# New physics effects in charm meson decays involving \(c \rightarrow u l^+ l^- (l_i^{\mp } l_j^{\,\pm \,})\) transitions

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## Abstract

We study the effect of the scalar leptoquark and \(Z^\prime \) boson on the rare decays of the *D* mesons involving flavour changing transitions \(c \rightarrow u l^+ l^- (l^{\mp }_i l^{\,\pm \,}_j)\). We constrain the new physics parameter space using the branching ratio of the rare decay mode \(D^0 \rightarrow \mu ^+ \mu ^-\) and the \(D^0 - {\bar{D}}^0\) oscillation data. We compute the branching ratios, forward–backward asymmetry parameters and flat terms in \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^+ \mu ^-\) processes using the constrained parameters. The branching ratios of the lepton flavour violating *D* meson decays, such as \(D^0 \rightarrow \mu e, ~\tau e\) and \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^- e^+\) are also investigated.

## 1 Introduction

The rare *B* and *D* meson decay processes driven by a flavour changing neutral current (FCNC) transitions constitute a subject of great interest in the area of electroweak interactions and provide an excellent testing ground to look for new physics beyond the standard model (SM). The FCNC decays are highly suppressed in the SM and occur only at one-loop level. Of particular interest among the FCNC decays are the rare semileptonic *B* meson decays involving the transitions \(b \rightarrow s l^+ l^-\), where several anomalies at the level of few sigma have been observed recently in the LHCb experiment [1, 2, 3, 4]. To complement these results, efforts should also be made towards the search for new physics signal in the up quark sector, mainly in the rare charm meson decays involving \(c \rightarrow u l^+ l^-\) quark level transitions. Recently the LHCb experiment has searched for the branching ratio of the lepton flavour violating (LFV) \(D^0 \rightarrow \mu ^{\mp } e^{\,\pm \,}\) decays and put the limit as \(\mathrm{BR}(D^0 \rightarrow \mu ^{\mp } e^{\,\pm \,})~ <~ 1.3 \times 10^{-8}\) [5] at \(90\%\) confidence level (CL). On the other hand, both the Belle and the BaBar experiments have reported significant deviations on the measured branching fractions of \({\bar{B}} \rightarrow D^{(*)} \tau \nu _{\tau }\) processes from the corresponding SM predictions. The ratio of these branching fractions, the so-called \(R(D^{(*)})\), defined as \(R(D^{(*)}) = \mathrm{BR}({\bar{B}} \rightarrow D^{(*)} \tau \nu _{\tau })/ \mathrm{BR}({\bar{B}} \rightarrow D^{(*)} l \nu _l)\), where \(l=e, \mu \), exceed the SM prediction by \(3.5 \sigma \) [6], thus opening an excellent window to search for new physics (NP) in the up quark sector.

Mixing between a neutral meson and its anti-meson with a specific flavour provides an useful tool to deal with problems in flavour sector. For example, in the past the \(K^0-{\bar{K}}^0\) and \(B^0-{\bar{B}}^0\) oscillations, involving mesons made of up- and down-type quarks, have provided information as regards the charm and top quark mass scale, much before the discovery of these particles in the collider. On the other hand, the \(D^0 - {\bar{D}}^0\) system involves mesons with up-type quarks and in the SM the mixing rate is sufficiently small, so that the new physics component might play an important role in this case. The mixing parameters required to describe the \(D^0 - {\bar{D}}^0\) mixing are defined by \(x = \Delta M{/}\Gamma \) and \(y= \Delta \Gamma / 2 \Gamma \), where \(\Delta M\) (\(\Delta \Gamma \)) is the mass (width) difference between the mass eigenstates.

In this paper, we focus on the analysis of rare charm meson decays induced by \(c \rightarrow u \mu ^+ \mu ^-\) and \(c \rightarrow u \mu ^{\mp } e^{\,\pm \,}\) FCNC transitions. We calculate the branching ratios, forward–backward asymmetry parameters and the flat terms in \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^+ \mu ^-\) processes both in the scalar leptoquark (LQ) and the generic \(Z^\prime \) model. These processes suffer from resonance background through \(c \rightarrow u M \rightarrow u l^+ l^-\), where *M* denotes \(\eta ^{(')}\) (pseudoscalar), \(\rho , \phi , \omega \) (vector) mesons. However, to reduce the background coming from these resonances, we work in the low and high \(q^2\) regimes, i.e., \( q^2 \in [0.0625, 0.275]\,\mathrm{GeV}^2\) and \( q^2 \in [1.56, 4.00]\,\mathrm{GeV}^2\), which lie outside the mass square range of the resonant mesons. We also compute the branching ratios of lepton flavour violating \(D^0 \rightarrow \mu e, \tau e\) and \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^- e^+\) processes. These LFV processes have negligible contributions from the SM, as they proceed through the box diagrams with tiny neutrino masses in the loop. However, they can occur at tree level in the LQ and \(Z^\prime \) models and are expected to have significantly large branching ratios. Leptoquarks are hypothetical colour triplet bosonic particles, which couple to quarks and leptons simultaneously and contain both baryon and lepton quantum numbers. It is interesting to study flavour physics with leptoquarks as they allow quark–lepton transitions at tree level, thus explaining several observed anomalies, e.g., the lepton non-universality (LNU) parameter \(R_K|_{q^2 \in [1,6]\,\mathrm{GeV}^2}=\mathrm{BR}(B \rightarrow K \mu ^+ \mu ^-)/\mathrm{BR}(B \rightarrow K e^+ e^-)\) in rare *B* decays. The existence of the scalar leptoquark is predicted in the extended SM theories, such as grand unified theory [7, 8, 9, 10, 11], the Pati–Salam model, the extended technicolour model [12, 13] and the composite model [14]. In this work, we consider the model which conserves baryon and lepton numbers and does not allow proton decay. Here we would like to see how this model affects the leptonic and semieptonic decays of the \(D^0\) meson induced by \(c \rightarrow u l^+ l^-\) transitions. The phenomenology of scalar leptoquarks and their implications to the *B* and *D* sector has been extensively studied in the literature [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

The \(Z^\prime \) boson is a colour singlet vector gauge boson and it is electrically neutral in nature. By adding an additional \(U(1)^\prime \) gauge symmetry, the new \(Z^\prime \) gauge boson could be naturally derived from the extension of electroweak symmetry of the SM, such as superstring theories, grand unified theories and theories with large extra dimensions. The processes mediated via \(c \rightarrow u\) FCNC transitions could be induced by the generic \(Z^\prime \) model at tree level. The theoretical framework of the heavy new \(Z^\prime \) gauge boson has been studied in the literature [52, 53, 56, 57]. In this paper, we investigate the \(Z^\prime \) contribution to the rare \(D^0\) meson decay processes within the parameter space constrained by \(D^0 - {\bar{D}}^0\) mixing and \(D^0 \rightarrow \mu ^+ \mu ^-\) processes.

The paper is organized as follows. In Sect. 2, we discuss the effective Hamiltonian describing \(\Delta C=1\) transitions i.e., \(c \rightarrow u l^+ l^-\), and \(\Delta C=2\) transition, which is responsible for \(D^0 - {\bar{D}}^0\) mixing. The new physics contribution to \(c \rightarrow u\) transitions and the constraint on leptoquark couplings from the \(D^0 - {\bar{D}}^0\) oscillation and the process \(D^0 \rightarrow \mu ^+ \mu ^-\) are discussed in Sect. 3. We calculate the constraint on \(Z^\prime \) couplings from \(D^0 - {\bar{D}}^0\) mixing and leptonic \(D^0 \rightarrow \mu ^+ \mu ^-\) decays in Sect. 4. In Sect. 5, we compute the branching ratios, forward–backward asymmetry parameters and the flat terms of the process \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^+ \mu ^-\) in both these models. The lepton flavour violating \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^- e^+\) and \(D^0 \rightarrow \mu e, \tau e \) processes are discussed in Sects. 6 and 7. Finally we summarize our findings in Sect. 8.

## 2 Effective Hamiltonian for \(\Delta C=1\) and \(\Delta C=2\) transitions

Though the rare charm decays are affected by large non-perturbative effects, the short distance structure of FCNC transitions can be investigated well theoretically. The change in charm quantum number for rare FCNC charm meson decays is either of two units or one unit, and hence, they involve either \(\Delta C=2\) or \(\Delta C=1\) transitions. The \(D^0 - {\bar{D}}^0\) mixing takes place via a \(\Delta C=2\) transition and the decay processes with \(\Delta C=1\) transitions are \(c \rightarrow u l^+ l^-\) and \(c \rightarrow u \gamma \).

*M*, an effective Hamiltonian in the form of a series of operators of increasing dimensions can be obtained. However, the operators of dimension \(d=6\) have important contributions to charm meson decays or mixing. In general, one can write the complete basis of these effective operators in terms of chiral quark fields for both \(D^0 - {\bar{D}}^0\) mixing and the process \(D^0 \rightarrow l^+ l^-\) as [52, 53]

*G*has inverse-mass squared dimensions, \(C_i\) are the Wilson coefficients.

^{1}

*M*are given by [52, 53]

## 3 New physics contribution due to the exchange of scalar leptoquarks

The presence of leptoquarks can modify the SM effective Hamiltonian of \(c \rightarrow u\) transitions, giving appreciable deviation from the SM values. These colour triplet bosons can be either scalars or vectors. There exist three scalar and four vector relevant leptoquark states which potentially contribute to the \(c \rightarrow u l^+ l^-\) transitions and are invariant under the SM gauge group \(SU(3)_C \times SU(2)_L \times U(1)_Y\), where the hypercharge *Y* is related to the electric charge and weak isospin (*I*) through \(Y= Q-I_3\). Out of three possible scalar leptoquarks with the quantum numbers \((3,3,-1/3)\), \((3,1,-1/3)\) and (3, 2, 7 / 6) [49, 50], only the leptoquark with multiplet (3, 2, 7 / 6) conserves both baryon and lepton numbers and, thus, avoids rapid proton decay at the electroweak scale. Similarly out of the vector multiplets (3, 3, 2 / 3), (3, 1, 5 / 3), (3, 2, 1 / 6) and \((3,2,-5/6)\), only the first two leptoquark states do not allow baryon number violation and can be considered to study the observed anomalies in flavour sector. In this work we consider the baryon number conserving \(X=(3,2,7/6)\) scalar leptoquark which induces the interaction between the up-type quarks and charged leptons and, thus, contributes to the semileptonic decay amplitudes.

*Q*and

*L*and \(u_R(l_R)\) is the right-handed quark (charged-lepton) singlet. We use the basis where CKM and PMNS rotations are assigned to down-type quarks and neutrinos, i.e., \(d_L \rightarrow V_{CKM} d_L\) and \(\nu _L \rightarrow V_{PMNS} \nu _L\). Here \(Y^{L}\) and \(Y^R\) are the leptoquark couplings in the mass basis of the up-type quarks and charged leptons. Now writing the leptoquark doublets in terms of its components as \(\Delta =(\Delta ^{(5/3)}, \Delta ^{(2/3)})^T\), where the superscripts denote the electric charge of the LQ components and expanding the terms in Eq. (6), one can obtain the interaction Lagrangian for different components of LQs given as [50]

### 3.1 Constraint on leptoquark couplings from \(D^0 - {\bar{D}}^0\) mixing

*W*-boson exchange and the weak interaction boxes are suppressed due to the GIM mechanism because of the smallness of the down quark mass in comparison to the weak scale. In the LQ model, there will be a contribution to the \(D^0 - {\bar{D}}^0\) mass difference from the box diagrams with the leptoquark and leptons flowing in the loop. Since the SM contribution to the mass difference is very small, we consider its value to be saturated by new physics contributions. Furthermore, the couplings to the left-handed quarks are considered to be zero in order to avoid strict constraints in the down-type quark sector. Thus, considering only right-handed couplings, one can write the effective Hamiltonian due to the leptoquark

*X*(3, 2, 7 / 6) and charged lepton/neutrinos in the loop as [25, 26]

*I*(

*x*) is given as

*l*denotes the charged-lepton flavours. In our analysis, the mass of the \(D^0\) meson is taken from [60], the value of the decay constant \(f_D=222.6 {\,\pm \,} 16.7^{+2.3}_{-2.4}\) MeV [62] and \(B_D (3\,\mathrm{GeV}) = 0.757(27)(4)\) [63]. To obtain the bound on the leptoquark coupling, we assume that the individual leptoquark contribution to the mass difference does not exceed the \(1\sigma \) range of the experimental value. Since we are interested in obtaining the bounds on the \(Y_{\mu c}^{R} Y_{\mu u}^{R*}\) couplings, here we assume that the leptoquark has dominant coupling to muons and its coupling to electron or tau is negligible. The SM contribution to the mass difference is very small and hence can be neglected. The corresponding experimental value is given by [60]

### 3.2 Constraint from \(D^0 \rightarrow \mu ^+ \mu ^- (e^+ e^-)\) process

*B*meson decay processes, the only non-perturbative quantity involved is the decay constant of the

*D*meson, which can be reliably calculated using non-perturbative methods such as QCD sum rules, lattice gauge theory and so on. The branching ratio of the process \(D^0 \rightarrow l^+ l^-\) is given by [48, 49]

*D*meson are [60]

If we impose chirality on the scalar leptoquarks i.e., they couple to either left-handed or right-handed quarks, but not to both, then the \(C_{S, P}^{(')}\) Wilson coefficients will vanish and we get only the additional contribution of the \(C_{9,10}^{(') LQ}\) Wilson coefficients to the SM. Now comparing the theoretical and experimental branching ratios, the allowed range of \(\tilde{C}_{10}^{(\prime ) LQ}\) Wilson coefficients are given in Table 1.

The allowed values of the Wilson coefficients obtained from the upper bound of the process \(D^0 \rightarrow \mu ^+ \mu ^-(e^+ e^-)\). The constraint on the \(\tilde{C}_i^{ LQ}\) coefficients can also be applicable to \(\tilde{C}_i^{\prime { LQ}}\) Wilson coefficients

Wilson coefficient | \(D^0\rightarrow \mu ^+ \mu ^-\) | \(D^0 \rightarrow e^+ e^-\) |
---|---|---|

\(\tilde{C}_{10}^{ LQ}\) | 0.8 | 600 |

\(\tilde{C}_S^{LQ}\) | 0.053 | 0.186 |

\(\tilde{C}_P^{LQ}\) | 0.053 | 0.186 |

*B*and

*K*physics, we consider a numerically tuned example as discussed in [49]. We assume that the \(Y^R\) coupling is perturbative, i.e., \(|Y^R| ~ <~ \sqrt{4\pi }\). In particular, we consider a large value for \(Y_{c\mu }^R\) coupling, e.g., \(Y_{c\mu }^R=3.5\). We compute the bound on \(Y_{u\mu }^R\) coupling by using the constraint on the \(\tilde{C}_{10}^{\prime LQ}\) Wilson coefficients from \(D^0 \rightarrow \mu ^+ \mu ^-\) process, which is found to be comparatively small, \(Y_{u\mu }^R~<~8.76 \times 10^{-3}\). Now we instigate a non-zero coupling to the left-handed quark \(Y_{u\mu }^L\), which along with the large \(Y_{c\mu }^R\) coupling provides non-zero values for the \(C_{S, P}\) and \(C_{T, T_5}\) coefficients. However, the process \(D^0 \rightarrow \mu ^+ \mu ^-\) imposes a strong bound on the coefficient \(C_S\), which, together with the large \(Y_{c\mu }^R\) coupling, limits the left-handed coupling to \(Y_{\mu u}^L ~<~1.14 \times 10^{-3}\). Thus, from the above discussion we observe that

*B*,

*K*physics [66].

## 4 New physics contribution in \(Z^\prime \) model

After having obtained the possible \(Z^\prime \) couplings with quarks and leptons, we proceed to constrain the new parameter space using the results from charm sector, e.g., the experimental data on \(D^0 - {\bar{D}}^0\) mixing and the branching ratios of \(D^0 \rightarrow l^+ l^-\) processes. The constraint on the coupling of \(Z^\prime \) with the leptonic part is obtained from the upper limit on the branching ratio of the lepton flavour violating \(\tau (\mu )^- \rightarrow e^- e^+ e^-\) processes.

### 4.1 Constraint from \(D^0 - {\bar{D}}^0\) mixing

### 4.2 Constraint from \(D^0 \rightarrow \mu ^+ \mu ^-\) process

*Z*boson to leptons as discussed in [53], i.e.,

*g*is the gauge coupling of

*Z*boson and \(\theta _W\) is the Weinberg mixing angle. Now using the experimental upper limit on the branching ratio \(\mathrm{BR}(D^0 \rightarrow \mu ^+ \mu ^-) ~<~6.2 \times 10^{-9}\) [60], we obtain

It should be noted that the constraint on \(Z^\prime \) couplings from the \(D^0 \rightarrow \mu ^+ \mu ^-\) decay process and the \(D^0 - {\bar{D}}^0\) mixing data have been computed in [53]. Similarly, the constraint on the couplings from \(D^0 - {\bar{D}}^0\) oscillation are obtained in Ref. [52]. We found that our constraints are consistent with the above predictions, if we use the updated values of various input parameters.

### 4.3 Constraints on \(g_{Z'1}^{\prime }\) from \(\tau ^-(\mu ^-) \rightarrow e^- e^+ e^-\) process

## 5 \(D^{+ (0)} \rightarrow \pi ^{+ (0)} \mu ^+ \mu ^-\) process

*D*meson and the final \(\pi \) meson can be parametrized in terms of three form factors \(f_0\), \(f_+\) and \(f_T\) [49]:

*k*are the four momenta of the

*D*and \(\pi \) mesons, respectively, and \(q=p_D-k\) is the momentum transfer. The form factors for \(D^0 \rightarrow \pi ^0\) are scaled as \(f_i \rightarrow f_i/\sqrt{2}\) by isospin symmetry. For the \(q^2\)-dependence of the form factors, we use the parameterization from Refs. [69, 70], as

*D*meson and the negatively charged lepton in the rest frame of the dilepton. The functions

*V*,

*A*,

*S*and

*P*are defined in terms of the Wilson coefficients as

*l*is given by [49, 72]

*D*meson from [60]. For the CKM matrix elements, we use the Wolfenstein parametrization with values \(A=0.814_{-0.024}^{+0.023}\), \(\lambda =0.22537{\,\pm \,} 0.00061\), \({\bar{\rho }} =0.117 {\,\pm \,} 0.021\) and \({\bar{\eta }} =0.353 {\,\pm \,} 0.013\) [60]. With these input parameters, we compute the resonant/non-resonant branching ratios of the process \(D^{+} \rightarrow \pi ^{+} \mu ^+ \mu ^-\) by integrating the decay distribution with respect to \(q^2\). We parametrize the contributions from the resonances with the Breit–Wigner shapes for \(C_9 \rightarrow C_9^\mathrm{res}\), for \( \rho , \omega , \phi \) (vector) and \(C_P \rightarrow C_P^\mathrm{res}\) for \(\eta ^{(\prime )}\) (pseudoscalar) mesons [48, 49]:

*M*, where

*M*corresponds to \( \eta ^{(\prime )}, \rho , \omega , \phi \) mesons. With the approximation of \(\mathrm{BR}(D^+ \rightarrow \pi ^+ M(\rightarrow \mu ^+ \mu ^-)) \simeq \mathrm{BR}(D^+ \rightarrow \pi ^+ M) \mathrm{BR}(M \rightarrow \mu ^+ \mu ^-)\) and considering the experimental upper bound from [60], the magnitudes of the Breit–Wigner parameters are given by [48]

The predicted branching ratios for \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^+ \mu ^-\) processes in both the low \(q^2\) and the high \(q^2\) region in the scalar *X*(3, 2, 7 / 6) LQ and \(Z^\prime \) model. This also contains the resonant and non-resonant SM branching ratios

Decay process | \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\) | \(D^0 \rightarrow \pi ^0 \mu ^+ \mu ^-\) |
---|---|---|

Low \(q^2\) | ||

Non-resonant SM | \((3.02 {\,\pm \,} 0.483) \times 10^{-13}\) | \( (1.19 {\,\pm \,} 0.19) \times 10^{-13}\) |

Resonant SM | \((1.36 - 2.4) \times 10^{-10}\) | \( (5.66-9.89) \times 10^{-11}\) |

LQ model | \((2.6-8.68) \times 10^{-10}\) | \((1.02-3.4) \times 10^{-10}\) |

\(Z^\prime \) model | \(~(0.65-1.18) \times 10^{-12}\) | \((2.55-4.62) \times 10^{-13}\) |

Expt. limit (\(90\%\) CL) | \(2 \times 10^{-8}\) [73] | \(\cdots \) |

High \(q^2\) | ||

Non-resonant SM | \((5.14{\,\pm \,} 0.82) \times 10^{-13}\) | \((2 {\,\pm \,} 0.32) \times 10^{-13}\) |

Resonant SM | \((1.25 - 3.29) \times 10^{-10}\) | \( (0.456-1.24) \times 10^{-10}\) |

LQ model | \((1.32-3.36) \times 10^{-9}\) | \((0.513-1.3)\times 10^{-9}\) |

\(Z^\prime \) model | \((1.4-2.78) \times 10^{-12}\) | \((0.545-1.08) \times 10^{-12}\) |

Expt. limit (\(90\%\) CL) | \(2.6 \times 10^{-8}\) [73] | \(\cdots \) |

With all the input parameters from [60] along with the SM Wilson coefficients [58, 59], we present in Table 2 the predicted values of the branching ratios for the \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^+ \mu ^-\) processes by integrating the decay distribution in low and high \(q^2\) bins. Here we have used the \(q^2\) regimes as \(q^2 \in [0.0625, 0.275]\,\mathrm{GeV}^{2}\) and \(q^2 \ge 1.56\,\mathrm{GeV}^2\) to reduce the background coming from the dominant resonances. The theoretical uncertainties in the SM are associated with the lifetime of the *D* meson, the CKM matrix elements and the hadronic form factors. In Fig. 3, the variation of the SM branching ratios of the process \(D^{+} \rightarrow \pi ^{+} \mu ^+ \mu ^-\) in the very low and high \(q^2\) regimes are shown in red dashed lines and the green bands represent the SM theoretical uncertainties.

## 6 \(D^{+ (0)} \rightarrow \pi ^{+ (0)} \mu ^- e^+\)

*D*and \(\mu ^-\) in the \(\mu -e\) rest frame) is given as

The predicted branching ratios for \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^- e^+\) lepton flavour violating processes in the scalar *X*(3, 2, 7 / 6) LQ and \(Z^\prime \) model. The present upper limit on the branching ratio \(\mathrm{BR}(D^{0} \rightarrow \pi ^{0} \mu ^{\mp } e^{\,\pm \,})= \mathrm{BR}(D^{0} \rightarrow \pi ^{0} \mu ^- e^+ + \pi ^0 \mu ^+ e^-) < 8.6 \times 10^{-5}\) [60]

Decay process | \(D^+ \rightarrow \pi ^+ \mu ^- e^+\) | \(D^0 \rightarrow \pi ^0 \mu ^- e^+\) |
---|---|---|

LQ model | \( (1.67-3.72) \times 10^{-11}\) | \( (0.56-1.4) \times 10^{-11}\) |

\(Z^\prime \) model | \((2.95-7.8)\times 10^{-12}\) | \((1.15-3.04) \times 10^{-12}\) |

Experimental limit | \({<}2.9 \times 10^{-6}\) [60] | \(\cdots \) |

## 7 \(D^0 \rightarrow \mu ^- e^+ (\tau ^- e^+)\) LFV decay process

## 8 Conclusion

In this paper we have studied the rare decays of the *D* meson in both scalar leptoquark and generic \(Z^\prime \) models. We have considered the simple renormalizable scalar leptoquark model with the requirement that proton decay would not be induced in perturbation theory. The leptoquark parameter space is constrained using the present upper limit on branching ratio of \(D^0 \rightarrow \mu ^+ \mu ^-\) process and the \(D^0 - {\bar{D}}^0\) oscillation data. For the \(Z^\prime \) model, the constraints on the \(Z^\prime \) couplings are obtained from the mass difference of \(D^0 - {\bar{D}}^0\) mixing, the process \(D^0 \rightarrow \mu ^+ \mu ^-\) and the lepton flavour violating \(\tau ^-(\mu ^-) \rightarrow e^- e^+ e^-\) processes. Using the constrained parameter space, we estimated the branching ratios, forward–backward asymmetry parameters and the flat terms in the \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^+ \mu ^-\) processes. The branching ratios in the LQ model are found to be \(\sim \mathcal{O}(10^{-10})\), which are larger than the corresponding SM predictions in the very low and very high \(q^2\) regimes. If these branching ratios will be observed in near future they would provide indirect hints of leptoquark signal. Furthermore, we estimated the branching ratios of lepton flavour violating \(D^{+(0)} \rightarrow \pi ^{+(0)} \mu ^- e^+\) and \(D^0 \rightarrow \mu (\tau )^- e^+ \) processes, which are found to be rather small. We also estimated the forward–backward asymmetry parameter and the flat term for the LFV decays.

## Footnotes

- 1.
We denote the Wilson coefficients for \(\Delta C =2\) operators as \({c_i}\) and those for \(\Delta C =1\) operators as \({C_i}\) throughout this work

## Notes

### Acknowledgements

We would like to thank the Science and Engineering Research Board (SERB), Government of India for financial support through Grant No. SB/S2/HEP-017/2013.

### References

- 1.R. Aaij et al.,[ LHCb Collaboration], JHEP
**1406**, 133 (2014). arXiv:1403.8044 - 2.
- 3.
- 4.
- 5.R. Aaij et.al., [LHCb collaboration], Phys. Lett. B
**754**, 167 (2016). arXiV:1512.00322 - 6.Heavy Flavour Averaging Group, http://www.slac.stanford.edu/xorg/hfag/semi/eps15/eps15-dtaunu.html
- 7.H. Georgi, S.L. Glashow, Phys. Rev. Lett.
**32**, 438 (1974)ADSCrossRefGoogle Scholar - 8.J.C. Pati, A. Salam, Phys. Rev. D
**10**, 275 (1974)ADSCrossRefGoogle Scholar - 9.H. Georgi, AIP Conf. Proc.
**23**, 575 (1975)ADSCrossRefGoogle Scholar - 10.H. Fritzsch, P. Minkowski, Ann. Phys.
**93**, 193 (1975)ADSCrossRefGoogle Scholar - 11.P. Langacker, Phys. Rep.
**72**, 185 (1981)ADSCrossRefGoogle Scholar - 12.B. Schrempp, F. Shrempp, Phys. Lett. B
**153**, 101 (1985)ADSCrossRefGoogle Scholar - 13.
- 14.D.B. Kaplan, Nucl. Phys. B
**365**, 259 (1991)ADSCrossRefGoogle Scholar - 15.J.M. Arnold, B. Fornal, M.B. Wise, Phys. Rev. D
**88**, 035009 (2013). arXiv:1304.6119 ADSCrossRefGoogle Scholar - 16.
- 17.
- 18.
- 19.
- 20.
- 21.
- 22.
- 23.S. Sahoo, R. Mohanta, A.K. Giri, Phys. Rev. D
**95**, 035027 (2017). arXiv:1609.04367 ADSCrossRefGoogle Scholar - 24.M. Duraisamy, S. Sahoo, R. Mohanta, Phys. Rev. D
**95**, 035022 (2015). arXiv:1610.00902 ADSCrossRefGoogle Scholar - 25.S. Davidson, D. C. Bailey and B. A. Campbell, Z. Phys. C
**61**, 613 (1994). arXiv:hep-ph/9309310 - 26.J.P. Saha, B. Misra, A. Kundu, Phys. Rev. D
**81**, 095011 (2010). arXiv:1003.1384 ADSCrossRefGoogle Scholar - 27.I. Dorsner, S. Fajfer, J.F. Kamenik, N. Kosnik, Phys. Lett. B
**682**, 67 (2009). arXiv:0906.5585 ADSCrossRefGoogle Scholar - 28.
- 29.R. Benbrik, M. Chabab, G. Faisel, arXiv:1009.3886
- 30.I. Dorsner, S. Fajfer, A. Greljo, J.F. Kamenik, N. Kosnik, Phys. Rep.
**641**, 1 (2016). arXiv:1603.04993 ADSMathSciNetCrossRefGoogle Scholar - 31.I. Dorsner, J. Drobnak, S. Fajfer, J.F. Kamenik, N. Kosnik, JHEP
**11**, 002 (2011). arXiv:1107.5393 ADSCrossRefGoogle Scholar - 32.F.S. Queiroz, K. Sinha, A. Strumia, Phys. Rev. D
**91**, 035006 (2015). arXiv:1409.6301 ADSCrossRefGoogle Scholar - 33.B. Allanach, A. Alves, F.S. Queiroz, K. Sinha, A. Strumia, Phys. Rev. D.
**92**, 055023 (2015). arXiv:1501.03494 ADSCrossRefGoogle Scholar - 34.L. Calibbi, A. Crivellin, T. Ota, Phys. Rev. Lett.
**115**, 181801 (2015). arXiv:1506.02661 ADSCrossRefGoogle Scholar - 35.Ivo de M. Varzielas, G. Hiller, JHEP
**1506**, 072 (2005) arXiv:1503.01084 - 36.M. Bauer, M. Neubert, Phys. Rev. Lett.
**116**, 141802 (2016). arXiv:1511.01900 ADSCrossRefGoogle Scholar - 37.
- 38.
- 39.D. Aristizabal Sierra, M. Hirsch, S.G. Kovalenko, Phys. Rev. D
**77**, 055011 (2008). arXiv:0710.5699 ADSCrossRefGoogle Scholar - 40.
- 41.
- 42.S. Fajfer, J.F. Kamenik, I. Nisandzic, J. Zupan, Phys. Rev. Lett.
**109**, 161801 (2012). arXiv:1206.1872 ADSCrossRefGoogle Scholar - 43.K. Cheung, W.-Y. Keung, P.-Y. Tseng, Phys. Rev. D
**93**, 015010 (2016). arXiv:1508.01897 ADSCrossRefGoogle Scholar - 44.D.A. Camargo. arXiv:1509.04263
- 45.
- 46.C. Hati, G. Kumar, N. Mahajan, JHEP
**01**, 117 (2016). arXiv:1511.03290 [hep-ph]ADSCrossRefGoogle Scholar - 47.B. Gripaios, M. Nardecchia, S.A. Renner, JHEP
**1505**, 006 (2015). arXiv:1412.1791 ADSCrossRefGoogle Scholar - 48.
- 49.
- 50.I. Dorsner, S. Fajfer, N. Kosnik, I. Nisandzic, JHEP
**11**, 084 (2013). arXiv:1306.6493 ADSCrossRefGoogle Scholar - 51.
- 52.E. Golowich, J. Hewett, S. Pakvasa, A.A. Petrov, Phys. Rev. D
**76**, 095009 (2007). arXiv:0705.3650 ADSCrossRefGoogle Scholar - 53.E. Golowich, J. Hewett, S. Pakvasa, A.A. Petrov, Phys. Rev. D
**79**, 114030 (2009). arXiv:0903.2830 ADSCrossRefGoogle Scholar - 54.S. Fajfer, P. Singer, J. Zupan Eur. Phys. J. C
**27**, 201 (2003)ADSCrossRefGoogle Scholar - 55.S. Fajfer, N. Kosnik, Phys. Rev. D
**87**, 054026 (2013)ADSCrossRefGoogle Scholar - 56.A. Arhrib, K. Cheung, C. Chiang, T.-C. Yuan, Phys. Rev. D
**73**, 075015 (2006). arXiv:hep-ph/0602175 ADSCrossRefGoogle Scholar - 57.X.-G. He, G. Valencia, Phys. Lett. B
**651**, 135 (2007). arXiv:hep-ph/0703270 ADSCrossRefGoogle Scholar - 58.
- 59.C. Greub, T. Hurth, M. Misiak, D. Wyler, Phys. Lett. B
**382**, 415 (1996). arXiv:hep-ph/9603417 ADSCrossRefGoogle Scholar - 60.K.A. Olive et al., (Particle Data Group), Chin. Phys. C
**38**, 090001 (2014)Google Scholar - 61.R.-M. Wang, J.-H. Sheng, J. Zhu, Y.-Y. Fan, Y.-D. Yang, Int. J. Mod Phys. A
**30**, 1550063 (2015). arXiv:1409.0181 ADSCrossRefGoogle Scholar - 62.M. Artuso et al. [CLEO Collaboration], Phys. Rev. Lett.
**95**, 251801 (2005). arXiv:hep-ex/0508057 - 63.N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz, G.C. Rossi, S. Simula, C. Tarantino [ETM Collaboration], Phys. Rev. D
**92**, 034516 (2015). arXiv:1505.06639 - 64.G. Burdman, E. Golowich, J. Hewett, S. Pakvasa, Phys. Rev. D
**66**, 014009 (2002). arXiv:hep-ph/0112235 ADSCrossRefGoogle Scholar - 65.J.P. Lees et al., (Fermilab Lattice, MILC), Phys. Rev. D
**85**, 091107 (2012). arXiv:1110.6480 - 66.M. Carpentier, S. Davidson, Eur. Phys. J. C
**70**, 1071 (2010). 1008.0280ADSCrossRefGoogle Scholar - 67.A. Crivellin, L. Hofer, J. Matias, U. Nierste, S. Pokorski, J. Rosiek, Phys. Rev. D
**92**, 054013 (2015). arXiv:1504.07928 ADSCrossRefGoogle Scholar - 68.D. Becirevic, O. Sumensari, R.Z. Funchal, Eur. Phys. J. C
**76**, 134 (2016). arXiv:1602.00881 ADSCrossRefGoogle Scholar - 69.D. Becirevic, A.B. Kaidalov, Phys. Lett. B
**478**, 417 (2000). arXiv:hep-ph/9904490 ADSCrossRefGoogle Scholar - 70.
- 71.H. Na, C.T.H. Davies, E. Follana, J. Koponen, G.P. Lepage, J. Shigemitsu, Phys. Rev. D
**84**, 114505 (2011). arXiv:1109.1501 ADSCrossRefGoogle Scholar - 72.C. Bobeth, G. Hiller, G. Piranishvili, JHEP
**12**, 040 (2007). arXiv:0709.4174 ADSCrossRefGoogle Scholar - 73.R. Aaij et al., [LHCb Collaboration], Phys. Lett. B
**724**, 203 (2013). arXiv:1304.6365 - 74.S. Fajfer, S. Prelovsek, P. Singer, Phys. Rev. D
**64**, 114009 (2001). arXiv:hep-ph/0106333 ADSCrossRefGoogle Scholar - 75.S. Fajfer, N. Kosnik, S. Prelovsek, Phys. Rev. D
**76**, 074010 (2007). arXiv:0706.1133 [hep-ph]ADSCrossRefGoogle Scholar - 76.C. Delaunay, J.F. Kamenik, G. Perez, L. Randall, JHEP
**1301**, 027 (2013). arXiv:1207.0474 [hep-ph]ADSCrossRefGoogle Scholar - 77.A. Paul, A. De La Puente, I.I. Bigi, Phys. Rev. D
**90**, 014035 (2014). arXiv:1212.4849 [hep-ph]ADSCrossRefGoogle Scholar - 78.S. Davidson, G. Isidori, S. Uhlig, Phys. Lett. B
**663**, 73 (2008). arXiv:0711.3376 ADSCrossRefGoogle Scholar - 79.

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