j A M / V 3 x c 5 k c Z k M r K b k s D Y L H s z 8 V 8 v k k v J M G y E O V d J C k z R R f A w F R h i P G s G D 7 h m F E R m C a G a 2 9 8 x H R N N K N j + K r Y U f 7 m C V d K + q P t e 3 b + 9 r D U b R T 1 l d I x O 0 B n y 0 R V q o h v U Q g G i K E P P 6 A W 9 O k / O m / P u f C x W S 0 5 x U 0 V / 4 H z + A K w Y m 5 k = < /l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t m 2 g j Y W r c S 4 h 1 r C E 6 c t b 5 T H s x 1 8 = " > A A A C D H i c b V D L S s N A F J 3 U V 6 2 v a J d u B o v g x p K I Y J c F N y 4 r m F Z o Q 5 l M p + 3 Q m U m Y u R F D 6 C / 4 C 2 5 1 7 0 7 c + g 9 u / R K n b R b a e u D C 4 Z x 7 O Z c T J Y I b 8 L w v p 7 S 2 v r G 5 V d 6 u 7 O z u 7 R + 4 h 0 d t E 6 e a s o D G I t b 3 E T F M c M U C 4 C D Y f a I Z k Z F g n W h y P f M 7 D 0 w b H q s 7 y B I W S j J S f M g p A S v 1 3 W r Q A / Y I + b l J u M K C S w 7 T v l v z 6 t 4 c e J X 4 B a m h A q 2 + + 9 0 b x D S V T A E V x J i u 7 y U Q 5 k Q D p 4 J N K 7 3 U s I T Q C R m x r q W K S G b C f P 7 8 F J 9 a Z Y C H s b a j A M / V 3 x c 5 k c Z k M r K b k s D Y L H s z 8 V 8 v k k v J M G y E O V d J C k z R R f A w F R h i P G s G D 7 h m F E R m C a G a 2 9 8 x H R N N K N j + K r Y U f 7 m C V d K + q P t e 3 b + 9 r D U b R T 1 l d I x O 0 B n y 0 R V q o h v U Q g G i K E P P 6 A W 9 O k / O m / P u f C x W S 0 5 x U 0 V / 4 H z + A K w Y m 5 k = < / l a t e x i t > LHCb 2022 < l a t e x i t s h a 1 _ b a s e 6 4 = " g 6 d h n w B 9 d Z 0 e n i b i o w x H i c D + 5 w z S i I o S W E a m 5 v x 7 R P N K F g U 8 z b U P z 5 C B Z J / a z k e y X / 5 q J Q K c / i y a E j d I x O k I 8 u U Q V d o y q q I Y o e 0 T N 6 Q a / O k / P m v D s f 0 9 Y l Z z Z z g P 7 A + f w B n i O d p g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B T O 0 e n i b i o w x H i c D + 5 w z S i I o S W E a m 5 v x 7 R P N K F g U 8 z b U P z 5 C B Z J / a z k e y X / 5 q J Q K c / i y a E j d I x O k I 8 u U Q V d o y q q I Y o e 0 T N 6 Q a / O k / P m v D s f 0 9 Y l Z z Z z g P 7 A + f w B n i O d p g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "B T O n i D N h 2 A 6 4 K h 5 W r + i 3 C G X E + b M = " > A A A C E n i c b V D L S g N B E J z 1 G e N r 1 Z N 4 G Q w B L 4 Z d F c w x 4 M V j B P O A 7 B J m J 5 N k y M z s M t M r h j X 4 E / 6 C V 7 1 7 E 6 / + g F e / x M n j o I k F D U V V N 9 1 d U S K 4 A c / 7 c p a W V 1 b X 1 n M b + c 2 t 7 Z 1 d d 2 + / b u J U U 1 a j s Y h 1 M y K G C a 5 Y D T g I 1 k w 0 I z I S r B E N r s Z + 4 4 5 p w 2 N 1 C 8 O E h Z L 0 F O 9 y S s B K b f e w F g C 7 h + z U J F y N H t p Z k E h 8 7 g X F U d s t e C V v A r x I / B k p o B m q b f c 7 6 M Q 0 l U w B F c S Y l u 8 l E G Z E A 6 e C j f J B a l h C 6 I D 0 W M t S R S Q z Y T Z 5 Y Y S L V u n g b q x t K c A T 9 f d E R q Q x Q x n Z T k m g b + a 9 s f i v F 8 m 5 z d A t h x l X S Q p M0 e n i b i o w x H i c D + 5 w z S i I o S W E a m 5 v x 7 R P N K F g U 8 z b U P z 5 C B Z J / a z k e y X / 5 q J Q K c / i y a E j d I x O k I 8 u U Q V d o y q q I Y o e 0 T N 6 Q a / O k / P m v D s f 0 9 Y l Z z Z z g P 7 A + f w B n i O d p g = = < / l a te x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B T O n i D N h 2 A 6 4 K h 5 W r + i 3 C G X E + b M = " >A A A C E n i c b V D L S g N B E J z 1 G e N r 1 Z N 4 G Q w B L 4 Z d F c w x 4 M V j B P O A 7 B J m J 5 N k y M z s M t M r h j X 4 E / 6 C V 7 1 7 E 6 / + g F e / x M n j o I k F D U V V N 9 1 d U S K 4 A c / 7 c p a W V 1 b X 1 n M b + c 2 t 7 Z 1 d d 2 + / b u J U U 1 a j s Y h 1 M y K G C a 5 Y D T g I 1 k w 0 I z I S r B E N r s Z + 4 4 5 p w 2 N 1 C 8 O E h Z L 0 F O 9 y S s B K b f e w F g C 7 h + z U J F y N H t p Z k E h 8 7 g X F U d s t e C V v A r x I / B k p o B m q b f c 7 6 M Q 0 l U w B F c S Y l u 8 l E G Z E A 6 e C j f J B a l h C 6 I D 0 W M t S R S Q z Y T Z 5 Y Y S L V u n g b q x t K c A T 9 f d E R q Q x Q x n ZT k m g b + a 9 s f i v F 8 m 5 z d A t h x l X S Q p M 0 e n i b i o w x H i c D + 5 w z S i I o S W E a m 5 v x 7 R P N K F g U 8 z b U P z 5 C B Z J / a z k e y X / 5 q J Q K c / i y a E j d I x O k I 8 u U Q V d o y q q I Y o e 0 T N 6 Q a / O k / P m v D s f 0 9 Y l Z z Z z g P 7 A + f w B n i O d p g = = < / l a t e x i t > U -spin|modif.< l a t e x i t s h a 1 _ b a s e 6 4 = " 5 o w 0 5 8 J T 5 N r W w i H A j 5 I c 2 l Q m N x s = " > A A A C F H i c b V A 9 S w N B E N 3 z M 8 a v q G W a x S D Y G O 5 E M G X A x j K C l w R y 4 d j b 7 C V L d v e O 3 T k x n C n 8 E / 4 F W + 3 t x N b e 1 l / i 5 q P r x M v D m p o D k a Y e k 7 6 C U 0 k 0 w B F c S Y j u e m 0 M 2 J B k 4 F G x e D z L C U 0 C H p s 4 6 l i k h m u v n 0 i T E + s U o P x 4 m 2 p Q B P 1 d 8 TO Z H G j G R k O y W B g V n 0 J u K / X i Q X N k N c 6 + Z c p R k w R W e L4 0 x g S P A k I d z j m l E Q I 0 s I 1 d z e j u m A a E L B 5 l i 0 o X i L E S y T 5 n n V c 6 v e z U W l X p v H U 0 B l d I x O k Y c u U R 1 d o w b y E U W P 6 B m 9 o F f n y X l z 3 p 2 P W e u K M 5 8 5 Q n / g f P 4 A I t O f F Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 o w 0 5 8 J T 5 N r W w i H A j 5 I c 2 l Q m N x s = " > A A A C F H i c b V A 9 S w N B E N 3 z M 8 a v q G W a x S D Y G O 5 E M G X A x j K C l w R y 4 d j b 7 C V L d v e O 3 T k x n C n 8 E / 4 F W + 3 t x N b e 1 l / i 5 q P r x M v D m p o D k a Y e k 7 6 C U 0 k 0 w B F c S Y j u e m 0 M 2 J B k 4 F G x e D z L C U 0 C H p s 4 6 l i k h m u v n 0 i T E + s U o P x 4 m 2 p Q B P 1 d 8 TO Z H G j G R k O y W B g V n 0 J u K / X i Q X N k N c 6 + Z c p R k w R W e L4 0 x g S P A k I d z j m l E Q I 0 s I 1 d z e j u m A a E L B 5 l i 0 o X i L E S y T 5 n n V c 6 v e z U W l X p v H U 0 B l d I x O k Y c u U R 1 d o w b y E U W P 6 B m 9 o F f n y X l z 3 p 2 P W e u K M 5 8 5 Q n / g f P 4 A I t O f F Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 o w 0 5 8 J T 5 N r W w i H A j 5 I c 2 l Q m N x s = " > A A A C F H i c b V A 9 S w N B E N 3 z M 8 a v q G W a x S D Y G O 5 E M G X A x j K C l w R y 4 d j b 7 C V L d v e O 3 T k x n C n 8 E / 4 F W + 3 t x N b e 1 l / i 5 q P r x M v D m p o D k a Y e k 7 6 C U 0 k 0 w B F c S Y j u e m 0 M 2 J B k 4 F G x e D z L C U 0 C H p s 4 6 l i k h m u v n 0 i T E + s U o P x 4 m 2 p Q B P 1 d 8 TO Z H G j G R k O y W B g V n 0 J u K / X i Q X N k N c 6 + Z c p R k w R W e L4 0 x g S P A k I d z j m l E Q I 0 s I 1 d z e j u m A a E L B 5 l i 0 o X i L E S y T 5 n n V c 6 v e z U W l X p v H U 0 B l d I x O k Y c u U R 1 d o w b y E U W P 6 B m 9 o F f n y X l z 3 p 2 P W e u K M 5 8 5 Q n / g f P 4 A I t O f F Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 o w 0 5 8 J T 5 N r W w i H A j 5 I c 2 l Q m N x s = " > A A A C F H i c b V A 9 S w N B E N 3 z M 8 a v q G W a x S D Y G O 5 E M G X A x j K C l w R y 4 d j b 7 C V L d v e O 3 T k x n C n 8 E / 4 F W + 3 t x N b e 1 l / i 5 q P r x M v D m p o D k a Y e k 7 6 C U 0 k 0 w B F c S Y j u e m 0 M 2 J B k 4 F G x e D z L C U 0 C H p s 4 6 l i k h m u v n 0 i T E + s U o P x 4 m 2 p Q B P 1 d 8 T O Z H G j G R k O y W B g V n 0 J u K / X i Q X N k N c 6 + Z c p R k w R W e L 4 0 x g S P A k I d z j m l E Q I 0 s I 1 d z e j u m A a E L B 5 l i 0 o X i L E S y T 5 n n V c 6 v e z U W l X p v H U 0 B l d I x O k Y c u U R 1 d o w b y E U W P 6 B m 9 o F f n y X l z 3 p 2 P W e u K M 5 8 5 Q n / g f P 4 A I t O f F Q = = < / l a t e x i t >BM I< l a t e x i t s h a 1 _ b a s e 6 4 = " a n x U t Q e B I D t + p X w U R + 2 l 0 l s F g q M = "> A A A C A n i c b V C 7 S g N B F L 3 r M 8 Z X 1 N J m M A h W Y V c E U w Z t t B A i m A d k l z A 7 m S R D 5 r H M z A p h S e c v 2 G p v J 7 b + i K 1 f 4 i T Z Q h M P X D i c c y / n c u K E M 2 N 9 / 8 t b W V1 b 3 9 g s b B W 3 d 3 b 3 9 k s H h 0 2 j U k 1 o g y i u d D v G h n I m a c M y y 2 k 7 0 R S L m N N W P L q e + q 1 H q g 1 T 8 s G O E x o J P J C s z w i 2 T u p k o R b o 6 i 5 E t 5 N u q e x X / B n Q M g l y U o Y c 9 W 7 p O + w p k g o q L e H Y m E 7 g J z b K s L a M c D o p h q m h C S Y j P a Y g 4 k W 4 6 / S O D J 1 r p w y A S u k I F p + r v i R R x K c f c 1 5 0 c q a G c 9 y b i v 5 7 P 5 z a r o O q m N I w T R U I 8 W x w k D K o I T k K C f S o I V m y s C c K C 6 t s h H i K B s N J R F n U o 9 n w E i 6 R d K d t W 2 b 4 9 L 9 W q e T w F c A i O w S m w w Q W o g Q Z o g h b A 4 B E 8 g x f w a j w Z b 8 a 7 8 T F r X T L y m Q P w B 8 b n D x g i n k I = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x 1 L M u u D z a M 3 8 z Q G 3 y K b + T a U J e u I = " > A A A C F 3 i c b V D L S s N A F J 3 4 r P

FIG. 3 :
FIG. 3: g4 F d 1 /M Z as a function of ∆ FR/F d 1 and dπ 0.1TeV 2 .The red area represents the excluded 95 % C.L. region by D-mixing (9).The parameter space of models BM I, II, and III (with G = 0) from Table I accounting for the experimental results of ∆ACP and a d π − π + within its 1σ range are shown in magenta, teal, and brown, respectively.BM IV has the same parameter space as BM III (with G = 0).The shaded bands include an additional 30% of hadronic uncertainty.The semianalytical expressions of these regions are: g4F d 1 /M Z = c (∆ FR/F d 1 ) −1/2 with factor c = 0.160 ± 0.012 TeV −1 (magenta), 0.149 ± 0.015 TeV −1 (teal), and 0.133 ± 0.003 TeV −1 (brown).The black dashed lines illustrate different values of a d π − π + .
) directly excludes BM I and II, and dictates a strong quark and lepton charge hierarchy in BM III (as |G| 1.3 • 10 −3 |F |) and IV.Next we study effects from kinetic mixing.If the Z does not couple directly to electrons and muons as in BM III with color factor N ψ C = 3 for quarks and N ψ C = 1 otherwise.F ψ L(R) denotes the U (1) charge of the lefthanded (right-handed) fermion ψ.Due to the low mass

FIG. 6 :
FIG. 6: Smoking gun signature of benchmark models at e + e −machines for |F ν/χ | |F ψ | where ψ = Q, u, d, L, e, ν.A q q-pair is produced via an s-channel photon and radiates of a Z invisibly decaying to χ χ or ν ν, leading to a final state containing hadrons in association with missing energy.

F 2 d
+ F 2 e = 2F 2 u .(C13) Two aspects are worth noting from this equation: (a) For any Pythagorean triple (m, n, p), i.e. integers which solve Pythagoras equation m 2 + n 2 = p 2 , one can find integer triples (F d , F e , F u ) which solve Eq. (C13) by substituting F d = m + n, F e = m − n, and F u = p.

TABLE I :
Benchmarks for anomaly-free U (1) -extensions of the SM+3 νR.BM I, II and IV avoid Z − Z mixing at one loop, while BM III does not.Note, |G/F | 1 due to (8), and G = 0 is also possible.BM IV may or may not contain right-handed neutrinos, in which case Fν = 0.It also can feature integer charges with hierarchy |Fe| |Fu|, |F d | in which case it becomes leptophobic, see App.C for details and construction.Due to sizable couplings to electrons or muons the BMs I, II are excluded C, with |F e | |F u,d |.The kinetic mixing is natural in this model and can be switched off or made feebly small, see App.D. B) BM III is electro-and muo-phobic.It has couplings to taus and ν R .The RG evolution of the kinetic mixing parameter reads

TABLE II :
Tree-level branching fractions in % for the different Z decay modes to fermion-antifermion pairs.Results for BM III and BM III-s are given in the limit G F .In BM IV, branching ratios depend on the different charge assignments F u,d,e,ν , see main text for details.The branching ratios shown in this table are obtained from Fu = 985, F d = 1393, Fe = 1 in (C30)