New Physics effects in charm meson decays involving $c \to u l^+ l^- (l_i^\mp l_j^\pm)$ transitions

TL;DR

This paper investigates how scalar leptoquarks and Z' bosons could influence rare charm meson decays involving flavor-changing neutral currents, providing constraints and predictions for branching ratios and asymmetries.

Contribution

It introduces a new analysis of scalar leptoquark and Z' effects on charm meson decays, constraining parameters with experimental data and predicting observable effects.

Findings

Constrained new physics parameters using D0 → μ+μ− and oscillation data.

Predicted branching ratios and asymmetries for rare D meson decays.

Explored lepton flavor violating decay modes with potential experimental signatures.

Abstract

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

Tables

Table 1. Table 1: The allowed values of Wilson coefficients obtained from the upper bound of D 0 → μ + ​ μ − ​ ( e + ​ e − ) → superscript 𝐷 0 superscript 𝜇 superscript 𝜇 superscript 𝑒 superscript 𝑒 D^{0}\to\mu^{+}\mu^{-}(e^{+}e^{-}) process. The constraint on C ~ i L ​ Q superscript subscript ~ 𝐶 𝑖 𝐿 𝑄 \tilde{C}_{i}^{LQ} coefficients can also be applicable to C ~ i ′ ⁣ L ​ Q superscript subscript ~ 𝐶 𝑖 ′ 𝐿 𝑄 \tilde{C}_{i}^{\prime{LQ}} Wilson coefficients.

Wilson coefficient	$D^0\to {\mu }^{+}{\mu }^{-}$	$D^0\to e^{+}{e}^{-}$
${\tilde{C}}_{10}^{LQ}$	$0.8$	$600$
${\tilde{C}}_S^{LQ}$	$0.053$	$0.186$
${\tilde{C}}_P^{LQ}$	$0.053$	$0.186$

Table 2. Table 2: The predicted branching ratios for D + ( 0 ) → π + ( 0 ) ​ μ + ​ μ − → superscript 𝐷 0 superscript 𝜋 0 superscript 𝜇 superscript 𝜇 D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-} processes in both the low q 2 superscript 𝑞 2 q^{2} and high q 2 superscript 𝑞 2 q^{2} region in the scalar X ​ ( 3 , 2 , 7 / 6 ) 𝑋 3 2 7 6 X(3,2,7/6) LQ and Z ′ superscript 𝑍 ′ Z^{\prime} model. This also contains the resonant and nonresonant SM branching ratios.

	Decay process	$D^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-}$	$D^0\to {\pi }^0{\mu }^{+}{\mu }^{-}$
	Nonresonant SM	$\left(3.02\pm 0.483\right)\times {10}^{-13}$	$\left(1.19\pm 0.19\right)\times {10}^{-13}$
	Resonant SM	$\left(1.36-2.4\right)\times {10}^{-10}$	$\left(5.66-9.89\right)\times {10}^{-11}$
low $q^2$	LQ model	$\left(2.6-8.68\right)\times {10}^{-10}$	$\left(1.02-3.4\right)\times {10}^{-10}$
	$Z^{\prime }$ model	$\left(0.65-1.18\right)\times {10}^{-12}$	$\left(2.55-4.62\right)\times {10}^{-13}$
	Expt. limit ($90\%$ CL)	$2\times {10}^{-8}$ LHCb-exp	$\cdots$
	Nonresonant SM	$\left(5.14\pm 0.82\right)\times {10}^{-13}$	$\left(2\pm 0.32\right)\times {10}^{-13}$
	Resonant SM	$\left(1.25-3.29\right)\times {10}^{-10}$	$\left(0.456-1.24\right)\times {10}^{-10}$
high $q^2$	LQ model	$\left(1.32-3.36\right)\times {10}^{-9}$	$\left(0.513-1.3\right)\times {10}^{-9}$
	$Z^{\prime }$ model	$\left(1.4-2.78\right)\times {10}^{-12}$	$\left(0.545-1.08\right)\times {10}^{-12}$
	Expt. limit ($90\%$ CL)	$2.6\times {10}^{-8}$ LHCb-exp	$\cdots$

Table 3. Table 3: The predicted branching ratios for D + ( 0 ) → π + ( 0 ) ​ μ − ​ e + → superscript 𝐷 0 superscript 𝜋 0 superscript 𝜇 superscript 𝑒 D^{+(0)}\to\pi^{+(0)}\mu^{-}e^{+} lepton flavour violating processes in the scalar X ​ ( 3 , 2 , 7 / 6 ) 𝑋 3 2 7 6 X(3,2,7/6) LQ and Z ′ superscript 𝑍 ′ Z^{\prime} model. The present upper limit on the branching ratio BR ​ ( D 0 → π 0 ​ μ ∓ ​ e ± ) = BR ​ ( D 0 → π 0 ​ μ − ​ e + + π 0 ​ μ + ​ e − ) < 8.6 × 10 − 5 BR → superscript 𝐷 0 superscript 𝜋 0 superscript 𝜇 minus-or-plus superscript 𝑒 plus-or-minus BR → superscript 𝐷 0 superscript 𝜋 0 superscript 𝜇 superscript 𝑒 superscript 𝜋 0 superscript 𝜇 superscript 𝑒 8.6 superscript 10 5 {\rm BR}(D^{0}\to\pi^{0}\mu^{\mp}e^{\pm})={\rm BR}(D^{0}\to\pi^{0}\mu^{-}e^{+}+\pi^{0}\mu^{+}e^{-})<8.6\times 10^{-5} pdg .

Decay process	$D^{+}\to {\pi }^{+}{\mu }^{-}{e}^{+}$	$D^0\to {\pi }^0{\mu }^{-}{e}^{+}$
LQ model	$\left(1.67-3.72\right)\times {10}^{-11}$	$\left(0.56-1.4\right)\times {10}^{-11}$
$Z^{\prime }$ model	$\left(2.95-7.8\right)\times {10}^{-12}$	$\left(1.15-3.04\right)\times {10}^{-12}$
Experimental limit	pdg	$\cdots$

Equations

$$⟨f|\mathcal{H}|i⟩=G\sum _{i=1}{\mathrm{C}}_i(\mu )⟨f|Q_i|i⟩(\mu ),$$

$$\displaystyle\begin{array}[]{l}Q_{1}=(\overline{u}_{L}\gamma_{\mu}c_{L})\ (\overline{u}_{L}\gamma^{\mu}c_{L})\ ,\\ Q_{2}=(\overline{u}_{L}\gamma_{\mu}c_{L})\ (\overline{u}_{R}\gamma^{\mu}c_{R})\ ,\\ Q_{3}=(\overline{u}_{L}c_{R})\ (\overline{u}_{R}c_{L})\ ,\\ Q_{4}=(\overline{u}_{R}c_{L})\ (\overline{u}_{R}c_{L})\ ,\end{array}\qquad\begin{array}[]{l}Q_{5}=(\overline{u}_{R}\sigma_{\mu\nu}c_{L})\ (\overline{u}_{R}\sigma^{\mu\nu}c_{L})\ ,\\ Q_{6}=(\overline{u}_{R}\gamma_{\mu}c_{R})\ (\overline{u}_{R}\gamma^{\mu}c_{R})\ ,\\ Q_{7}=(\overline{u}_{L}c_{R})\ (\overline{u}_{L}c_{R})\ ,\\ Q_{8}=(\overline{u}_{L}\sigma_{\mu\nu}c_{R})\ (\overline{u}_{L}\sigma^{\mu\nu}c_{R})\ \ ,\end{array}$$

$${\mathcal{H}}_{eff}={\lambda }_d{\mathcal{H}}^d+{\lambda }_s{\mathcal{H}}^s+{\lambda }_b{\mathcal{H}}^{\mathrm{peng}},$$

$${\mathcal{H}}^{\mathrm{peng}}=-\frac{4G_F}{\sqrt{2}}\left(\sum _{i=3,\cdots 10,S,P}{C}_i{\mathcal{O}}_i+\sum _{i=7,\cdots 10,S,P}{C}_i^{\prime }{\mathcal{O}}_i^{\prime }\right),$$

$${\mathcal{O}}_7^{{(}^{\prime })}=\frac{e}{16{\pi }^2}{m}_c(\bar{u}{\sigma }_{\mu \nu }R(L)c)F^{\mu \nu },$$

$${\mathcal{O}}_9^{{(}^{\prime })}=\frac{e^2}{16{\pi }^2}(\bar{u}{\gamma }_{\mu }L(R)c)(\bar{\ell }{\gamma }^{\mu }\ell ),{\mathcal{O}}_{10}^{{(}^{\prime })}=\frac{e^2}{16{\pi }^2}(\bar{u}{\gamma }_{\mu }L(R)c)(\bar{\ell }{\gamma }^{\mu }{\gamma }_5\ell ),$$

$${\mathcal{O}}_S^{{(}^{\prime })}=\frac{e^2}{16{\pi }^2}(\bar{u}R(L)c)(\bar{\ell }\ell ),{\mathcal{O}}_P^{{(}^{\prime })}=\frac{e^2}{16{\pi }^2}(\bar{u}R(L)c)(\bar{\ell }{\gamma }_5\ell ),$$

$${\mathcal{O}}_T=\frac{e^2}{16{\pi }^2}(\bar{u}{\sigma }_{\mu \nu }c)(\bar{\ell }{\sigma }^{\mu \nu }\ell ),{\mathcal{O}}_{T_5}=\frac{e^2}{16{\pi }^2}(\bar{u}{\sigma }_{\mu \nu }c)(\bar{\ell }{\sigma }^{\mu \nu }{\gamma }_5\ell ).$$

$$\mathcal{L}={\bar{l}}_R{Y}^L{\Delta }^{†}Q+{\bar{u}}_R{Y}^R{\tilde{\Delta }}^{†}L+h.c.,$$

$${\mathcal{L}}^{(2/3)}$$

$${\mathcal{L}}^{(5/3)}$$

$$C_9^{\mathrm{LQ}}=C_{10}^{\mathrm{LQ}}=-\frac{\pi }{2\sqrt{2}{G}_F{\alpha }_{em}{\lambda }_b}\frac{Y_{\mu c}^L{Y}_{\mu u}^{L\ast }}{m_{\Delta }^2},$$

$$C_9^{\prime \mathrm{LQ}}=-C_{10}^{\prime \mathrm{LQ}}=-\frac{\pi }{2\sqrt{2}{G}_F{\alpha }_{em}{\lambda }_b}\frac{Y_{c\mu }^{R\ast }{Y}_{u\mu }^R}{m_{\Delta }^2},$$

$$C_S^{\mathrm{LQ}}=C_P^{\mathrm{LQ}}=-\frac{\pi }{2\sqrt{2}{G}_F{\alpha }_{em}{\lambda }_b}\frac{Y_{\mu u}^{L\ast }{Y}_{c\mu }^{R\ast }}{m_{\Delta }^2},$$

$$C_S^{\prime \mathrm{LQ}}=-C_P^{\prime \mathrm{LQ}}=-\frac{\pi }{2\sqrt{2}{G}_F{\alpha }_{em}{\lambda }_b}\frac{Y_{\mu c}^L{Y}_{u\mu }^R}{m_{\Delta }^2},$$

$$C_T^{\mathrm{LQ}}=-\frac{\pi }{8\sqrt{2}{G}_F{\alpha }_{em}{\lambda }_b}\frac{Y_{u\mu }^R{Y}_{\mu c}^L+{Y_{c\mu }^R}^{\ast }{Y_{\mu u}^L}^{\ast }}{m_{\Delta }^2},$$

$$C_{T_5}^{\mathrm{LQ}}=-\frac{\pi }{8\sqrt{2}{G}_F{\alpha }_{em}{\lambda }_b}\frac{-Y_{u\mu }^R{Y}_{\mu c}^L+{Y_{c\mu }^R}^{\ast }{Y_{\mu u}^L}^{\ast }}{m_{\Delta }^2},$$

$${\mathcal{H}}_{eff}=\sum _l\frac{(Y_{lc}^R{Y}_{lu}^{R\ast }{)}^2}{128{\pi }^2}\left[\frac{1}{M_{\Delta }^2}I\left(\frac{m_l^2}{M_{\Delta }^2}\right)+\frac{1}{M_{\Delta }^2}\right](\bar{c}{\gamma }^{\mu }{P}_{R}u)(\bar{c}{\gamma }_{\mu }{P}_{R}u),$$

$$I(x)=\frac{1-x^2+2x\log x}{(1-x{)}^2},$$

$$⟨{\bar{D}}^0|(\bar{c}{\gamma }^{\mu }{P}_{R}u)(\bar{c}{\gamma }_{\mu }{P}_{R}u)|D^0⟩=\frac{2}{3}{B}_D{f}_D^2{M}_D^2,$$

$$M_{12}^{LQ}=\frac{1}{2M_D}⟨{\bar{D}}^0|{\mathcal{H}}_{eff}|D^0⟩=\frac{{\sum }_l(Y_{lc}^R{Y}_{lu}^{R\ast }{)}^2}{192{\pi }^2{M}_{\Delta }^2}{B}_D{f}_D^2{M}_D$$

$$\Delta M_D=2|M_{12}^{LQ}|=\frac{2}{3}{M}_D{f}_D^2{B}_D\frac{{\left|{\sum }_l{Y}_{lc}^R{Y_{lu}^R}^{\ast }\right|}^2}{64{\pi }^2{M}_{\Delta }^2},$$

$$\Delta M_D=0.0095_{-0.0044}^{+0.0041}{\mathrm{ps}}^{-1}.$$

$$\displaystyle 7.73\times 10^{-3}\Big{(}\frac{M_{\Delta}}{1~{}{\rm TeV}}\Big{)}\leq|Y_{\mu c}^{R}{Y_{\mu u}^{R}}^{*}|\leq 1.26\times 10^{-2}\Big{(}\frac{M_{\Delta}}{1~{}{\rm TeV}}\Big{)},$$

$$\displaystyle 0.1~{}\Big{(}\frac{M_{\Delta}}{1~{}{\rm TeV}}\Big{)}\leq\lambda_{b}{C}_{9}^{\prime LQ}=-\lambda_{b}{C}_{10}^{\prime LQ}\leq 0.17~{}\Big{(}\frac{M_{\Delta}}{1~{}{\rm TeV}}\Big{)}.$$

$$\mathrm{BR}\left(D^0\to l^{+}{l}^{-}\right)$$

$$\mathrm{BR}\left(D^0\to {\mu }^{+}{\mu }^{-}\right)<6.2\times 10^{-9},\mathrm{BR}\left(D^0\to e^{+}{e}^{-}\right)<7.9\times 10^{-8}.$$

$${\tilde{C}}_i^{{(}^{\prime })LQ}={\lambda }_b{C}_i^{{(}^{\prime })LQ},$$

$$\displaystyle\Big{|}\tilde{C}_{S}^{LQ}-\tilde{C}_{S}^{\prime LQ}\Big{|}~{}\leqslant~{}0.06,~{}~{}~{}~{}\Big{|}\tilde{C}_{S}^{LQ}+\tilde{C}_{S}^{\prime LQ}\Big{|}~{}\leqslant~{}0.06,$$

$$\displaystyle\Big{|}\tilde{C}_{S}^{LQ}-\tilde{C}_{S}^{\prime LQ}\Big{|}~{}\leqslant~{}0.2,~{}~{}~{}~{}\Big{|}\tilde{C}_{S}^{LQ}+\tilde{C}_{S}^{\prime LQ}\Big{|}~{}\leqslant~{}0.2.$$

$${\tilde{C}}_9^{LQ}=-{\tilde{C}}_{10}^{\prime LQ}=0.8,{\tilde{C}}_S^{LQ}={\tilde{C}}_P^{LQ}=4{\tilde{C}}_T^{LQ}=4{\tilde{C}}_{T_5}^{LQ}=-\mathrm{0.053.}$$

$${\mathcal{H}}_{Z^{\prime }}^{\mathrm{FCNC}}={\mathcal{H}}_{Z^{\prime }}^q=g_{Z^{\prime }1}{\overline{u}}_L{\gamma }_{\mu }{c}_L{Z}^{\prime \mu }+g_{Z^{\prime }2}{\overline{u}}_R{\gamma }_{\mu }{c}_R{Z}^{\prime \mu }.$$

$${\mathcal{H}}_{Z^{\prime }}^L=g_{Z^{\prime }1}^{\prime }{\overline{\ell }}_L{\gamma }_{\mu }{\ell }_L{Z}^{\prime \mu }+g_{Z^{\prime }2}^{\prime }{\overline{\ell }}_R{\gamma }_{\mu }{\ell }_R{Z}^{\prime \mu }.$$

$$\Delta M_{\mathrm{D}}^{({\mathrm{Z}}^{\prime })}=\frac{f_D^2{M}_D{B}_D}{2M_{Z^{\prime }}^2}\left[\frac{2}{3}\left(c_1(m_c)+c_6(m_c)\right)-\left(\frac{1}{2}+\frac{\eta }{3}\right)c_2(m_c)+\left(\frac{1}{12}+\frac{\eta }{2}\right)c_3(m_c)\right].$$

$$c_1(m_c)=r(m_c,M_{Z^{\prime }})g_{Z^{\prime }1}^2,c_3(m_c)=\frac{4}{3}\left[r(m_c,M_{Z^{\prime }}{)}^{1/2}-r(m_c,M_{Z^{\prime }}{)}^{-4}\right]g_{Z^{\prime }1}{g}_{Z^{\prime }2},$$

Full text

I introduction

The rare $B$ and $D$ meson decay processes driven by 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 $b\to sl^{+}l^{-}$, where several anomalies at the level of few sigma have been observed recently in the LHCb experiment lhcb10 . To compliment 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 $c\to ul^{+}l^{-}$ quark level transitions. Recently LHCb experiment has searched for the branching ratio of the lepton flavour violating (LFV) $D^{0}\to\mu^{\mp}e^{\pm}$ decays and put the limit as $\rm{BR}(D^{0}\to\mu^{\mp}e^{\pm})~{}\textless~{}1.3\times 10^{-8}$ D0exp at $90\%$ confidence level (CL). On the other hand, both Belle and BaBar experiments have reported significant deviations on the measured branching fractions of $\bar{B}\to D^{(*)}\tau\nu_{\tau}$ processes from the corresponding SM predictions. The ratio of these branching fractions, the so-called $R(D^{(*)})$, defined as $R(D^{(*)})={\rm BR}(\bar{B}\to D^{(*)}\tau\nu_{\tau})/{\rm BR}(\bar{B}\to D^{(*)}l\nu_{l})$, where $l=e,\mu$, exceed the SM prediction by $3.5\sigma$ RDstar , thus open 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 up of down type quarks, had provided information about the charm and top quark mass scale, much before the discovery of these particles in the collider. On the other hand, $D^{0}-\bar{D}^{0}$ 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 $c\to u\mu^{+}\mu^{-}$ and $c\to u\mu^{\mp}e^{\pm}$ FCNC transitions. We calculate the branching ratios, forward-backward asymmetry parameters and the flat terms in $D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-}$ processes both in the scalar leptoquark (LQ) and generic $Z^{\prime}$ model. These processes suffer from resonance background through $c\to uM\to ul^{+}l^{-}$, where $M$ denotes $\eta^{(^{\prime})}$ (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., $q^{2}\in[0.0625,0.275]{\rm GeV}^{2}$ and $q^{2}\in[1.56,4.00]{\rm GeV}^{2}$, which lie outside the mass square range of the resonant mesons. We also compute the branching ratios of lepton flavour violating $D^{0}\to\mu e,\tau e$ and $D^{+(0)}\to\pi^{+(0)}\mu^{-}e^{+}$ processes. These LFV processes have negligible contributions from 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 color 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 explain several observed anomalies, e.g., lepton non-universality (LNU) parameter $R_{K}|_{q^{2}\in[1,6]~{}{\rm GeV}^{2}}={\rm BR}(B\to K\mu^{+}\mu^{-})/{\rm BR}(B\to Ke^{+}e^{-})$ in rare $B$ decays. The existence of scalar leptoquark is predicted in the extended SM theories, such as grand unified theory georgi ; georgi2 , Pati-Salam model, extended technicolor model schrempp and the composite model kaplan . 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 $D^{0}$ meson induced by $c\to ul^{+}l^{-}$ transitions. The phenomenology of scalar leptoquarks and their implications to the $B$ and $D$-sector has been extensively studied in the literature Arnold ; mohanta1 ; davidson ; leptoquark ; Hiller ; kosnik ; kosnik2 ; D0-D0bar ; ansatz1 ; Hiller3 .

The $Z^{\prime}$ boson is a color singlet vector gauge boson and 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 $c\to u$ FCNC transitions could be induced by generic $Z^{\prime}$ model at tree level. The theoretical framework of the heavy new $Z^{\prime}$ gauge boson has been studied in the literature Pakvasa ; Pakvasa2 ; Arhrib . 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 $D^{0}\to\mu^{+}\mu^{-}$ processes.

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

II 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 or one unit, and hence, involve either $\Delta C=2$ or $\Delta C=1$ transitions. The $D^{0}-\bar{D}^{0}$ mixing takes place via $\Delta C=2$ transition and the decay processes with $\Delta C=1$ transitions are $c\to ul^{+}l^{-}$ and $c\to u\gamma$.

If we integrate out the heavy degrees of freedom associated with the new interactions at a scale $M$, 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 $D^{0}\to l^{+}l^{-}$ process as Pakvasa ; Pakvasa2

[TABLE]

where $G$ has inverse-mass squared dimensions, $C_{i}$ are the Wilson coefficients 111We denote the Wilson coefficients for $\Delta C=2$ operators as ${c_{i}}$ and those for $\Delta C=1$ operators as ${C_{i}}$ through out this work.

The effective operators for $D^{0}-\bar{D}^{0}$ mixing at the heavy mass scale $M$ are given by Pakvasa ; Pakvasa2

[TABLE]

where $q_{L(R)}=L(R)q$ are the chiral quark fields with $L(R)=(1\mp\gamma_{5})/2$ as the projection operators.

In the standard model, the effective weak Hamiltonian for $c\to u$ transitions at the scale $\mu=m_{c}$, can be written as the sum of three contributions as kosnik ; faj-ham

[TABLE]

where $\lambda_{q}=V_{uq}V_{cq}^{*}$ is the product of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The explicit form of $\mathcal{H}^{\rm peng}$, which basically responsible for the $c\to ul^{+}l^{-}$ transition is given by

[TABLE]

where $G_{F}$ is the Fermi constant, $C_{i}$’s are the Wilson coefficients evaluated at the charm quark mass scale $(\mu=m_{c})$ at Next-Next-to-Leading-Order (NNLO) wilson . We use the two loop result of Ref. wilson2 for the $C_{7}^{\rm eff}(m_{c})$ Wilson coefficients, $V_{cb}^{*}V_{ub}C_{7}^{\rm eff}=V_{cs}^{*}V_{us}(0.007+0.020i)(1\pm 0.2)$ and the corresponding effective operators for $c\to ul^{+}l^{-}$ transitions are given as kosnik

[TABLE]

The contributions from the primed operators as well as the scalar, pseudoscalar and tensor operators are absent in the SM and arise only in beyond the standard model scenarios. The renormalization group running does not affect the $\mathcal{O}_{10}$ operator, i.e., $C_{10}(m_{c})=C_{10}(M_{W})\propto(m_{d,s}^{2}/m_{W}^{2})$ and hence, the Wilson coefficient $C_{10}$ is negligible in the SM.

III New physics contribution due to the exchange of scalar leptoquarks

The presence of leptoquarks can modify the SM effective Hamiltonian of $c\to u$ transitions, giving appreciable deviation from the SM values. These color triplet bosons can be either scalars or vectors. There exist three scalar and four vector relevant leptoquark states which potentially contribute to the $c\to ul^{+}l^{-}$ 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 $(3,2,7/6)$ kosnik ; kosnik2 , 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 first two leptoquark states don’t 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.

The interaction Lagrangian of $X=(3,2,7/6)$ scalar leptoquark with the SM bilinears is given by kosnik ; kosnik2

[TABLE]

where $\tilde{\Delta}=i\tau_{2}\Delta^{*}$ represents the conjugate state. The transition of weak basis to mass basis divides the Yukawa couplings to two part of couplings pertinent for the upper and lower doublet components. The left handed quark and lepton doublets are represented by $Q$ 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., $d_{L}\to V_{CKM}d_{L}$ and $\nu_{L}\to V_{PMNS}\nu_{L}$. 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 Eqn. (14), one can obtain the interaction Lagrangian for different components of LQs given as kosnik2

[TABLE]

Thus, one can see from (15), that only $\Delta^{(5/3)}$ component mediates the interaction between up-type quarks and charged lepton. Now applying the Fierz transformation, we obtain additional contributions to the SM Wilson coefficients for $c\to u\mu^{+}\mu^{-}$ transition as kosnik

[TABLE]

where $\alpha_{em}$ is the fine structure constant. After having an idea about the new Wilson coefficients, we now proceed to constrain the combination of LQ couplings using the experimental data on $D^{0}-\bar{D}^{0}$ mixing and $D^{0}\to l^{+}l^{-}$ process, where $l=\mu,e$.

III.1 Constraint on leptoquark couplings from $D^{0}-\bar{D}^{0}$ mixing

In the standard model, $D^{0}-\bar{D}^{0}$ mixing proceeds through the box diagrams with an internal down-type quarks and $W$-boson exchange and the weak interaction boxes are suppressed due to GIM mechanism because of the smallness of down-quark mass in comparison to the weak scale. In the LQ model, there will be 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 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 D0-D0bar

[TABLE]

where the first term is due the charged lepton and second term is due to neutrinos in the loop (ignoring the effect of neutrino mixing). The loop function $I(x)$ is given as

[TABLE]

which is very close to 1, i.e., $I(0)=1$. Using the relation

[TABLE]

we obtain the contribution due to leptoquark exchange as

[TABLE]

Since $\Delta M_{D}=2|M_{12}|$, we get

[TABLE]

where $l$ denotes the charged lepton flavours. In our analysis, the mass of $D^{0}$ meson is taken from pdg , the value of the decay constant $f_{D}=222.6\pm 16.7^{+2.3}_{-2.4}$ MeV cleo and $B_{D}(3~{}{\rm GeV})=0.757(27)(4)$ BD . To obtain the bound on the leptoquark coupling, we assume that individual leptoquark contribution to the mass difference does not exceed the $1\sigma$ range of the experimental value. Since we are interested to obtain the bounds on $Y_{\mu c}^{R}Y_{\mu u}^{R*}$ couplings, here we assume that 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 pdg

[TABLE]

Now comparing the mass difference with the $1\sigma$ range of experimental data, the bound on leptoquark coupling for a TeV scale LQ is given by

[TABLE]

which can be translated with Eqn. (16) to give the constraint on new Wilson coefficients as

[TABLE]

III.2 Constraint from $D^{0}\to\mu^{+}\mu^{-}(e^{+}e^{-})$ process

The rare leptonic $D^{0}\to\mu^{+}\mu^{-}(e^{+}e^{-})$ processes, mediated by FCNC transitions $c\to ul^{+}l^{-}$ at the quark level, are highly suppressed in SM due to negligible $C_{10}$ Wilson coefficient and also suffer from CKM suppression. These processes occur only at one-loop level and are considered as some of the most powerful channels to constrain the new physics parameter space in the charm-sector. Analogous to the leptonic $B$ meson decay processes, the only non-perturbative quantity involved is the decay constant of $D$ 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 $D^{0}\to l^{+}l^{-}$ process is given by Hiller ; kosnik

[TABLE]

The $D^{0}\to\mu^{+}\mu^{-}$ process has dominant intermediate $\gamma^{*}\gamma^{*}$ state in the SM, which is electromagnetically converted to a $\mu^{+}\mu^{-}$ pair. After including the contribution of $\gamma^{*}\gamma^{*}$ intermediate state, the predicted branching ratio of this process is ${\rm BR}(D^{0}\to\mu^{+}\mu^{-})\simeq 2.7\times 10^{-5}\times{\rm BR}(D^{0}\to\gamma\gamma)$ D0-gamma . Using the upper bound ${\rm BR}(D^{0}\to\gamma\gamma)<2.2\times 10^{-6}$ at $90\%$ CL reported in BaBar , the estimated limit on branching ratio is ${\rm BR}\left(D^{0}\to\mu^{+}\mu^{-}\right)^{SM}\lesssim 10^{-10}$ kosnik . The present experimental limits on the branching ratios of dileptonic decays of $D$ meson are pdg

[TABLE]

Using the above experimental bounds, the constraint on leptoquark coupling can be obtained by imposing the condition that individual leptoquark contribution to the branching ratio does not exceed the experimental limit. In this analysis, we neglect the new physics contribution to the Wilson coefficient $C_{10}$, as the scalar and pseudoscalar Wilson coefficients will be dominating due to the additional multiplication factor $M_{D}/m_{l}$ as noted from Eqn. (25). Now, redefining the Wilson coefficients as

[TABLE]

we show in Fig. 1, the allowed region in $\tilde{C}_{S}^{LQ}-\tilde{C}_{S}^{\prime LQ}$, $\tilde{C}_{S}^{LQ}+\tilde{C}_{S}^{\prime LQ}$ plane, obtained from $D^{0}\to\mu^{+}\mu^{-}$ (left panel) and $D^{0}\to e^{+}e^{-}$ processes (right panel). Here we have used the relations $C_{S}^{LQ}=C_{P}^{LQ}$ and $C_{S}^{{}^{\prime}LQ}=-C_{P}^{{}^{\prime}LQ}$ from Eqn. (16). From the figure, we found the allowed range for the above combinations of Wilson coefficients from $D^{0}\to\mu^{+}\mu^{-}$ process as

[TABLE]

whereas the bounds obtained from $D^{0}\to e^{+}e^{-}$ process is rather weak, i.e.,

[TABLE]

It is obvious that the bounds obtained in Eqns. (28) and (29) could not give us proper information about the bounds on individual $\tilde{C}_{S}^{LQ}$ and $\tilde{C}_{S}^{\prime LQ}$ coefficients. Therefore, we consider only one Wilson coefficient at a time to extract the upper bound on individual coefficients. In Table I, we report the constraint on $\tilde{C}_{S,P}^{LQ}$ Wilson coefficients obtained from the experimental bound on the branching fraction of $D^{0}\to\mu^{+}\mu^{-}(e^{+}e^{-})$ process. The bounds on $\tilde{C}_{i}^{\prime LQ}$ Wilson coefficients will be same as those for $\tilde{C}_{i}^{LQ}$.

If we impose chirality on scalar leptoquarks i.e., they couple to either left-handed or right-handed quarks, but not to both then the $C_{S,P}^{(^{\prime})}$ Wilson coefficients will vanish and we get only the additional contribution of $C_{9,10}^{(^{\prime})LQ}$ 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 I.

In order to evade the strict bounds in the down type quark sector, we consider the leptoquark couplings to the left handed quarks ($Y^{L}$) as zero. Therefore, the only contribution to the rare charm decays comes from $\tilde{C}_{9}^{\prime}=-\tilde{C}_{10}^{\prime}$ Wilson coefficients, which are related to the right-handed quark couplings. Now to include (pseudo)scalar and (pseudo)tensor Wilson coefficients and to extract respective upper bound complying with the constraints from $B$ and $K$ physics, we consider a numerically tuned example as discussed in kosnik . We assume that the $Y^{R}$ coupling is perturbative, i.e., $|Y^{R}|~{}\textless~{}\sqrt{4\pi}$. 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 $\tilde{C}_{10}^{\prime LQ}$ Wilson coefficients from $D^{0}\to\mu^{+}\mu^{-}$ process, which is found to be small comparatively, $Y_{u\mu}^{R}~{}\textless~{}8.76\times 10^{-3}$. Now we instigate a nonzero coupling to the left handed quark $Y_{u\mu}^{L}$, which along with the large $Y_{c\mu}^{R}$ coupling provides nonzero values for $C_{S,P}$ and $C_{T,T_{5}}$ coefficients. However, the $D^{0}\to\mu^{+}\mu^{-}$ process imposes strong bound on $C_{S}$ coefficient, which together with large $Y_{c\mu}^{R}$ coupling, limits the left handed coupling as $Y_{\mu u}^{L}~{}\textless~{}1.14\times 10^{-3}$. Thus, from the above discussion we observe that

[TABLE]

Our predicted bound on leptoquark coupling are in agreement with the constraints obtained in Refs. kosnik ; Hiller and also with the constraints obtained from $B$, $K$ physics carp .

IV New physics contribution in $Z^{\prime}$ model

The new heavy $Z^{\prime}$ gauge boson can exist in many extended SM scenarios and can mediate the FCNC transitions among the fermions in the up-quark sector at tree level. The most general Hamiltonian for $c\to u$ transition in the $Z^{\prime}$ model is given as Pakvasa2

[TABLE]

Analogously, one can write the Hamiltonian for the leptonic sector ${\cal H}^{l}_{Z^{\prime}}$ as

[TABLE]

Here $g_{Z^{\prime}i}$ and $g_{Z^{\prime}i}^{\prime}$ are the couplings of $Z^{\prime}$ boson with the quarks and leptons respectively, where $i=1$ or 2 for the $Z^{\prime\mu}$ vector boson coupled to left handed or right-handed currents.

After knowing 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 $D^{0}\to l^{+}l^{-}$ processes. The constraint on the coupling of $Z^{\prime}$ with the leptonic part is obtained from the upper limit on branching ratio of lepton flavour violating $\tau(\mu)^{-}\to e^{-}e^{+}e^{-}$ processes.

IV.1 Constraint from $D^{0}-\bar{D}^{0}$ mixing

In this subsection, we calculate the constraint on $Z^{\prime}$ couplings from the mass difference of charm meson mass eigenstates, which characterizes the $D^{0}-\bar{D}^{0}$ mixing phenomena. The $D^{0}-\bar{D}^{0}$ oscillation arises from $|\Delta C=2|$ transition that generates off-diagonal terms in the mass matrix for $D^{0}$ and $\bar{D}^{0}$ mesons. The mass difference of $D^{0}-\bar{D}^{0}$ mixing at the scale $\mu=m_{c}$ is given by Pakvasa2

[TABLE]

At the charm mass scale, the Wilson coefficients in terms of $Z^{\prime}$ couplings are expressed as

[TABLE]

where $r(m_{c},M_{Z^{\prime}})$ is the RG factor at the heavy mass scale and $r(m_{c},M_{Z^{\prime}})=0.72~{}(0.71)$ for $Z^{\prime}$ mass, $M_{Z^{\prime}}=1(2)$ TeV Pakvasa .

Now we consider two possible cases to constrain the couplings $g_{Z^{\prime}1}$ and $g_{Z^{\prime}2}$. One with only left handed coupling present, i.e., ($g_{Z^{\prime}2}=0$) and the second where both left handed and right handed couplings are present with equal strength ($g_{Z^{\prime}1}=g_{Z^{\prime}2}=g_{Z^{\prime}}$). Here, we make the simple assumption that the NP part dominates over the SM contribution in $D^{0}-\bar{D}^{0}$ mixing. Thus, for the first case, substitution of $g_{Z^{\prime}2}=0$ in Eqns. (33) and (34), the mass difference becomes

[TABLE]

Now varying the mass difference $\Delta M_{\rm D}$ within its $1\sigma$ allowed range pdg , we obtain

[TABLE]

and for a representative $Z^{\prime}$ mass $M_{Z^{\prime}}=1$ TeV, the value of the coupling is found to be

[TABLE]

Analogously for the second case, i.e., $g_{Z^{\prime}1}=g_{Z^{\prime}2}=g_{Z^{\prime}}$, the constraint obtained as

[TABLE]

IV.2 Constraint from $D^{0}\to\mu^{+}\mu^{-}$ process

The effective Hamiltonian for $D^{0}\to l^{+}l^{-}$ process in the $Z^{\prime}$ model is given by Pakvasa2

[TABLE]

where the operators $\widetilde{Q}_{1,2}$ are

[TABLE]

and $\widetilde{Q}_{6,7}$ can be obtained from $\widetilde{Q}_{1,2}$ by the substitutions of $q_{L}\to q_{R}$ and $q_{R}\to q_{L}$.

Comparing Eqn. (39) with the SM effective Hamiltonian (12), yields the additional contributions to Wilson coefficients $C_{9,10}^{(\prime)Z^{\prime}}$ as

[TABLE]

The branching ratio for $D^{0}\to\mu^{+}\mu^{-}$ process in the $Z^{\prime}$ model is given as Pakvasa2

[TABLE]

For simplicity, we consider here $g_{Z^{\prime}2}=0$. Now considering the couplings of $Z^{\prime}$ boson to the final leptons as the same form as the SM-like diagonal couplings of $Z$ boson to leptons as discussed in Pakvasa2 , i.e.,

[TABLE]

where $g$ is the gauge coupling of $Z$ boson and $\theta_{W}$ is the weak mixing angle. Now using the experimental upper limit on branching ratio ${\rm BR}(D^{0}\to\mu^{+}\mu^{-})~{}\textless~{}6.2\times 10^{-9}$ pdg , we obtain

[TABLE]

For $M_{Z^{\prime}}=1$ TeV, the constraint on $g_{Z^{\prime}1}$ coupling is

[TABLE]

which is rather weak compared to the constraint obtained from $D^{0}-\bar{D}^{0}$ mixing.

It should be noted that the constraint on $Z^{\prime}$ couplings from the $D^{0}\to\mu^{+}\mu^{-}$ decay process and the $D^{0}-\bar{D}^{0}$ mixing data have been computed in Pakvasa2 . Similarly, the constraint on the couplings from $D^{0}-\bar{D}^{0}$ oscillation are obtained in Ref. Pakvasa . We found that our constraints are consistent with the above predictions, if we use the updated values of various input parameters.

IV.3 Constraints on $g_{Z1}^{\prime}$ from $\tau^{-}(\mu^{-})\to e^{-}e^{+}e^{-}$ process

Considering only the left handed coupling to the $Z^{\prime}$ boson, the branching ratio of $\mu^{-}\to e^{-}e^{+}e^{-}$ process in the $Z^{\prime}$ model is given by matias ; mu-to-3e

[TABLE]

where we have explicitly shown the indices in the couplings. The experimental upper limit on branching ratio of this mode is ${\rm BR}(\mu^{-}\to e^{-}e^{+}e^{-})<10^{-12}$ pdg . For the analysis, we use the mass and lifetime of muon from pdg and consider the coupling ${g^{\prime}}_{ee}^{L}=g_{Z^{\prime}1}^{\prime}$ as SM-like with its value as presented in Eqn.(43). Thus, using the experimental upper limit, we get the bound on ${g^{\prime}}_{\mu e}^{L}$ coupling as

[TABLE]

Analogously using the branching ratio of $\tau^{-}\to e^{-}e^{+}e^{-}$ process, ${\rm BR}(\tau^{-}\to e^{-}e^{+}e^{-})<2.7\times 10^{-8}$ pdg , the constraint on lepton flavour violating ${g^{\prime}}_{\tau e}^{L}$ coupling is found to be

[TABLE]

V $D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-}$ process

In this section, we study the rare semileptonic decay process $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$, which is mediated by the quark level transition $c\to u\mu^{+}\mu^{-}$ and constitutes a suitable tool to search for new physics. The dominant resonance contributions come from the $\phi$, $\rho$ and $\omega$ vector mesons and the effects of $\eta^{(^{\prime})}$ mesons are comparatively negligible. These decay modes are recently studied in Refs. Hiller ; kosnik ; rim in various new physics scenarios and it is found that the model with scalar/vector leptoquarks and minimal supersymmetric model with R-parity violation can give significant contributions. The matrix elements of various hadronic currents between the initial $D$ meson and the final $\pi$ meson can be parametrized in terms of three form factors $f_{0}$, $f_{+}$ and $f_{T}$ kosnik as

[TABLE]

[TABLE]

where $p_{D}$ and $k$ are the four momenta of the $D$ and $\pi$ mesons respectively and $q=p_{D}-k$ is the momentum transfer. The form factors for $D^{0}\to\pi^{0}$ are scaled as $f_{i}\to f_{i}/\sqrt{2}$ by isospin symmetry. For the $q^{2}$-dependence of the form factors, we use the parameterization from Refs. formfactor1 ; formfactor2 , as

[TABLE]

where $x=q^{2}/m_{pole}^{2}$ with $m_{pole}=1.90(8)$ GeV, $a=0.28(14)$ and $b=1.27(17)$ are the shape parameters kosnik measured from $D\to\pi l\nu$ decay process and $f_{+}(0)=0.67(3)$ fplus . The parameters in the $f_{T}$ form factor are: $x_{T}=q^{2}/M_{D^{*}}^{2}$, $f_{T}(0)=0.46(4)$ and $a_{T}=0.18(16)$ formfactor2 . Thus, one can write the transition amplitude for $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process as Hiller2 ; kosnik

[TABLE]

where $\theta$ is the angle between the $D$ 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

[TABLE]

Here $C_{9,10}^{(^{\prime})NP}$, $C_{S,P}^{(^{\prime})NP}$ and $C_{T,T_{5}}^{NP}$ are the new Wilson coefficients arising from either the scalar leptoquark model or the generic $Z^{\prime}$ model. Using Eqn. (52), the double differential decay distribution with respect to $q^{2}$ and $\theta$, for the lepton flavour $l$ is given by kosnik ; Hiller2

[TABLE]

where

[TABLE]

with

[TABLE]

and

[TABLE]

Thus, the branching ratio is given by

[TABLE]

The forward-backward asymmetry ($A_{FB}$) is another useful observable to look for new physics, which is defined as kosnik

[TABLE]

Since the coefficient $b_{l}$ depends only on scalar and pseudoscalar Wilson coefficients the forward-backward asymmetry is zero in SM. However, the additional new physics contribution can give non-zero contribution to the forward-backward asymmetry parameter. Another interesting observable is the flat term, defined as Hiller2

[TABLE]

where the uncertainties get reduced due to the cancelation between the numerator and denominator.

For numerical evaluation, we take the particle masses and the lifetime of $D$ meson from pdg . 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 $\bar{\eta}=0.353\pm 0.013$ pdg . With these input parameters, we compute the resonant/non-resonant branching ratios of $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process by integrating the decay distribution with respect to $q^{2}$. We parametrize the contributions from the resonances with the Breit-Wigner shapes for $C_{9}\to C_{9}^{\rm res}$, for $\rho,\omega,\phi$ (vector) and $C_{P}\to C_{P}^{\rm res}$ for $\eta^{(\prime)}$ (pseudoscalar) mesons as Hiller ; kosnik

[TABLE]

Here $m_{M}~{}(\Gamma_{M})$ denotes the mass (total decay width) of the resonant state $M$, where $M$ corresponds to $\eta^{(\prime)},\rho,\omega,\phi$ mesons. With the approximation of ${\rm BR}(D^{+}\to\pi^{+}M(\to\mu^{+}\mu^{-}))\simeq{\rm BR}(D^{+}\to\pi^{+}M){\rm BR}(M\to\mu^{+}\mu^{-})$ and considering the experimental upper bound from pdg , the magnitudes of the Breit-Winger parameters are given by Hiller

[TABLE]

The detailed procedure of SM resonant contributions to $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process can be found in formfactor2 ; kosnik ; Hiller . In Fig. 2, we show the $q^{2}$ variation of branching ratio of $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process including the resonant contribution in the SM. The band in the figure is due to the uncertainties associated with the $a_{M}$ parameters as given in (V) and the random variation of relative phases within $-\pi$ and $\pi$. For simplicity we have assumed the same phase for all the resonances. From the figure, one can observe that in the low and high $q^{2}$ regions the long distance resonant contributions are approximately one order of magnitude below the current experimental sensitivity, and hence these regions are suitable to look for new physics beyond the SM. Thus, both in the SM and in the leptoquark and $Z^{\prime}$ models, we study the $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process only at the very low and high $q^{2}$ regimes. However, it should be emphasized that the uncorrelated variation of the unknown resonant phases affects the branching ratio in the low $q^{2}$ region significantly, which makes it quite difficult to infer the possible role of new physics.

With all the input parameters from pdg along with the SM Wilson coefficients wilson ; wilson2 , we present in Table II, the predicted values of branching ratios for the $D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-}$ processes by integrating the decay distribution in low and high $q^{2}$ bins. Here we have used the $q^{2}$ regimes as $q^{2}\in[0.0625,0.275]~{}{\rm GeV}^{2}$ and $q^{2}\geq 1.56~{}{\rm GeV}^{2}$ to reduce the background coming from the dominant resonances. The theoretical uncertainties in the SM are associated with the lifetime of $D$ meson, CKM matrix elements and the hadronic form factors. In Fig. 3, the variation of SM branching ratios of $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process in the very low and high $q^{2}$ regimes are shown in red dashed lines and the green bands represent the SM theoretical uncertainties.

Now using the constraint on the leptoquark parameter space obtained in section III, we show in Fig. 3, the $q^{2}$ variation of branching ratio of $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process in low $q^{2}$ (left panel) and high $q^{2}$ (right panel) both in the scalar leptoquark and $Z^{\prime}$ models. Here the orange (blue) band represents the contributions from the scalar leptoquark ($Z^{\prime}$) model. The $90\%$ CL experimental upper bounds on the branching ratios from LHCb-exp

[TABLE]

are shown in thick black lines. In Table II, we present the integrated branching ratios of $D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-}$ processes in both the low and high $q^{2}$ regions in the leptoquark and $Z^{\prime}$ models. We find that the predicted branching ratios in the leptoquark model have significant deviations from the corresponding SM values due to the effect of scalar leptoquark and are well below the experimental upper limits. However the effect of $Z^{\prime}$ boson to the branching ratios of $D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-}$ processes is very marginal.

In the leptoquark model, the variation of forward-backward asymmetry for $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process in low $q^{2}$ (left panel) and high $q^{2}$ (right panel) is presented in Fig. 4. The forward-backward asymmetry depends on the combinations of $C_{S}^{(\prime)}$ and $C_{T,T_{5}}$ Wilson coefficients, thus have zero value in the SM. However, the additional contributions of $C_{S,P}^{{}^{\prime}}$ Wilson coefficients due to scalar leptoquark exchange give non-zero contribution to the forward-backward asymmetry, though it is not so significant. The integrated forward-backward asymmetry for $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process is given as

[TABLE]

The $Z^{\prime}$ model provides additional contributions only to the $C_{9,10}$ Wilson coefficients, and there are no new contributions to scalar or tensor terms. Thus, the forward-backward asymmetry vanishes in the $Z^{\prime}$ model. In both the LQ and $Z^{\prime}$ model, the plot for flat term of $D^{+}\to\pi^{+}\mu^{+}\mu^{-}$ process with respect to low $q^{2}$ (left panel) and high $q^{2}$ (right panel) is given in Fig. 5. The predicted values in low $q^{2}$ range are

[TABLE]

and in the region of high $q^{2}$

[TABLE]

In addition to the leptoquark and $Z^{\prime}$ models, the rare charm meson decays mediating by the $c\to u$ transitions have also been investigated in various new physics models such as, Minimal Supersymmetric Standard Model D0-gamma ; rim ; Fajfer2 ; Fajfer3 , two Higgs doublet model Fajfer2 , warped extra dimensions model Fajfer3 and the up vector like quark singlet model Delaunay . In the Ref. Hiller ; kosnik , the $D\to\pi\mu^{+}\mu^{-}$ process is studied in the context of both scalar and vector leptoquark models. Our predicted results are found to be consistent with the literature.

VI $D^{+(0)}\to\pi^{+(0)}\mu^{-}e^{+}$

Since the individual lepton flavour number is conserved in the standard model, the observation of lepton flavour violation in the near future will provide unambiguous signal of new physics beyond the SM. The observation of neutrino oscillation implies the violation of lepton flavour in neutral sector and it is expected that there could be FCNC transitions in the charged lepton sector as well, such as $l_{i}\to l_{j}\gamma$, $l_{i}\to l_{j}l_{k}\bar{l}_{k}$, $B\to l_{i}^{\mp}l_{j}^{\pm}$ and $B\to K^{(*)}l_{i}^{\mp}l_{j}^{\pm}$ etc. The LFV decay modes proceed through box diagrams with tiny neutrino masses in the loop, thus become very rare in the SM. However, these modes can occur at tree level in the leptoquark and $Z^{\prime}$ models, thus can provide observable signature in the high luminosity experiments. In this section, we would like to study the lepton flavour violating semileptonic decay process $D^{+}\to\pi^{+}\mu^{-}e^{+}$. Due to the absence of intermediate states, these LFV processes have no long distance QCD contributions and dominant $\phi$, $\omega$ resonance backgrounds. The general expression for the transition amplitude of $D^{+}\to\pi^{+}\mu^{-}e^{+}$ process in a generalized new physics model, is given by

[TABLE]

where the functions $F_{i}$, $i=V,A,S,P$ are defined as

[TABLE]

Here the Wilson coefficients $(K_{i}^{NP})$ involve the combination of LQ couplings as $Y_{\mu c}^{L}Y_{eu}^{L/R*}$ instead of $Y_{\mu c}^{L}Y_{\mu u}^{L/R*}$ in Eqn. (8). Now using Eqn. (67), the differential decay distribution for the $D^{+}\to\pi^{+}\mu^{-}e^{+}$ process with respect to $q^{2}$ and $\cos\theta$ ($\theta$ is the angle between the $D$ and $\mu^{-}$ in the $\mu-e$ rest frame) is given as

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

For numerical estimation in the leptoquark model, we use the constrained leptoquark couplings obtained from $D^{0}\to\mu^{+}\mu^{-}$ process and assume that the coupling between different generation of quarks and leptons follow the simple scaling laws, i.e. $Y_{ij}^{L(R)}/Y_{ii}^{L(R)}=(m_{i}/m_{j})^{1/2}$ with $j>i$. As discussed in ansatz1 ; ansatz2 , such pattern of ansatz can explain the decay widths of radiative LFV decay $\mu\to e\gamma$. Now using such ansatz, the variation of branching ratio with respect to $q^{2}$ for $D^{+(0)}\to\pi^{+(0)}\mu^{-}e^{+}$ process in the leptoquark model is shown in left panel of Fig. 6 and the corresponding integrated value is given in Table III. In this mode, the forward backward asymmetry depends on $K_{9,10}^{(^{\prime})NP}$ Wilson coefficients which give nonzero contribution. The left panel of Fig. 7 shows the $q^{2}$ variation of the forward backward asymmetry and the corresponding integrated value is found to be $(0.039\to 0.047)$. The variation of the flat term with respect to $q^{2}$ is presented in the left panel of Fig. 8 and the integrated value is $(0.137\to 0.33)$.

For the $Z^{\prime}$ model, we consider the constraint on the coupling of $Z^{\prime}$ boson to the quarks, obtained from the $D^{0}-\bar{D}^{0}$ mixing and $D^{0}\to\mu^{+}\mu^{-}$ process as given in Eqn. (37) and (45). For the lepton flavour violating coupling, the constraint is taken from $\mu^{-}\to e^{-}e^{+}e^{-}$ process, as discussed in section IV. Thus, using Eqn (37), (45) and (47), the predicted branching ratio of $D^{+(0)}\to\pi^{+(0)}\mu^{-}e^{+}$ process in the $Z^{\prime}$ model is given in Table III and the $q^{2}$ variation of $D^{+}\to\pi^{+}\mu^{-}e^{+}$ process is shown in Fig. 6 (right panel). The forward-backward asymmetry variation is shown in right panel of Fig. 7 and the predicted value is $-1.15\times 10^{-3}$, which is very small. In Fig. 8 (right panel), we show the plot for $q^{2}$ variation of the flat term and the integrated value is $0.158$.

From Table III, one can note that the predicted branching ratios are well below the present experimental limit for the $D^{+}\to\pi^{+}\mu^{-}e^{+}$ process. Although there is no experimental bound on $D^{0}\to\pi^{0}\mu^{-}e^{+}$ process so far, the experimental upper limit on branching ratios of $D^{0}\to\pi^{0}\mu^{\mp}e^{\pm}$ process is known, which is given as ${\rm BR}(D^{0}\to\pi^{0}\mu^{\mp}e^{\pm})={\rm BR}(D^{0}\to\pi^{0}\mu^{-}e^{+}+\pi^{0}\mu^{+}e^{-})<8.6\times 10^{-5}$. Our results for $D^{0}\to\pi^{0}\mu^{-}e^{+}$ process in both the leptoquark and $Z^{\prime}$ models are found to be within the above experimental bound.

VII $D^{0}\to\mu^{-}e^{+}(\tau^{-}e^{+})$ LFV decay process

Recently LHCb put the upper limit on branching ratio of the $D^{0}\to\mu^{\mp}e^{\pm}$ lepton flavour violating decay mode as D0exp

[TABLE]

Neglecting the mass of electron, the branching ratio of $D^{0}\to\mu^{-}e^{+}$ process is given by Hiller

[TABLE]

We use the scaling ansatz as discussed in the previous section to compute the required leptoquark coupling for $D^{0}\to\mu^{-}e^{+}$ process and the predicted branching ratio is found to be

[TABLE]

Now using Eqns. (37), (45) and (47), the predicted branching ratio of this LFV process in the $Z^{\prime}$ model is

[TABLE]

The predicted branching ratio is although small, but can be searched at LHCb experiment. The exploration/observation of this decay mode would definitely shed some light in the leptoquark scenarios.

Similarly using the new Wilson coefficient generated via leptoquark exchange, the branching ratio for $D^{0}\to\tau^{-}e^{+}$ process is found to be

[TABLE]

However, there is no experimental observation of lepton flavour violating $D^{0}\to\tau^{-}e^{+}$ process. The constraint on $Z^{\prime}$ coupling to tau and electron is obtained from the $\tau^{-}\to e^{-}e^{+}e^{-}$ process. Using Eqn. (48), the branching ratio for $D^{0}\to\tau^{-}e^{+}$ process in $Z^{\prime}$ model is given as

[TABLE]

So far there is no experimental evidence on the LFV $D^{0}\to\tau^{-}e^{+}$ decay process. Our results for $D^{0}\to\tau^{-}(\mu^{-})e^{+}$ process is comparable with Hiller ; Pakvasa2 ; Hiller3 .

VIII conclusion

In this paper we have studied the rare decays of $D$ 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 $D^{0}\to\mu^{+}\mu^{-}$ process and the $D^{0}-\bar{D}^{0}$ oscillation data. For the $Z^{\prime}$ model, the constraints on $Z^{\prime}$ couplings are obtained from the mass difference of $D^{0}-\bar{D}^{0}$ mixing, $D^{0}\to\mu^{+}\mu^{-}$ process and the lepton flavour violating $\tau^{-}(\mu^{-})\to e^{-}e^{+}e^{-}$ processes. Using the constrained parameter space, we estimated the branching ratios, forward backward asymmetry parameters and the flat terms in $D^{+(0)}\to\pi^{+(0)}\mu^{+}\mu^{-}$ processes. The branching ratios in the LQ model are found to be $\sim{\cal O}(10^{-10})$, 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 $D^{+(0)}\to\pi^{+(0)}\mu^{-}e^{+}$ and $D^{0}\to\mu(\tau)^{-}e^{+}$ processes, which are found to be rather small. We also estimated the forward-backward asymmetry parameter and the flat term for the LFV decays.

Acknowledgments

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