Searching for New Physics in $D^0\rightarrow \mu^+\mu^-,\; e^+e^-, \;\mu^{\pm}e^{\mp}$ at BES and/or Super Charm-Tau Factory

In contrast with $B^0-\bar B^0$, $B_s-\bar B_s$ mixing where the standard model (SM) contributions overwhelm that of new physics beyond standard model (BSM), a measured relatively large $D^0-\bar D^0$ mixing where the SM contribution is negligible, definitely implies the existence of new physics BSM. It is natural to consider that the rare decays of D meson might be more sensitive to new physics, and the rare decay $D^0\to \mu^+\mu^-$ could be an ideal area to search for new physics because it is a flavor changing process. In this work we look for a trace of new physics BSM in the leptonic decays of $D^0$, concretely we discuss the contributions of unparticle or an extra gauge boson $Z'$ while imposing the constraints set by fitting the $D^0-\bar D^0$ mixing data. We find that the long-distance SM effects for $D^0\to l\bar l$ still exceed those contributions of the BSM under consideration, but for a double-flavor changing process such as $D^0\to \mu^{\pm}e^{\mp}$, the new physics contribution would be significant.


I. INTRODUCTION
One of the main goals of the study on lower energy processes is to look for traces of new physics BSM and it is mutually complementary with the very high energy processes at LHC.It is believed that the SM is very successful that its predictions are well consistent with all the present experimental data.But the SM is still an effective theory.The consistency is because at lower energy scales the contributions from new physics BSM are much smaller than that of SM which dominates all the processes.Even though the effects of new physics are small, they may manifest in some precise measurements and leave traces.Generally, BSM effects may show up at rare processes where the SM contributions are forbidden or strongly suppressed.Therefore more theorists and experimentalists have growing interests in the rare decays of heavy flavor mesons and baryons.Such studies may find traces of BSM and provide valuable information to LHC for designing new experiments.
As is well understood, the SM dominates the B 0 − B0 and B s − Bs mixing due to an enhancement factor m 2 t /M 2 W in the box-diagrams, thus contributions of new physics BSM are much smaller than that of SM.By contrary, the SM contributions to D 0 − D0 mixing are negligible because the intermediate quarks in the box are b and s which are much lighter than M W .The first evidence of D 0 − D0 oscillation is presented by the BaBar [1] and Belle [2] Collaborations and later further confirmed by the CDF Collaboration [3] in 2007.The relatively large mixing implies the existence of new physics BSM.There have been many models which offer a flavor-changingneutral current (FCNC) and enhance the mixing to the observational level.For example, the Littlest Higgs Model [4], the fourth generation [5], non-universal Z ′ [6] and unparticle [7] etc., can result in a larger D 0 − D0 mixing.Thus it motivates people to look for rare decay processes where the SM contributions are suppressed, so that the new physics effects would not be buried in the SM background.Taking into account the constraints set by D 0 − D0 mixing, we turn to investigate new physics contributions to the rare decays D 0 → l ′+ l − .
Recently, an intensive study on the leptonic decays of B 0 ( B0 ) and B s ( Bs ) is carried out.It seems that no evidence of new physics BSM is needed to explain the present data obtained by LHCb [8,9] and CMS [10].One may wonder if D is more sensitive to new physics as it happens to the D 0 − D0 mixing.As the existence of a flavor changing neutral current can explain the D 0 − D0 mixing, the same mechanism should apply to the leptonic decays D 0 → µ + µ − , e + e − , and it might cause sizable effects to enhance the rates of the leptonic decays.Definitely, such mechanism would also apply to leptonic decays of B 0 and B s even though they do not manifest for the B − B mixing.
In this work we calculate the decay rates of D 0 → µ + µ − , e + e − in terms of both the unparticle model and an extra gauge boson Z ′ .Our numerical results indicate that the contributions of the new physics of concern to the decay rates do not exceed that coming from the long-distance SM effects.But it is not the end of the story, as we proceed to study the lepton-flavor changing decay D 0 → µ + e − (µ − e + ) which is double-flavor changing process, the new physics may be significant.Moreover, when we consider the possible CP violation, the role of new physics might also be important.
Our strategy is that we employ the model parameters obtained by fitting the data of D 0 − D0 mixing for both unparticle and extra gauge boson Z ′ scenarios, then apply them to estimate the decay rates under consideration.This work is organized as follows: after this short introduction, we formulate the decay rates of D 0 → µ + µ − , e + e − and µ ± e ∓ in section II.In section III, we present our numerical results along with all the input model parameters.In section IV, we discuss possible measurement schemes on the leptonic decays and the lepton-flavor-changing decay, and the last section is devoted to our conclusion and a brief discussion.

II. CONTRIBUTIONS OF NEW PHYSICS BSM TO
The SM contribution to the decay of D 0 → l ′+ l − has been estimated as the short distance contribution to B D 0 →µ + µ − is of order 10 −19 ∼ 10 −18 [11][12][13], while taking into the long distance contributions, the branching ratio can reach a level of 10 −13 [12,13].The branching ratio of D 0 → e + e − is of order 10 −23 [13], and the decay mode D 0 → µ ± e ∓ is deeply suppressed in SM.Obviously, these rates are too small to be detected by the present facilities.Our goal of this work is to investigate if the new physics BSM would result in larger rates for those decays.In this work we only let ourselves concentrate on two possible models: unparticle [14] and non-universal boson Z ′ [15][16][17][18][19][20].These models have been thoroughly discussed in literature, so that first we briefly show how to extract model parameters from D 0 − D0 mixing data, then we formulate the new physics contributions to the rare decays D 0 → l ′+ l − .
A. Determination of the new physics parameters by fitting D 0 − D0 mixing A detectable D 0 − D0 mixing has been measured, but as indicated the SM contribution cannot induce a detectable mixing.There are two crucial parameters for the D 0 − D0 mixing which are experimentally measured via the D 0 − D0 oscillation.The physical eigen-states are and the measurable parameters x, y are defined as , where Γ = (Γ 1 + Γ 2 )/2.Experimentally, the "rotated" parameters x ′ , y ′ are also used (for more details, see, e.g., [21]).The updated Belle results [22] are x = (0.56 ± 0.19 +0.03+0.06−0.09−0.09)%, y = (0.30 ± 0.15 +0.04+0.03−0.05−0.06)%.No evidence of CP violation was observed at Belle so far and it is consistent with the observed results at LHCb [23,24].

Constraints on the parameters of unparticle scenario
The scale invariant unparticle scenario was proposed by Georgi [14], which has a non-integral scale dimension d U below a typical energy scale Λ U .In the scenario of unparticle, different flavors can be coupled to unparticle, so that FCNC can be induced at tree level.The scalar, vector unparticle fields are denoted as O U , O µ U .The propagator of scalar unparticle is [25][26][27] where The vector unparticle propagator is [28] d U −1 The unitarity bounds on the non-integral scale dimension d U below the typical energy scale Λ U are that d U ≥ 1 for scalar unparticle and d U ≥ 3 for vector unparticle [28].
The mass and width differences are related to the mixing elements, ∆m In the case of CP conservation, the scalar unparticle's contribution to the mass difference (for more, see e.g.[7,29]) is For vector unparticle, the result is Here the Wick contraction factors have been taken into consideration.f D is the decay constant, f D ≃0.2 GeV, and BD is a factor related to non-perturbative QCD with order of unity, BD ≃ 1 corresponding to the vacuum saturation [30].m D is D 0 meson mass, and Λ U is of order TeV.c S , c V are the coupling parameters.
For the mixing induced by unparticle, the relation As the contributions to the mass and width differences are totally from unparticle, i.e. ignoring the contributions from the SM and other BSMs, ∆m U D ∼ ∆m D and ∆Γ U D ∼ ∆Γ D .The measurement values of x, y can be used to determine the unparticle parameters and then applied to calculate the rates of D 0 → l ′+ l − .

Constraints on the parameters of the non-universal Z ′
Instead of unparticle scenario, let us turn to another possible BSM.In this scenario, a tree-level FCNC is induced by the new non-universal gauge boson Z ′ .Some phenomenological applications of the non-universal Z ′ have been widely studied [15][16][17][18][19][20].It was applied to the D 0 − D0 mixing by the authors of [6].The flavor-changing couplings of Z ′ to quarks and leptons are in the form where g is the SU(2) L coupling, and θ W is the Weinberg angle, as in SM. θ R is related to the right-handed interaction strength, and ξ Z parameterizes the Z − Z ′ mixing angle.V u,d R ij are the matrix rotating the right-handed up(down)-type quarks from their weak eigen-states to their mass eigen-states.
The bound set by the LEP-II measurements can be approximated in a relation form [31,32], Supposing that the measured x is fully determined by the contribution of Z ′ , the D 0 − D0 mixing constrains the matrix element This bound will used for evaluating the rates of D 0 → l ′+ l − decays.
For the mixing, as shown in Eqs.( 5),( 6), the vector unparticle's contribution is more suppressed by a factor ( m D Λ U ) 2d U compared with the scalar unparticle.The unparticle effect on B s → µ + µ − was discussed in Ref. [33].The leptonic decay D 0 → l ′+ l − is similar.Therefore, here we just consider the scalar unparticle contribution, and the Feynman diagram is presented in Fig. 1.The effective interaction of scalar unparticle with quarks and/or leptons is where c q ′ q S , c l ′ l S are the coupling constants for quarks and leptons respectively.Including contributions of SM and unparticle, the decay width of where H SM , H U are SM and unparticle Hamiltonians respectively.H U is in the form where the relation P 2 = m 2 D has been used.Let us first consider only the unparticle contribution to the decay rate.The decay width is Taking ∆m U D ,∆Γ U D into the above formula (13), we have C. Non-universal Z ′ contribution to D 0 → l ′+ l − In the limit of the mixing angle ξ Z ∼ 0, we only consider the contribution of Z ′ .The decay width of D 0 → µ + µ − can be formulated as [6] For the process D 0 → e + e − , the decay width is proportional to the lepton mass square, so it is suppressed compared to D 0 → µ + µ − .
In formula (7), the rotation only applies to the quark sector, one may naturally generalize the lagrangian to involve a rotation at the lepton sector.The lagrangian can be re-written as where V l c ,ν R ij are matrix elements rotating the lepton weak eigen-states to the mass eigen-states, moreover, this lagrangian allows flavor changes as i is not necessary to be equal to j.In this case, the lepton-flavor-changing interaction induced by Z ′ would occur at tree level.The decay width of D 0 → µ + e − can be obtained, In the following computations, we are simply going to employ the model parameters obtained by others and will list them in next section.
In the following, we present our numeral results of the decay D 0 → l ′+ l − based on the new physics BSM, both unparticle and non-universal Z ′ .

A. Unparticle
First we discuss the unparticle contribution to the decays D 0 → l ′+ l − .Relevant parameters are input as m c = 1.275 ± 0.025 GeV, m D = 1.86486 ± 0.00013 GeV, and the mean lifetime of D 0 meson (410.1 ± 1.5) × 10 −15 s [34].The updated Belle results [22] are used to constrain the new physics contributions, taking the central values, x ∼ 0.056, y ∼ 0.030, and x 2 + y 2 ∼ 4.0 × 10 −5 .Though with a large uncertainty of x 2 + y 2 , it should be taken as an upper bound of unparticle contribution.The branching ratios B U D 0 →l ′+ l − with the contributions from only unparticle are As is well recognized, due to the large experimental errors, only the order of magnitude of these theoretical evaluations are meaningful.
The lagrangian determines that l(q) can be equal or unequal to l ′ (q ′ ), thus, it is natural to assume the couplings to be universal, namely a coupling takes a value for all the same flavors and another value for all different flavors, as discussed in Ref. [35], where κ >1.To estimate the branching ratios, κ = 3 is taken as suggested by the authors of Ref. [35].The branching ratios B U D 0 →l ′+ l − are B. Non-universal Z ′ Next let us turn to the non-universal Z ′ contribution to the decays D 0 → l ′+ l − .Taking m Z ′ ∼ 500 GeV [6], and just accounting the contributions from non-universal Z ′ , with tan θ R ∼ 0.088, the branching ratios B D 0 →µ + µ − ,e + e − are For the lepton flavor violation case, we take the bound given in Ref. [36] for our discussions.That is in unit of GeV −1 .The constraint is or The branching ratio IV. THE D 0 → l ′+ l − DECAY SEARCH AT BESIII AND FUTURE CHARM-TAU FAC-TORY Since the first effort on limiting the branching fraction of FCNC process D 0 → µ + µ − was carried out by the European Muon Collaboration [37] in 1985, there have been many experimental groups searchings for D 0 → µ + µ − , D 0 → µ ± e ∓ , and D 0 → e + e − during the past thirty years.Table I summarizes their results, where the 1 st column refers the name of the experiments; the 2 nd column is for the year when the results were published; the 3 rd to 5 th columns present the Upper Limit of the branching fractions; the 6 th column shows the experiment style, i.e. fixed target, leptonic collider, hadronic collider, or heavy ion collider; and the last two columns correspond to the center-of-mass energies and data samples in use.Most of the measurements suffered from high background contaminations, and so the detection efficiency is rather low.The important task for gaining meaningful conclusion is to enhance the ability of distinguishing background and signal events.While, in the experiments whose center-of-mass energy is near the D 0 D0 threshold, the neutral charm mesons are produced in pairs, one can measure the di-lepton decays absolutely based on a technical treatment namely double tagging method (i.e. to properly reconstruct double D mesons).In the e + e − annihilation experiment around 3.773 GeV, which is just above the D D production threshold, D D pair is produced via a decay of the resonance ψ(3770) (ψ(3770) → D D).If we only identify a fully reconstructed D meson in one event, called as a singly tagged D meson, there must exist a D meson at the recoiling side.And if we reconstructed the whole D D pair in the analysis procedure, the event will be called as a doubly tagged event.Thus, with the data sample consisting of the identified singly tagged D0 events, the di-leptonic final states from decay of neutral D mesons can be indubitably selected, and the absolute branching fractions would be well measured.The advantage of the double tagging method can extremely reduce the background by tagging the D meson pairs.Historically, there were only two measurements of D 0 → e + e − and D 0 → µ ± e ∓ using the threshold data by the MARK3 Collaboration, while they proceeded the analysis with single tagging method (i.e.reconstruct only one D meson) with a large background, the threshold data did not bring up any advantages at all.Till now, the BESIII collaboration has accumulated 2.92 fb −1 [60] ψ(3770) data samples near its production threshold during 11 month's data taking.There is about 2.15 × 10 7 neutral D mesons among 3.84 × 10 7 D mesons assuming σ obs D D = 6.57nb [61].And we can eventually have more than 20 fb −1 ψ(3770) data according to the data taking plan of the experiment, resulting a D 0 sample of about 1.47 × 10 8 .Then, the key issue will be, how many singly tagged D0 events we can reconstruct, and how well we can carry out the measurement.To answer this question, here we present a full simulation of searching for di-leptonic decays at the BESIII experiment with the Monte Carlo method to discuss the experimental sensitivities that can be reached in the future.

Experiment
Year The Monte Carlo samples are obtained with the BESIII offline Software System [62], where the particle trajectories are simulated with a GEANT4 [63] based package [64] for the BESIII detector [65] at the BEPC-II collider.The events used in this discussion, named as generic MC events, are generated as e + e − → ψ(3770) → D D at the c.m. energy √ s = 3.773 GeV with the D D mesons decaying into all possible final states with the branching fractions cited from PDG [34].Totally ∼ 1.31 × 10 8 D D events are produced at √ s = 3.773 GeV, corresponding to an integrated luminosity of ∼ 20 fb −1 ψ(3770) data assuming σ obs D D = 6.57nb [61], which contains ∼ 7.35 × 10 7 D 0 D0 pairs, ∼ 6.08 × 10 7 D + D − pairs.The singly tagged D0 events are reconstructed in 4 golden hadronic decays of D0 → K + π − (69%), D0 → K + π − π 0 (35%), D0 → K + π − π − π + (39%), and D0 → K + π − π − π + π 0 (14%), constituting more than 30% of all D0 decays, where the numbers in brackets are reconstruction efficiencies.Tagged D0 events are filtered by two kinematic variables based on the principles of energy and momentum conservations: (1) Difference in energy where E f is the total energy of the daughter particles from D0 in one event and E b is the e + /e − beam energy for the experiment, is recorded to describe the deviation from energy conservation caused by experimental errors.(2) Beam-constrained mass is calculated to reduce the uncertainty caused by experimental errors when measuring the momenta of the produced particles.In this definition, the energy E f in the expression of for the D invariant mass is replaced by E b = E c.m. /2, where E c.m. is the c.m. energy that D 0 D0 produced.The total energy and momentum of all the daughter particles in D0 decays must satisfy the Energy Conservation (EC) principle, generally one needs to introduce a kinematic fit, including energy and momentum constraints and some correlated corrections, to reject those not satisfying the EC which are caused by the uncertainty of experimental measurement.This replacement of the real invariant mass by M BC partly plays the role.Moreover, events are rejected if they fail to satisfy the selection constraint |∆E| < 3 × σ ∆E , which is tailored for each individual decay mode, and σ ∆E is the standard deviation of the ∆E distribution.If the D0 events were correctly tagged, a peak in M BC spectrum would emerge at the nominal mass of D0 .Thus, if there are more than one combinations in one tagged event, the one with the smallest |∆E| is retained.After considering the detection efficiencies of each tag mode, 16856207 ± 8874 tagged D0 events have been obtained based on simulated sample of about 20 fb −1 .
With the tagged D0 mesons, the D 0 decays into a lepton pair is reconstructed in the recoiling side, i.e.D 0 → l + l ′− , where two charged track are identified as electrons or muons.To suppress the contamination from γ conversion, the angle between electron and another charged tracks should be greater than 30 • .And it is required that the ∆E distribution of the lepton pairs should fall into the range of |∆E| < 3 × σ ∆E , where σ ∆E is obtained by fitting the ∆E distribution determined by the signal MC events.And the valid signals would produce a peak at the D 0 nominal mass within 3σ M BC at the M BC spectra.For the processes of D 0 → µ + µ − , D 0 → µ ± e ∓ , and D 0 → e + e − , the numbers of estimated background events are found to be all zero, by counting the signal window of |M e + e − − M D 0 | < 3σ M BC .
To examine the sensitivities of the measurement, we evaluate the upper limits of the possible observed signal events, s 90 , at 90% confidence level, based on the expected background events assuming zero signals.The upper limits are obtained by using the Poissonian Limit Estimator (POLE) program [66], which is developed with an extended version [66] of the Feldman-Cousins method [67].Thus, the upper limit on the branching fractions are calculated to be respectively, with , by inserting the s 90 , the detection efficiencies ǫ(31% for D 0 → µ + µ − , 43% for D 0 → µ ± e ∓ , 55% for D 0 → e + e − ), and the number of singly tagged D0 events N tag D0 .The detection efficiencies are obtained by analyzing the simulated events which are generated as D 0 → l + l ′− and D0 → anything with the same procedure to the generic MC events.
The BEPCII collider is designed to work at the c.m. energy of √ s = 3.773 GeV with an instantaneous luminosity of 10 33 cm −2 s −1 .As a conservative estimate, a data sample with the integrated luminosity of about 20 fb −1 can be collected during less than 10 years' running.This world largest threshold data sample will deliver an experimental sensitivity for searching di-leptonic decays of D 0 meson of about 10 −7 level.It seems that there will be a desperate running time for the threshold experiment to challenge the sensitivities from experiments at higher energies (e.g. 10 −8 at BELLE), however, it will not be a problem if one can have a τ -charm factory with an increasing of the luminosity of more than 100 times.

V. CONCLUSIONS
In this article we give some discussions about the search of flavor-changing interactions caused by new physics in D 0 leptonic decays.Considering the constraints set by the D 0 − D0 mixing, we derive the new physics contributions: unparticle and non-universal Z ′ concerned in this work, to the decay modes D 0 → µ + µ − , e + e − , µ + e − , and estimate the numerical results of the rare decays D 0 → l ′+ l − .The theoretical predictions of branching ratios are shown in Table II, including contributions from SM and new physics from unparticle and non-universal Z ′ .
For the decay D 0 → µ + µ − , it is shown that the long-distance effect of SM still exceeds the contributions from unparticle and non-universal Z ′ , therefore the two models do not manifest in the decays.But if the leptonic decay D 0 → µ + µ − is observed with larger branching ratio (larger that 10 −13 ), it indicates that there exist BSM contributions, but not from unparticle or non-universal Z ′ .Since D 0 → e + e − suffers from the helicity suppression in the SM, so that the new physics contribution may exceed the SM contribution, but this branching ratio is very  small to be observed with the present facilities.A simple analysis indicates that the decay mode D 0 → µ ± e ∓ is much suppressed in SM.Therefore a sizable or at least observable mode D 0 → µ ± e ∓ must be due to new physics contributions.
As discussed in this work, even though the leptonic decays of D 0 are sensitive to the new physics as implied by the measured D 0 − D0 mixing, the contributions from unparticle and nonuniversal Z ′ cannot exceed the SM contribution.The favorable modes which may distinguish between the SM and BSM contributions are the lepton-flavor violation processes which are much suppressed in the SM.However, the branching ratio of such modes are very small, even though some BSM mechanisms such as unparticle and non-universal Z ′ are taken into account.They are far below the reach of any presently available facilities.In fact there are many new physics models which might cause a larger branching ratio (other schemes, see e.g.[68]).The measurement on the leptonic decays D 0 → µ + µ − is worthwhile and one might find a trace of new physics.Meanwhile D 0 → µ + e − (µ − e + ) is a much better place to look for new physics.
Even though the present facilities cannot provide large amount of D 0 , one may expect that the future super charm-tau factory and LHC may do the job.

TABLE I :
Historical measurements on searching for dilepton decays.

TABLE II :
The branching ratio predictions in D 0 → l ′+ l − decays, with the contributions from SM and new physics unparticle, non-universal Z ′ .