Decay ${\mathit{D} \to} {{\mathit K}^{(*)}}{{\mathit \ell}^{+}}{{\mathit \nu}_{{{{\mathit \ell}}}}}$ in covariant quark model

TL;DR

This paper calculates form factors and decay rates for D-meson leptonic and semileptonic decays using a covariant quark model with infrared confinement, providing results consistent with isospin symmetry and experiments.

Contribution

It introduces a covariant quark model with infrared confinement to compute form factors across the full kinematic range for D-meson decays, enhancing theoretical predictions.

Findings

Form factors computed across entire kinematic range.

Branching ratios consistent with experimental data.

Ratios of partial widths close to isospin invariance expectations.

Abstract

We study the leptonic and semileptonic $D$-meson decays (${{\mathit D} \to} {{\mathit \ell}^{+}}{{\mathit \nu}_{{{{\mathit \ell}}}}}$ and ${\mathit{D} \to} {{\mathit K}^{(*)}}{{\mathit \ell}^{+}}{{\mathit \nu}_{{{{\mathit \ell}}}}}$) in the framework of covariant quark model with built-in infrared confinement. We compute the required form factors in the entire kinematical momentum transfer region. The calculated form factors are used to evaluate the branching fractions of these transitions. We determine the following ratios of the partial widths: $\Gamma ({{\mathit D}^{0}} \rightarrow {{\mathit K}^{-}}{{\mathit e}^{+}}{{\mathit \nu}_{{e}}})/\Gamma({{\mathit D}^{+}} \rightarrow {{\overline{\mathit K}}^{0}}{{\mathit e}^{+}}{{\mathit \nu}_{{e}}}) = 1.02$, $\Gamma({{\mathit D}^{0}} \rightarrow {{\mathit K}^{-}}{{\mathit \mu}^{+}}{{\mathit \nu}_{{\mu}}})/\Gamma({{\mathit D}^{+}} \rightarrow {{\overline{\mathit K}}^{0}}{{\mathit \mu}^{+}}{{\mathit \nu}_{{\mu}}}) = 0.99$ and $\Gamma({{\mathit D}^{+}} \rightarrow {{\overline{\mathit K}}^{0}}{{\mathit \mu}^{+}}{{\mathit \nu}_{{\mu}}}) / \Gamma({{\mathit D}^{+}} \rightarrow {{\overline{\mathit K}}^{0}}{{\mathit e}^{+}}{{\mathit \nu}_{{e}}}) = 0.97$ which are in close resemblance with the iso-spin invariance and experimental results.

Tables

Table 1. Table 1: Leptonic decay constants f H subscript 𝑓 𝐻 f_{H} (in MeV)

$f_H$	Present	Data	Reference
$f_D$	206.1	204.6 $\pm$ 5.0	PDG pdg2016
		207.4 (3.8)	LQCD Carrasco:2015
		210 $\pm$ 11	QCDSR Wang:2015
$f_{D^{\ast }}$	244.3	263 $\pm$ 21	QCDSR Wang:2015
		278 $\pm$ 13 $\pm$ 10	LQCD Becirevic:2012
$f_{D_s}$	257.5	257.5 $\pm$ 4.6	PDG pdg2016
		254 (2) (4)	LQCD Yang:2015
		250.2 $\pm$ 3.6	LQCD Na:2010
		247.2 (4.1)	LQCD Carrasco:2015
		259 $\pm$ 10	QCDSR Wang:2015
$f_{D_s^{\ast }}$	272.0	308 $\pm$ 21	QCDSR Wang:2015
		311 $\pm$ 9	LQCD Becirevic:2012
$f_{D_s}/f_D$	1.249	1.258 $\pm$ 0.038	PDG pdg2016
		1.192 (0.22)	LQCD Carrasco:2015
		1.23 $\pm$ 0.07	QCDSR Wang:2015
$f_K$	156.0	155.0 (1.9)	LQCD Carrasco:2015
		155.37 (34)	LQCD Dowdall:2013
		157.9 $\pm$ 1.5	LQCD Na:2010
$f_{K^{\ast }}$	226.8	217 $\pm$ 7	PDG pdg2016
$f_{\pi }$	130.3	132.3 $\pm$ 1.6	LQCD Na:2010
		130.39 (20)	LQCD Dowdall:2013

Table 2. Table 2: Leptonic D + superscript 𝐷 D^{+} -decay branching fraction ( τ D + = 1.040 × 10 − 12 subscript 𝜏 superscript 𝐷 1.040 superscript 10 12 \tau_{D^{+}}=1.040\times 10^{-12} s pdg2016 )

Channel	Present	Data	Reference
$D^{+}\to e^{+}{\nu }_e$	$8.953\times {10}^{-9}$		PDG pdg2016
$D^{+}\to {\mu }^{+}{\nu }_{\mu }$	$3.803\times {10}^{-4}$	$\left(3.71\pm 0.19\right)\times {10}^{-4}$	BESIII ablikim:2014
		$\left(3.82\pm 0.32\right)\times {10}^{-4}$	CLEO-c Eisenstein:2008
$D^{+}\to {\tau }^{+}{\nu }_{\tau }$	$1.013\times {10}^{-3}$		PDG pdg2016

Table 3. Table 3: Double pole parameters for the computation of form factors in Eq. ( 18 )

	$F_{+}$	$F_{-}$	$A_0$	$A_{+}$	$A_{-}$	$V$
$F\left(0\right)$	0.76	-0.39	2.07	0.67	-0.90	0.89
$a$	0.72	0.75	0.39	0.84	0.95	0.96
$b$	0.046	0.032	-0.10	0.087	0.13	0.13

Table 4. Table 4: Branching fractions of D → K ( ∗ ) ​ ℓ + ​ ν ℓ → 𝐷 superscript 𝐾 superscript ℓ subscript 𝜈 ℓ D\to K^{(*)}{\mathit{\ell}}^{+}\nu_{\ell} and D → π ​ ℓ + ​ ν ℓ → 𝐷 𝜋 superscript ℓ subscript 𝜈 ℓ D\to\pi\ell^{+}\nu_{\ell} (in %)

Channel	Present	Data	Reference
$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	8.84	8.60 $\pm$ 0.06 $\pm$ 0.15	BESIII ablikim:2017lks
		8.83 $\pm$ 0.10 $\pm$ 0.20	CLEO-c Besson:2009
$D^{+}\to {\overline{K}}^0{\mu }^{+}{\nu }_{\mu }$	8.60	8.72 $\pm$ 0.07 $\pm$ 0.18	BESIII ablikim:2016sqt
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$	0.619	0.363 $\pm$ 0.08 $\pm$ 0.05	BESIII ablikim:2017lks
		0.405 $\pm$ 0.016 $\pm$ 0.009	CLEO-c Besson:2009
$D^{+}\to {\pi }^0{\mu }^{+}{\nu }_{\mu }$	0.607	–	–
$D^{+}\to {\overline{K}}^{\ast }{\left(892\right)}^0{e}^{+}{\nu }_e$	8.35	–	–
$D^{+}\to {\overline{K}}^{\ast }{\left(892\right)}^0{\mu }^{+}{\nu }_{\mu }$	7.94	–	–
$D^0\to K^{-}{e}^{+}{\nu }_e$	3.46	3.538 $\pm$ 0.033	PDG pdg2016
		3.505 $\pm$ 0.014 $\pm$ 0.033	BESIII ablikim:2015
		3.50 $\pm$ 0.03 $\pm$ 0.04	CLEO-c Besson:2009
		3.45 $\pm$ 0.07 $\pm$ 0.20	Belle Widhalm:2006
$D^0\to K^{-}{\mu }^{+}{\nu }_{\mu }$	3.36	3.33 $\pm$ 0.13	PDG pdg2016
		3.505 $\pm$ 0.014 $\pm$ 0.033	BESIII
$D^0\to {\pi }^{-}{e}^{+}{\nu }_e$	0.239	0.2770 $\pm$ 0.0068 $\pm$ 0.0092	BABAR Lees:2015
		0.295 $\pm$ 0.004 $\pm$ 0.003	BESIII ablikim:2015
		0.288 $\pm$ 0.008 $\pm$ 0.003	CLEO-c Besson:2009
		0.255 $\pm$ 0.019 $\pm$ 0.016	Belle Widhalm:2006
$D^0\to {\pi }^{-}{\mu }^{+}{\nu }_{\mu }$	0.235	0.238 $\pm$ 0.024	PDG pdg2016
$D^0\to K^{\ast }{\left(892\right)}^{-}{e}^{+}{\nu }_e$	3.25	2.16 $\pm$ 0.16	PDG pdg2016
$D^0\to K^{\ast }{\left(892\right)}^{-}{\mu }^{+}{\nu }_{\mu }$	3.09	1.92 $\pm$ 0.25	PDG pdg2016

Table 5. Table 5: Ratios of the semileptonic decays of D 𝐷 D mesons

Ratio	Value
$\Gamma \left(D^0\to K^{-}{e}^{+}{\nu }_e\right)/\Gamma \left(D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e\right)$	1.02
$\Gamma \left(D^0\to K^{-}{\mu }^{+}{\nu }_{\mu }\right)/\Gamma \left(D^{+}\to {\overline{K}}^0{\mu }^{+}{\nu }_{\mu }\right)$	0.99
$\Gamma \left(D^{+}\to {\overline{K}}^0{\mu }^{+}{\nu }_{\mu }\right)/\Gamma \left(D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e\right)$	0.97

Table 6. Table 6: Averages of forward-backward asymmetry and convexity parameters

Channel	$\ell$	$⟨A_{FB}^{\ell }⟩$	$⟨C_F^{\ell }⟩$	$⟨C_F^h⟩$
$D\to K$	$e$	-4.27 $\times$ 10-6	-1.5	3
	$\mu$	-0.058	-1.32	3
$D\to K^{\ast }$	$e$	0.17	-0.45	0.91
	$\mu$	0.13	-0.37	0.89

Equations

$${\mathcal{L}}_{int}$$

$$F_M(x,x_1,x_2)=\delta \left(x-\sum _{i=1}^2{w}_i{x}_i\right){\Phi }_M\left((x_1-x_2{)}^2\right)$$

$${\tilde{\Phi }}_M(-p^2)=\exp (p^2/{\Lambda }_M^2)$$

$$Z_M=1-{\tilde{\Pi }}_M^{\prime }(m_M^2)=0,$$

$${\tilde{\Pi }}_M(p^2)$$

$$S_q(k+p)$$

$$\prod _{i=1}^n\underset{0}{\overset{\infty }{\int }}d{\alpha }_{i}f({\alpha }_1,\dots ,{\alpha }_n)$$

$$\underset{0}{\overset{\infty }{\int }}dt(\dots )\to \underset{0}{\overset{1/{\lambda }^2}{\int }}dt(\dots ),$$

$$M(D\to K^{(\ast )}{\ell }^{+}{\nu }_{\ell })=\frac{G_F}{\sqrt{2}}{V}_{cs}⟨K^{(\ast )}|\bar{s}{O}^{\mu }c|D⟩{\ell }^{+}{O}^{\mu }{\nu }_{\ell }$$

$$⟨K_{[\bar{d}s]}(p_2)|\bar{s}{O}^{\mu }c|D_{[\bar{d}c]}(p_1)⟩$$

$$=$$

$$⟨K_{[\bar{d}s]}^{\ast }(p_2,{ϵ}_{\nu })|\bar{s}{O}^{\mu }c|D_{[\bar{d}c]}(p_1)⟩$$

$$=$$

$$\mathcal{B}(D^{+}\to {\ell }^{+}{\nu }_{\ell })=\frac{G_F^2}{8\pi }{m}_D{m}_{\ell }^2{\left(1-\frac{m_{\ell }^2}{m_D^2}\right)}^2{f}_D^2|V_{cd}{|}^2{\tau }_D$$

$$F(q^2)=\frac{F(0)}{1-as+b{s}^2},s=\frac{q^2}{m_1^2}$$

$$A_0$$

$$A_{-}=\frac{2m_2(m_1+m_2)}{q^2}(A_3-A_0),V=V$$

$$A_0(0)$$

$$2m_2{A}_3(q^2)$$

$$\frac{d\Gamma (D\to K{\ell }^{+}{\nu }_{\ell })}{d{q}^2}$$

$${\mathcal{H}}_L=|H_0{|}^2,{\mathcal{H}}_S=|H_t{|}^2,{\mathcal{H}}_{SL}=Re(H_0{H}_t^{†})$$

$$H_t=\frac{1}{\sqrt{q^2}}(Pq{F}_{+}+q^2{F}_{-})$$

$$H_0=\frac{2m_1|p_2|}{\sqrt{q^2}}{F}_{+}$$

$$\frac{d\Gamma (D\to K^{\ast }{\ell }^{+}{\nu }_{\ell })}{d{q}^2}$$

$${\mathcal{H}}_U=|H_{+1+1}{|}^2+|H_{-1-1}{|}^2,{\mathcal{H}}_P=|H_{+1+1}{|}^2-|H_{-1-1}{|}^2,$$

$${\mathcal{H}}_L=|H_{00}{|}^2,{\mathcal{H}}_S=|H_{t0}{|}^2,{\mathcal{H}}_{SL}=Re(H_{00}{H}_{t0}^{†})$$

$$H_{t0}=\frac{1}{m_1+m_2}\frac{m_1|p_2|}{m_2\sqrt{q^2}}(Pq(-A_0+A_{+})+q^2{A}_{-})$$

$$H_{\pm 1\pm 1}=\frac{1}{m_1+m_2}(-Pq{A}_0\pm 2m_1|p_2|V)$$

$$H_{00}$$

$${\mathcal{A}}_{FB}(q^2)=-\frac{3}{4}\frac{{\mathcal{H}}_P+4{\delta }_{\ell }{\mathcal{H}}_{SL}}{(1+{\delta }_{\ell })({\mathcal{H}}_U+{\mathcal{H}}_L)+3{\delta }_{\ell }{\mathcal{H}}_S}$$

$$C_F^{\ell }=\frac{3}{4}\frac{(1-2{\delta }_{\ell })({\mathcal{H}}_U-2{\mathcal{H}}_L)}{(1+{\delta }_{\ell })({\mathcal{H}}_U+{\mathcal{H}}_L)+3{\delta }_{\ell }{\mathcal{H}}_S}$$

$$C_F^h=-\frac{3}{2}\frac{(1+{\delta }_{\ell })({\mathcal{H}}_U-2{\mathcal{H}}_L)-6{\delta }_{\ell }{\mathcal{H}}_S}{(1+{\delta }_{\ell })({\mathcal{H}}_U+{\mathcal{H}}_L)+3{\delta }_{\ell }{\mathcal{H}}_S}$$

Full text

confinement model, form factors, decay rates

I Introduction

The semileptonic decays involve strong as well as weak interactions. The extraction of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements from these exclusive decays can be parameterized by form factor calculations. As $|V_{cd}|$ and $|V_{cs}|$ are constrained by CKM unitarity, the calculation of semileptonic decays of $D$-mesons can also be an important test to look for new physics. The decay $D\rightarrow K^{(*)}\ell^{+}\nu_{\ell}$ provides accurate determination of $|V_{cs}|$. Thus, the theoretical prediction for the form factors and their $q^{2}$-dependence need to be tested. A comprehensive review of experimental and theoretical challenges in study of hadronic decays of $D$ and $D_{s}$ mesons along with required experimental and theoretical tools Anders:2012 provide motivation to look into semileptonic decays in this paper.

Recently, BESIII ablikim:2017lks ; ablikim:2016sqt ; ablikim:2015 ; Ablikim:2015qgt and BABAR Lees:2015 collaborations have reported precise and improved measurements on semileptonic form factors and branching fractions on decays of $D\rightarrow K\ell^{+}\nu_{\ell}$ and $D\rightarrow\pi\ell^{+}\nu_{\ell}$. A brief review of the earlier work and present experimental status of $D$-meson decays are given in Amhis:2016 . Also there are variety of theoretical models available in the literature for the computation of hadronic form factors. One of the oldest model is based on the quark model known as ISGW model for CP violation in semileptonic $B$ meson decays based on the nonrelativistic constituent quark picture Isgur:1989 . The advanced version (ISGW2 model Scora:1995 ) includes the heavy quark symmetry and has been used for semileptonic decays of $B_{(s)}$, $D_{(s)}$ and $B_{c}$ mesons. The form factors are also calculated in Lattice Quantum Chromodynamics (LQCD) Aubin:2004 ; Bernard:2009 ; Na:2010 ; AlHaydari:2010 ; Na:2011 ; Koponen:2013 , light-cone sum rules (LCSR) Khodjamirian:2000 ; Khodjamirian:2007 ; Khodjamirian:2009 and LCSR with heavy quark effective theory Wang:2003 . The form factor calculations from LCSR provide good results at low ($q^{2}\simeq 0$) and high ($q^{2}\simeq q_{max}^{2}$) momentum transfers. The form factors have also been calculated for the process $D\to K\ell\nu_{\ell}$ in the entire momentum transfer range Koponen:2013 using the LQCD. Also recently the Flavour Lattice Averaging Group (FLAG) have reported the latest lattice results for determination of CKM matrices within the standard modelAoki:2016 .

The form factors of $D,B\to P,V,S$ transitions with $P$, $V$ and $S$ corresponding to pseudoscalar, vector and scalar meson respectively have been evaluated in the light front quark model (LFQM) Verma:2011 . The form factors for $D\to P,V$ are also computed in the framework of chiral quark model ($\chi$QM) Palmer:2013 as well in the phenomenological model based on heavy meson chiral theory (HM$\chi$T) Fajfer:2004 ; Fajfer:2005 . The form factors of $B_{(s)},D_{(s)}\to\pi,K,\eta$ have been evaluated in three flavor hard pion chiral perturbation theory Bijnens:2011 . The form factors for $D\to\pi e^{+}\nu_{e}$ have been computed in the framework of “charm-changing current” Ananthanarayan:2011 . The authors of Rong:2014 ; Fang:2014 have determined the form factors $f_{+}^{K(\pi)}$ by globally analysing the available measurements of branching fractions for $D\to K(\pi)e^{+}\nu_{e}$. The vector form factors for $D\to K\ell\nu_{\ell}$ were also parameterized in DescotesGenon:2008 . The evaluation of transition form factors and decays of $B_{(s)},D_{(s)}\to f_{0}(980),K_{0}^{*}(1430)\ell\nu_{\ell}$ has been done in Yang:2005 ; Bediaga:2003 from QCD sum rules. The computation of differential branching fractions for $D_{(s)}\to(P,V,S)\ell\nu_{\ell}$ was also performed using chiral unitary approach Sekihara:2015 ; Xie:2014 , generalized linear sigma model Fariborz:2014 ; Fariborz:2011 and sum rules Wang:2010 . Various decay properties of $D_{(s)}$ and $B_{(s)}$ are also studied in the formalism of semi-relativistic Kher:2017 ; Devlani:2013 ; Devlani:2012 ; Devlani:2011 and relativistic Shah:2016 ; Shah:2014 ; Shah:2014d potential models.

In this paper, we employ the covariant constituent quark model (CQM) with built-in infrared confinement Efimov:1988 ; Efimov:1993 ; Ivanov:1999 ; Ivanov:2000 ; Faessler:2002 ; Branz:2010 to compute the leptonic and semileptonic decays. The form factors of these transitions are expressed through only few universal functions. One of the key feature of CQM is access to the entire physical range of momentum transfer. Our aim is to perform independent calculations of these decays including $q^{2}$ behavior of the transition form factors, leptonic decay constants of $D$ and $K$ mesons and ratios of branching fractions for the decay ${\mathit{D}\to}{{\mathit{K}}^{(*)}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{{{\mathit{\ell}}}}}}$ and ${\mathit{D}\to}{\mathit{\pi}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{{{\mathit{\ell}}}}}}$.

This paper is organised as follows. After a brief introduction of the semileptonic $D$-meson decays in Sec. I, in Sec. II we introduce the theoretical framework of CQM and also discuss the method of extracting the model parameters. In Sec. III, we give the definition of the form factors for the decays ${\mathit{D}\to}{{\mathit{K}}^{(*)}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{{{\mathit{\ell}}}}}}$. In Sec. IV for numerical results, we first compute the leptonic branching fractions of $D^{+}$-meson. Next we give numerical results of the form factors. We also parameterize the form factors using double pole approximation. From the form factors, we compute the differential branching fraction for the ${\mathit{D}\to}{{\mathit{K}}^{(*)}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{{{\mathit{\ell}}}}}}$, with $\ell=e$ and $\mu$ and the branching fractions. We also calculate the forward-backward asymmetry and convexity parameters. We compare our results with available experimental, lattice and other theoretical results. Finally, we summarize present work in Sec. V.

II Model

The CQM is an effective quantum field approach Efimov:1988 ; Efimov:1993 ; Ivanov:1999 ; Ivanov:2000 ; Faessler:2002 ; Branz:2010 for hadronic interactions based on an effective Lagrangian of hadrons interacting with their constituent quarks. In this paper, we restrict ourselves to weak decays of $D$-mesons only. The interaction Lagrangian describing the coupling of meson $M(q_{1}\bar{q}_{2})$ to the constituent quarks $q_{1}$ and $\bar{q}_{2}$ in the framework of CQM is given by

[TABLE]

where $\Gamma_{M}$ is the Dirac matrix and projects onto the spin quantum number of relevant mesonic field $M(x)$. $g_{M}$ is the coupling constant and $F_{M}$ is the vertex function that is related to the scalar part of the Bethe-Salpeter amplitude. $F_{M}$ also characterizes the finite size of the mesons. We choose the vertex function that satisfies the Lorentz invariance of the Lagrangian Eq. (1),

[TABLE]

with $\Phi_{M}$ is the correlation function of two constituent quarks with masses $m_{q_{1}}$ and $m_{q_{2}}$ and $w_{q_{i}}=m_{q_{i}}/(m_{q_{1}}+m_{q_{2}})$ such that ${\mathit{w_{1}}}+{\mathit{w_{2}}}=1$. We choose Gaussian function for vertex function as

[TABLE]

with the parameter $\Lambda_{M}$ characterized by the finite size of the meson. In the Euclidian space, we can write $p^{2}=-p^{2}_{E}$, so that the vertex function has the appropriate falloff behaviour so as to remove the ultraviolet divergence in the loop integral.

We use the compositeness conditions Salam:1962 ; Weinberg:1963 to determine the coupling strength $g_{M}$ in Eq. (5) that requires the renormalization constant $Z_{M}$ for the bare state to composite mesonic state $M(x)$ set to zero, i.e.,

[TABLE]

where $\tilde{\Pi}^{\prime}_{M}$ is the derivative of meson mass operator and $Z_{M}$ is the wave function renormalization constant of the meson $M$. Here, $Z_{M}^{1/2}$ is the matrix element between the physical state and the corresponding bare state. The above condition guarantees that the physical state does not contain any bare quark state i.e. bound state. The constituents are virtual and are introduced to realize the interaction and as a result the physical state turns dressed and its mass and wave function are renormalized.

The meson mass operator Fig. 1 for any meson is defined as

[TABLE]

where $N_{c}=3$ is the number of colors. $\Gamma_{1}$, $\Gamma_{2}$ are the Dirac matrices and for scalar, vector and pseudoscalar mesons, we choose the gamma matrices accordingly. $S^{\prime}s$ are the quark propagator and we use the free fermion propagator for the constituent quark. For the computation of loop integral in Eq. (5), we write the quark propagator in terms of Fock-Schwinger representation as

[TABLE]

where $k$ is the loop momentum and $p$ is the external momentum. The use of Fock-Schwinger representation allows to do the tensor integral in an efficient way since the loop momenta can be converted into the derivative of exponential function Branz:2010 . All the necessary trace evaluation and loop integrals are done in FORM Vermaseren:2008 . For the remaining integral over the Fock-Schwinger parameters $0\leq\alpha_{i}\leq\infty$, we use an additional integration converting the Fock-Schwinger parameters into a simplex. The transformation reads Feynman:1949

[TABLE]

For meson case $n$ = 2.

While the integral over $t$ in Eq. (II) is convergent below the threshold $p^{2}<(m_{q_{1}}+m_{q_{2}})^{2}$, its convergence above threshold $p^{2}\geq(m_{q_{1}}+m_{q_{2}})^{2}$ is guaranteed by augmenting the quark mass by an imaginary part, i.e. $m_{q}\rightarrow m_{q}-i\epsilon,\ \epsilon>0$, in the quark propagator Eq. (6). This makes it possible to rotate the integration variable $t$ to the imaginary axis $t\to it$. The integral Eq. (II) in turn becomes convergent but obtains an imaginary part corresponding to quark pair production. However, by reducing the scale of integration at the upper limit corresponding to the introduction of an infrared cutoff

[TABLE]

one can remove all possible thresholds present in the initial quark diagram Branz:2010 . Thus the infrared cutoff parameter $\lambda$ effectively guarantees the confinement of quarks within hadrons.

Before going for the semileptonic decays, we need to specify the independent model parameters namely size parameter of meson $\Lambda$ and constituent quark masses $m_{q_{i}}$. These model parameters are determined by fitting calculated decay constants of basic processes such as leptonic (Fig. 2) and radiative decays to available experimental data or LQCD for vector and pseudoscalar mesons. We use the updated least square fit performed in the recent papers of the model parameters dubnicka:2016 ; Gutsche:2015 ; Ganbold:2014 (all in GeV). We take the infrared cutoff parameter $\lambda$ to be the same throughout this study.

[TABLE]

and the size parameters

[TABLE]

We have listed our results for the leptonic decay constants of $D_{(s)}^{(*)}$, $K^{(*)}$ and $\pi$ mesons in the Table 1. The decay constants we use in our calculations match quite well with Particle Data Group (PDG), LQCD and QCD sum rules (QCDSR) results.

III Form factors

In the Standard Model of Particle Physics, semileptonic decays of any meson is caused by weak force in which one lepton and corresponding neutrino is produced in addition to one or more hadrons (Fig. 3).

The invariant matrix element for the semileptonic $D\rightarrow K^{(*)}\ell^{+}\nu_{\ell}$ decay can be written as

[TABLE]

where $O^{\mu}=\gamma^{\mu}(1-\gamma_{5})$ is the weak Dirac matrix with left chirality. The matrix elements for the above semileptonic transitions in the covariant quark model are written as

[TABLE]

[TABLE]

with $P=p_{1}+p_{2}$, $q=p_{1}-p_{2}$ and $\epsilon_{\nu}$ to be the polarization vector such that $\epsilon_{\nu}^{{\dagger}}\cdot p_{2}=0$ and on-shell conditions of particles require $p_{1}^{2}=m_{1}^{2}=m_{D}^{2}$ and $p_{2}^{2}=m_{2}^{2}=m_{K^{(*)}}^{2}$. Since there are three quarks involved in this transition, we use the notation $w_{ij}=m_{q_{j}}/(m_{q_{i}}+m_{q_{j}})$ ($i$, $j$ = 1, 2, 3) such that $w_{ij}+w_{ji}=1$.

IV Numerical Results

Having determined the necessary model parameters and form factors, we are now in position to present our numerical results. We first compute pure leptonic decays of $D^{+}$-meson and then using the form factors obtained in Sec. III, we compute branching fractions for semileptonic $D$-meson decays.

We compute the pure leptonic decays of $D^{+}\to\ell^{+}\nu_{\ell}$ within the Standard Model. The branching fraction for leptonic decay is given by

[TABLE]

where $G_{F}$ is the fermi coupling constant, $m_{D}$ and $m_{\ell}$ are the $D$-meson and lepton masses respectively and $\tau_{D}$ is the $D$-meson lifetime. $f_{D}$ is the leptonic decay constant of $D$-meson from Table 1. The resultant branching fractions for $\ell=\tau,\mu$ and $e$ are given in Table 2. It is important to note that the helicity flip factor $(1-m_{\ell}^{2}/m_{D}^{2})$ affects the leptonic branching fractions because of the different lepton masses. We also compare our results with the experimental data. The branching fraction for $D^{+}\to\mu^{+}\nu_{\mu}$ shows very good agreement with BESIII ablikim:2014 and CLEO-c Eisenstein:2008 data. The branching fractions for $D^{+}\to e^{+}\nu_{e}$ and $D^{+}\to\tau^{+}\nu_{\tau}$ also fulfill the experimental constraints.

In Figs. 4 and 5, we plot our calculated form factors as a function of momentum transfer squared in the entire range $0\leq q^{2}\leq q^{2}_{max}=(m_{D}-m_{K^{(*)}})^{2}$. The multi-dimensional integral (three-fold for semileptonic case) appearing in Eqs. (11) and (14) are computed numerically using Mathematica. Our form factor results are also well represented by the double-pole parametrization

[TABLE]

The numerical results of form factors and associated double-pole parameters are listed in Table 3.

In Fig. 4, we plot the form factor $F_{+}$ for $D\to K(\pi)\ell^{+}\nu_{\ell}$ decays in the entire kinematical range of momentum transfer. We compare our plot with the results from LCSR Ref. Khodjamirian:2009 , LFQM Ref. Verma:2011 , LQCD Ref. Aubin:2004 as well with the BESIII data Ref. ablikim:2015 . Our results at maximum recoil point $q^{2}\to 0$ are in very good agreement with the other approaches as well as with the experimental result. Similar plot can be obtained for form factor $F_{-}$. We also plot the vector form factors and for the comparison of the form factors for $D\to K^{*}\ell^{+}\nu_{\ell}$ transition with other approaches, we need to write our form factors Eq. (14) in terms of those used in Ref. Khodjamirian:2007 . The relations read

[TABLE]

[TABLE]

The form factors in Eq. (19) also satisfy the constraints

[TABLE]

Fig. 5 shows form factors from the present calculation along with the results from LFQM Verma:2011 , Chiral Quark Model ($\chi$QM) Palmer:2013 and with Heavy Meson Chiral Theory (HM$\chi$T) Fajfer:2005 . The plot shows that our results of the form factors $A_{0}$, $A_{1}$ and $A_{2}$ match with LFQM Verma:2011 and the vector form factors match with the $\chi$QM Palmer:2013 where the authors have used energy scaling parameters extracted from modified low energy effective theory in $H\to V$ transitions. Our results show little deviation from those obtained using HM$\chi$T Fajfer:2005 . In computation of form factors for $q^{2}=0$ using LCSR, the authors of Khodjamirian:2009 have used the $\overline{MS}$ scheme for $c$-quark mass and the computation of form factors for $q\geq 0$ is performed in the form of conformal mapping and series parametrization. In the LFQM Verma:2011 , the authors have used the method of double pole approximation, where as in BESIII ablikim:2015 and BABAR Lees:2015 experiment, the form factors are parameterized in terms of two and three parameters series expansion respectively.

The differential branching fractions for semileptonic $D\to K\ell^{+}\nu_{\ell}$ decay are computed using Ivanov:2015 ; Ivanov:2016

[TABLE]

where the helicity flip factor $\delta_{\ell}=m_{\ell}^{2}/2q^{2}$, $|p_{2}|=\lambda^{2}(m_{1}^{2},m_{2}^{2},q^{2})/2m_{1}$ is momentum of $K$ meson in the rest frame of $D$-meson and velocity-type parameter $v=1-m_{\ell}^{2}/q^{2}$.

The bilinear combinations of the helicity amplitudes $\mathcal{H}$ are defined as Faessler:2002 ,

[TABLE]

and the helicity amplitudes are expressed via the form factor in the matrix element as,

[TABLE]

[TABLE]

Similarly the differential branching fractions for semileptonic $D\to K^{*}\ell^{+}\nu_{\ell}$ decay is computed by Ivanov:2015 ; Ivanov:2016

[TABLE]

The bilinear combinations of the helicity amplitudes $\mathcal{H}$ are defined as Faessler:2002

[TABLE]

[TABLE]

here also the helicity amplitudes are expressed via the form factor in the matrix element as

[TABLE]

[TABLE]

[TABLE]

In Fig. 6, we present our results for differential branching fractions of $D\to K^{(*)}\ell^{+}\nu_{\ell}$ in the entire kinematical range of momentum transfer. The semileptonic branching fractions in Eqs. (21) and (IV) are computed by numerically integrating the differential branching fractions shown in Fig. 6. The branching fractions for $D\to K^{(*)}\ell^{+}\nu_{\ell}$ and $D\to\pi\ell^{+}\nu_{\ell}$ are presented in Table 4. We also compare our results with experimental results. The results for $\mathcal{B}({\mathit{D}}^{+}\rightarrow{{\overline{\mathit{K}}}^{0}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{\ell}}})$ and $\mathcal{B}({{\mathit{D}}^{0}}\rightarrow{{\mathit{K}}^{-}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{\ell}}})$, ($\ell=e$ and $\mu$) show excellent agreement with the recent BESIII data ablikim:2017lks ; ablikim:2016sqt ; ablikim:2015 as well with the other experimental collaborations. Also the ratios of the different semileptonic decay widths for the channels $D\to K\ell^{+}\nu_{\ell}$ are presented in Table 5 and our results are well within the isospin conservation rules given in Ref. Korner:1989 . We also present our results for $\mathcal{B}({{\mathit{D}}^{0}}\rightarrow{{\mathit{K}}^{*}{(892)}^{-}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{e}}})$ but our results overestimate the data given in PDG pdg2016 . This deviation of the present study within the Standard Model might be explained through hadronic uncertainty or ratios of differential distributions for longitudinal and transverse polarizations of these $K^{*}$ mesons Fajfer:2015 . The FOCUS Link:2002 and CLEO-c Shepherd:2006 experiments have also reported mixing of scalar amplitudes with dominant vector decays. These observations open up new possibilities of investigations in charm semileptonic decays. There have also been attempts to explain these exclusive decays using $R$-parity violating supersymmetric effects Wang:2014 and their direct correlation with possible supersymmetric signals expected from LHC and BESIII data. We predict the branching fractions for ${{\mathit{D}}^{+}}\rightarrow{{\overline{\mathit{K}}}^{*}{(892)}^{0}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{e}}}$ but we do not compare our results since no experimental results available for this channel.

We also present our results for branching fractions of ${\mathit{D}}^{+}\rightarrow{{\mathit{\pi}}^{0}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{\ell}}}$ and ${{\mathit{D}}^{0}}\rightarrow{{\mathit{\pi}}^{-}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{\ell}}}$ transitions. Our prediction for $\mathcal{B}({\mathit{D}}^{+}\rightarrow{{\mathit{\pi}}^{0}}{{\mathit{e}}^{+}}{{\mathit{\nu}}_{e}})$ is higher than BESIII ablikim:2017lks and CLEO-c data Besson:2009 while the trend is opposite in the case of $\mathcal{B}({{\mathit{D}}^{0}}\rightarrow{{\mathit{\pi}}^{-}}{{\mathit{e}}^{+}}{{\mathit{\nu}}_{{e}}})$. The deviation of the $\mathcal{B}({\mathit{D}}^{+}\rightarrow{{\mathit{\pi}}^{0}}{{\mathit{e}}^{+}}{{\mathit{\nu}}_{e}})$ from experimental and LQCD data might be attributed to the computed form factors. However, our $\mathcal{B}({\mathit{D}}^{0}\rightarrow{{\mathit{\pi}}^{-}}{{\mathit{e}}^{+}}{{\mathit{\nu}}_{e}})$ is in close proximity to that by Belle Widhalm:2006 and $\mathcal{B}({\mathit{D}}^{0}\rightarrow{{\mathit{\pi}}^{-}}{{\mathit{\mu}}^{+}}{{\mathit{\nu}}_{\mu}})$ is in excellent agreement with PDG data pdg2016 .

We also list some more physical observables in terms of helicity amplitudes. We have already shown the computed differential branching fractions in Fig. 6. Next, the helicity amplitudes defined above are used to plot the forward-backward asymmetry in Fig. 7 for $D\to K^{(*)}\ell^{+}\nu_{\ell}$ in the entire kinematical range of momentum transfer. We use the following relation for plotting the forward-backward asymmetry ($A_{FB}$) Gutsche:2015 ; Ivanov:2015

[TABLE]

It is evident from Fig. 7 that the $A_{FB}(q^{2})$ for $D\to K\ell^{+}\nu_{\ell}$ and $D\to K^{*}\ell^{+}\nu_{\ell}$ are similar for both $e$ and $\mu$ modes. $A_{FB}(q^{2})\to 0$ for in the both zero recoil and larger recoil limits because of the zero recoil relations of the helicity functions $\mathcal{H}_{P}=\mathcal{H}_{SL}=0$ and longitudinal dominance in the partial rates at the maximum recoil.

Also the lepton and hadron side convexity parameter are defined as Gutsche:2015 ; Ivanov:2015

[TABLE]

and

[TABLE]

The plot for the convexity parameters Eqs. (31) and (32) as a function of entire momentum transfer range can easily be obtained. In Table 6, we give the $q^{2}$ averages of the above observables. Note that in order to obtain the averages of these observables, we need to multiply the numerator and denominator by phase space factor $|p_{2}|q^{2}v^{2}$. Also in computation of leptonic and semileptonic branching fractions, forward-backward asymmetry and convexity parameters, the values of CKM matrices namely $|V_{cs}|$ and $|V_{cd}|$, meson masses, lepton masses and their lifetimes are taken from PDG pdg2016 .

V Conclusion

In this article, we have analysed the leptonic ($D^{+}\to{{\mathit{e}}^{+}}{{\mathit{\nu}}_{{e}}}$) and semileptonic ($D\to K^{(*)}\ell^{+}\nu_{\ell}$, $D\to\pi\ell^{+}\nu_{\ell}$) decays using covariant quark model with infrared confinement within the standard model framework. The ratios of the partial widths are found to be consistent with the isospin conservation holding within uncertainties in experimental data. It is interesting to note here that the $\mathcal{B}({\mathit{D}}^{+}\rightarrow{{\mathit{\pi}}^{0}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{\ell}}})$ deviate from existing data while $\mathcal{B}({{\mathit{D}}^{0}}\rightarrow{{\mathit{\pi}}^{-}}{{\mathit{\ell}}^{+}}{{\mathit{\nu}}_{{\ell}}})$ match well. Further exploration to this observation may lead to interesting outcome.

The deviation of branching fractions in case of $D\to K^{*}\ell\nu_{\ell}$ might be understood by underlying hadronic uncertainty or ratios of differential distributions for longitudinal and transverse polarizations of the $K^{*}$ mesons. We are looking forward to analyse $D\to K^{*}\ell^{+}\nu_{\ell}$ decay and expect the experimental facilities to throw more light on their form factor shapes in forthcoming attempts that will help in understanding the charm decays and possibly the dynamics of these systems beyond the standard model.

Acknowledgment

We thank Prof. Mikhail A. Ivanov for the continuous support through out this work and providing critical remarks for improvement of the manuscript. NRS would like to thank Bogoliubov Laboratory of Theoretial Physics, Joint Institute for Nuclear Research for warm hospitality during Helmholtz-DIAS International Summer School “Quantum Field Theory at the Limits: from Strong Field to Heavy Quarks” where this work was initiated. This work is done under Major Research Project F.No.42-775/2013(SR) with financial support from the University Grants Commission of India.