# Study of form factors and branching ratios for $D\rightarrow
S,Al\bar{\nu_{l}}$ with light-cone sum rules

Qi Huang1,2<EMAIL_ADDRESS>Yan-Jun Sun1,2<EMAIL_ADDRESS>Di
Gao1,2<EMAIL_ADDRESS>Guo-Hua Zhao1,2<EMAIL_ADDRESS>Bin Wang1,2
<EMAIL_ADDRESS>Wei Hong1,2<EMAIL_ADDRESS>1 College of
Physics and Electronic Engineering, Northwest Normal University, Lanzhou
730070, China
2 Lanzhou Center for Theoretical Physics, Lanzhou University Lanzhou 730070,
China

###### Abstract

We systematically study the semileptonic decay process of $D\rightarrow
S,Al\bar{\nu_{l}}(l=e,\mu)$ by light-cone sum rules (LCSR) with chiral
currents, calculate the form factors containing only the contribution of the
leading twist light-cone distribution amplitudes (LCDAs). For scalar mesons
$a_{0}(980)$ and $a_{0}(1450)$, we take them as $q\bar{q}$ states. For axial-
vector meson, we study $a_{1}(1260)(1^{3}p^{1})$ and
$b_{1}(1235)(1^{1}p^{1})$. Based on the results of these form factors, we
further present the branching ratios of these semileptonic decay processes.
The numerical results for $D\rightarrow a_{0}(980),b_{1}(1235)l\bar{\nu_{l}}$
are in good agreement with experiments and that for $D\rightarrow
a_{0}(1450),a_{1}(1260)l\bar{\nu_{l}}$ processes are expected to be tested
experimentally in the future.

##  I Introduction

The study of the $D$ meson semileptonic decay process can not only be used to
extract the CKM (Cabibbo-Kobayashi-Maskawa) matrix elements, but also play an
important role in determining weak interaction between quarks in the standard
model (SM). More specifically, the $D$ meson semileptonic decay process has a
relatively simple decay mechanism and final state interactions, making it an
ideal occasion for studying light scalar meson structures. To figure it out, a
series of experiments have been carried out by BES III Ablikim:2018ffp ;
Ablikim:2020agq ; Ablikim:2017lks ; Ablikim:2018evp , CLEOc Besson:2009uv ;
CLEO:2011ab , BarBar Lees:2014ihu , Belle Widhalm:2006wz , and so on.

For $D\rightarrow P,Vl\bar{\nu_{l}}$ ($P,V$ are pseudoscalar and vector mesons
respectively), a lot of research work on experiments Ablikim:2017lks ;
Ablikim:2018evp ; Besson:2009uv ; CLEO:2011ab ; Lees:2014ihu ; Widhalm:2006wz
and theories Palmer:2013yia ; Li:2019phv ; Bernard:1991bz ; Wang:2002zba ;
Wu:2006rd ; Li:2012gr ; Khodjamirian:2009ys ; Ball:2006yd ;
Khodjamirian:2000ds ; Ball:1993tp ; Faustov:2019mqr ; Soni:2018adu ;
Fu:2018yin ; Chang:2019mmh ; Fajfer:2005ug ; Huang:2008sn  have been carried
out, but less for $D\rightarrow S,Al\bar{\nu_{l}}$. From the theoretical point
of view, the decay channels of $D\rightarrow P,Vl\bar{\nu_{l}}$ can be studied
directly because the final state mesons are classical quark-antiquark states,
but for the scalar mesons below 1 GeV, there have been many controversy about
their internal structures. For example, QCD sum rules (QSR) has investigated
the possibility to distinguish the two-quark and tetra-quark picture for light
scalar meson Wang:2009azc  and LCSR has studied them as quark-antiquark
statesCheng:2017fkw . Lattice QCD has studied them as tetra-quark states
Alexandrou:2017itd , and the MIT bag model has studied them with a diquark-
diantiquark picture Jaffe:1976ig . Throughout this paper, we take scalar meson
$a_{0}(980)$ and $a_{0}(1450)$ as quark-antiquark state. For axial-vector
meson, there are two different nonets of $J^{P}=1^{+}$ in the quark model as
the orbital excitations of the $q\bar{q}$ system. In terms of the
spectroscopic notation ${}^{2s+1}L_{J}$, p-wave axial-vector meson can be
divided into $1^{3}p^{1}$ ($C+$) and $1^{1}p^{1}$ ($C-$) depending on the
$C$-quantum number. Experimentally, the $J^{PC}=1^{++}$ nonet consists of
$a_{1}(1260)$, $f_{1}(1285)$, $f_{1}(1420)$ and $K_{1A}$, while the
$J^{PC}=1^{+-}$ nonet has $b_{1}(1235)$, $h_{1}(1170)$, $h_{1}(1380)$ and
$K_{1B}$. For the present work, we study two mesons $a_{1}(1260)(1^{3}p^{1})$
and $b_{1}(1235)(1^{1}p^{1})$ with simple internal structure relative to other
axial-vector mesons, which cannot have mixing due to the opposite
$C$-parities.

The form factors which are important parameters for theoretical studies of the
$D$ meson semileptonic decay process, can be calculated by various methods
that are powerful in a certain region of the transfer momentum square $q^{2}$.
For example, the low energy effective theory (LEET) can be used in the region
$q^{2}\rightarrow 0$ Palmer:2013yia ; lattice QCD (LQCD) can be used in the
region of large momentum transfer $q^{2}\rightarrow\infty$ Li:2019phv ;
Bernard:1991bz ; the heavy quark effective theory (HQET) and the heavy-light
chiral perturbation theory can be used in the region $q^{2}=\infty$
Wang:2002zba ; Wu:2006rd .

In this paper, we study the form factors of semileptonic decay process
$D\rightarrow S,Al\bar{\nu_{l}}$ by using the LCSR method with the chiral
currents, in the region $0\leqslant q^{2}\leqslant(m_{D}-m_{M})^{2}$ ($M$ are
scale meson or axial-vector meson) . Experimentally, BES III Ablikim:2018ffp ;
Ablikim:2020agq  provide an ideal place to study $D\rightarrow
S,Al\bar{\nu_{l}}$. To compare with it, we also calculate the branching
ratios.

The paper is organized as follows: In Sec. II, the form factors and branching
ratios of the semileptonic decays for $D\rightarrow S,Al\bar{\nu_{l}}$ are
derived, and some simple relations between form factors are also obtained. In
Sec. III, we present our numerical analysis on values of the form factors and
branching ratios of the semileptonic decays, and a comparison is also made
between our results and the predictions of other methods and experiments.

##  II Form Factors

In the semileptonic decays, the hadron currents, which represent the strong
interaction between quarks, can be parameterized into form factors. The
hadronic matrix elements for $D\rightarrow S$ can be parameterized by form
factors as:

$\displaystyle\langle S(p)|\bar{q}_{2}\gamma_{\mu}\gamma_{5}c|D(p+q)\rangle$
$\displaystyle=-2if_{+}^{D\rightarrow S}(q^{2})p_{\mu}-i[f_{+}^{D\rightarrow
S}(q^{2})+f_{-}^{D\rightarrow S}(q^{2})]q_{\mu},$ (1) $\displaystyle\langle
S(p)|\bar{q}_{2}\sigma_{\mu\nu}(1+\gamma_{5})q^{\nu}c|D(p+q)\rangle$
$\displaystyle=[2p_{\mu}q^{2}-2q_{\mu}(p\cdot q)]\frac{-f_{T}^{D\rightarrow
S(q^{2})}}{m_{D}+m_{S}},$ (2)

where $f_{+}^{D\rightarrow S}(q^{2})$ and $f_{-}^{D\rightarrow S}(q^{2})$ are
the transition form factors of the $D\rightarrow S$, and $f_{T}^{D\rightarrow
S}(q^{2})$ is the penguin form factor. The hadronic matrix elements for
$D\rightarrow A$ can be parameterized by form factors as:

$\displaystyle\langle
A(p)|\bar{q}_{2}\gamma_{\mu}(1+\gamma_{5})c|D(p+q)\rangle=$
$\displaystyle\frac{2iA^{D\rightarrow
A}(q^{2})}{m_{D}-m_{A}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}(q+p)^{\alpha}p^{\beta}$
$\displaystyle-(m_{D}-m_{A})V_{1}^{D\rightarrow
A}(q^{2})\varepsilon^{*}_{\mu}$ $\displaystyle+\frac{V_{2}^{D\rightarrow
A}(q^{2})}{m_{D}-m_{A}}(\varepsilon^{*}\cdot{q})(2p+q)_{\mu}$
$\displaystyle+\frac{2(\varepsilon^{*}\cdot{q})m_{A}}{q^{2}}q_{\mu}[V_{3}^{D\rightarrow
A}(q^{2})-V_{0}^{D\rightarrow A}(q^{2})]$ (3)

and

$\displaystyle\langle
A(p)|\bar{q}_{2}\sigma_{\mu\nu}(1+\gamma_{5})q^{\nu}c|D(p+q)\rangle=$
$\displaystyle-2\epsilon_{\mu\nu\rho\sigma}\varepsilon^{*\nu}q^{\rho}p^{\sigma}T_{1}^{D\rightarrow
A}(q^{2}),$
$\displaystyle+i(\varepsilon^{*}\cdot{q})p_{\mu}[T_{2}^{D\rightarrow
A}(q^{2})+(\frac{q^{2}}{m_{D}^{2}-m_{V}^{2}}-1)T_{3}^{D\rightarrow A}(q^{2})]$
$\displaystyle+2i(\varepsilon^{*}\cdot{q})p_{\mu}(T_{2}^{D\rightarrow
A}(q^{2})+\frac{q^{2}}{m_{D}^{2}-m_{V}^{2}}T_{3}^{D\rightarrow A}(q^{2}))$
$\displaystyle-i\varepsilon^{*}_{\mu}(m_{D}^{2}-m_{A}^{2})T_{2}^{D\rightarrow
A}(q^{2}),$ (4)

where

$\displaystyle V_{3}^{D\rightarrow A}(q^{2})$
$\displaystyle=\frac{m_{D}-m_{A}}{2m_{A}}V_{1}^{D\rightarrow
A}(q^{2})-\frac{m_{D}+m_{A}}{2m_{A}}V_{2}^{D\rightarrow A}(q^{2}),$
$\displaystyle V_{0}^{D\rightarrow A}(q^{2})$
$\displaystyle=\frac{m_{D}-m_{A}}{2m_{A}}V_{1}^{D\rightarrow
A}(q^{2})-\frac{m_{D}+m_{A}}{2m_{A}}V_{2}^{D\rightarrow
A}(q^{2})+\frac{q^{2}}{2m_{A}(m_{D}-m_{A})}V_{2}^{D\rightarrow A}(q^{2}),$ (5)

and $\varepsilon$ is polarization vector for axial-vector meson.

To calculate the form factors, we construct the two-point correlation function
as

$\displaystyle\Pi_{1\mu}(p,q)$ $\displaystyle=i\int d^{4}xe^{iqx}\langle
M(p)|T\\{J_{1\mu}(x),J_{1}(0)\\}|0\rangle,$ (6) $\displaystyle\Pi_{2\mu}(p,q)$
$\displaystyle=i\int d^{4}xe^{iqx}\langle
M(p)|T\\{J_{2\mu}(x),J_{1}(0)\\}|0\rangle,$ (7)

The transfer momentum $q$ is defined as $q=p_{D}-p$, where $p_{D}$ and $p$ are
the four-momentum of the initial and final meson states. $J_{i\mu}$ and
$J_{i}$ $(i=1,2)$ are chiral currents, whose explicit forms are shown in Table
I, depending on the decay processes.

Table 1: Chiral currents for different decay processes. Mode |  $J_{i\mu}(x)$ |  $J_{i}(0)$
---|---|---
$D\rightarrow S$ |  $J_{1\mu}(x)=\bar{q}_{2}(x)\gamma_{\mu}(1-\gamma_{5})c(x)$ |  $J_{1}(0)=\bar{c}(0)i(1-\gamma_{5})q_{1}(0)$
|  $J_{2\mu}(x)=\bar{q}_{2}(x)\sigma_{\mu\nu}(1+\gamma_{5})c(x)$ |  $J_{2}(0)=\bar{c}(0)i(1-\gamma_{5})q_{1}(0)$
$D\rightarrow A$ |  $J_{1\mu}(x)=\bar{q}_{2}(x)\gamma_{\mu}(1-\gamma_{5})c(x)$ |  $J_{1}(0)=\bar{c}(0)i(1+\gamma_{5})q_{1}(0)$
|  $J_{2\mu}(x)=\bar{q}_{2}(x)\sigma_{\mu\nu}(1+\gamma_{5})c(x)$ |  $J_{2}(0)=\bar{c}(0)i(1+\gamma_{5})q_{1}(0)$

In general, the correlation function should be described by two ways:

(1) Inserting a complete set of intermediate hadronic states in the middle of
two currents in the correlation function to obtain the phenomenological side;

(2) Operator product expanding (OPE) for correlation functions to obtain the
theoretical side.

On the phenomenological side

The hadronic representations of correlators are achieved by inserting between
two currents a complete set of resonance states with the same quantum numbers
as the operator $J_{1}(0)$. On the desired pole contributions due to the
lowest pseudoscalar $D$ meson are insolated and we obtain the hadronic
repredentations

$\displaystyle\Pi_{1\mu}(p,q)$ $\displaystyle=\frac{\langle
M(p)|\bar{q}_{2}\gamma_{\mu}(1\pm\gamma_{5})c|D\rangle\langle
D|\bar{c}i\gamma_{5}q_{1}|0\rangle}{m^{2}_{D}-(p+q)^{2}}+higher\quad states,$
(8) $\displaystyle\Pi_{2\mu}(p,q)$ $\displaystyle=\frac{\langle
M(p)|\bar{q}_{2}\sigma_{\mu\nu}(1\pm\gamma_{5})q^{\nu}c|D\rangle\langle
D|\bar{c}i\gamma_{5}q_{1}|0\rangle}{m^{2}_{D}-(p+q)^{2}}+higher\quad states.$
(9)

It should be stressed that the correlation functions receive contributions
from the scalar resonances, in addition to the higher pseudoscalar ones, and
the ground-state scalar meson is a bit lighter than the pseudoscalar resonance
lying in the first excited state. The matrix element, $\langle
M(p)|\bar{q}_{2}\gamma_{\mu}(1\pm\gamma_{5})c|D\rangle$ and $\langle
M(p)|\bar{q}_{2}\sigma_{\mu\nu}(1\pm\gamma_{5})q^{\nu}c|D\rangle$, can be
parameterized by the form factors as in Eqs. (1)-(5). The second element is
expressed via $D$ meson decay constant as:

$\displaystyle\langle D|\bar{c}i\gamma_{5}q_{1}|0\rangle$
$\displaystyle=\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}.$ (10)

Substituting the matrix elements Eqs. (1)-(5) into Eqs. (6) and (7), we obtain
the phenomenological part of correlation function in terms of the form factors
and Lorentz structures. The correlation functions of $D\rightarrow S$ are
expressed as

$\displaystyle\Pi_{1\mu}(p,q)$ $\displaystyle=-\frac{2if_{+}^{D\rightarrow
S}(q^{2})p_{\mu}+i[f_{+}^{D\rightarrow S}(q^{2})+f_{-}^{D\rightarrow
S}(q^{2})]q_{\mu}}{m_{D}^{2}-(p+q)^{2}}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}+\int_{s_{0}}^{\infty}\frac{\rho(s)ds}{s-(p+q)^{2}},$
(11) $\displaystyle\Pi_{2\mu}(p,q)$
$\displaystyle=-\frac{[q^{2}2p_{\mu}-2q_{\mu}(p\cdot
q)]}{m_{D}^{2}-(p+q)^{2}}\frac{if_{T}^{D\rightarrow
S}}{m_{D}+m_{P}}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}+\int_{s_{0}}^{\infty}\frac{\rho(s)ds}{s-(p+q)^{2}},$
(12)

and the correlation functions of $D\rightarrow A$ as

$\displaystyle\Pi_{1\mu}(p,q)=$ $\displaystyle\\{\frac{2iA^{D\rightarrow
A}(q^{2})}{m_{D}-m_{A}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}(q+p)^{\alpha}p^{\beta}-(m_{D}-m_{A})V_{1}^{D\rightarrow
A}(q^{2})\varepsilon^{*}_{\mu}+\frac{V_{2}^{D\rightarrow
A}(q^{2})}{m_{D}-m_{A}}(\varepsilon^{*}\cdot{q})(2p+q)_{\mu}$
$\displaystyle+\frac{2(\varepsilon^{*}\cdot{q})m_{A}}{q^{2}}q_{\mu}[V_{3}^{D\rightarrow
A}(q^{2})-V_{0}^{D\rightarrow
A}(q^{2})]\\}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}+\int_{s_{0}}^{\infty}\frac{\rho(s)ds}{s-(p+q)^{2}},$
(13) $\displaystyle\Pi_{2\mu}(p,q)=$
$\displaystyle\\{-2\epsilon_{\mu\nu\rho\sigma}\varepsilon^{*\nu}q^{\rho}p^{\sigma}T_{1}^{D\rightarrow
A}(q^{2})+i(\varepsilon^{*}\cdot{q})p_{\mu}[T_{2}^{D\rightarrow
A}(q^{2})+(\frac{q^{2}}{m_{D}^{2}-m_{A}^{2}}-1)T_{3}^{D\rightarrow
A}(q^{2})]\\}$
$\displaystyle+2i(\varepsilon^{*}\cdot{q})p_{\mu}[T_{2}^{D\rightarrow
A}(q^{2})+\frac{q^{2}}{m_{D}^{2}-m_{A}^{2}}T_{3}^{D\rightarrow
A}(q^{2})]-i\varepsilon^{*}_{\mu}(m_{D}^{2}-m_{A}^{2})T_{2}^{D\rightarrow
A}(q^{2})\\}$
$\displaystyle\frac{1}{m_{D}^{2}-(p+q)^{2}}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}+\int_{s_{0}}^{\infty}\frac{\rho(s)ds}{s-(p+q)^{2}},$
(14)

where $s_{0}$ near the squared mass of the lowest scalar $D$ meson and
$\rho(s)$ is the spectral density of the higher excited states and continuum.
Since the contributions of the higher excited and continuum spectrum on the
phenomenological side cannot be calculated exactly, we apply quark-hadron
duality to express the integral of the higher excited and continuum
spectrumColangelo:2000dp :

$\displaystyle\int_{s_{0}}^{\infty}ds\frac{\rho(s)}{s-(p+q)^{2}}\simeq\int_{s_{0}}^{\infty}ds\frac{1}{\pi}\frac{Im\Pi_{\mu}^{pert}(p,q)}{s-(p+q)^{2}}.$
(15)

On the theoretical side

The calculation of the correlation function in the region of large space-like
momentum is based on the expansion of the T-product of the quark currents near
the light-cone $x^{2}\approx 0$, due to sufficiently large momentum transfer.
By contracting $c$ and $\bar{c}$ quark fields, we obtain

$\displaystyle\Pi_{1\mu}(p,q)$ $\displaystyle=i\int d^{4}xe^{iqx}\langle
M(p)|\\{\bar{q}_{2}(x)\gamma_{\mu}(1\pm\gamma_{5})S^{c}(x,0)i(1\pm\gamma_{5})q_{1}(0)\\}|0\rangle,$
(16) $\displaystyle\Pi_{2\mu}(p,q)$ $\displaystyle=i\int d^{4}xe^{iqx}\langle
M(p)|\\{\bar{q}_{2}(x)\sigma_{\mu\nu}(1\pm\gamma_{5})q^{\nu}S^{c}(x,0)i(1\pm\gamma_{5})q_{1}(0)\\}|0\rangle,$
(17)

where $S^{c}(x,0)$ is the propagator of the free-$c$ quark as

$\displaystyle S^{c}(x,0)$
$\displaystyle=-i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot
x}\frac{\not{k}+m_{c}}{m^{2}_{c}-k^{2}}.$ (18)

After rearrangement of the quantum fields and matrices, the correlation
function, is turned into a form including a matrix trace and a matrix element
of non-local operators between $M$ meson state and vacuum state,

$\displaystyle\Pi_{1\mu}(p,q)=$ $\displaystyle
i\int\frac{d^{4}xd^{4}k}{(2\pi)^{4}}\frac{e^{i(q-k)x}}{m_{c}^{2}-k^{2}}Tr{[\gamma_{\mu}(1\pm\gamma_{5})(\not{k}+m_{c})(1\pm\gamma_{5})]_{\delta\alpha}\langle
M(p)|\bar{q}_{2\delta}(x)q_{1\alpha}(0)}|0\rangle,$ (19)
$\displaystyle\Pi_{2\mu}(p,q)=$ $\displaystyle
i\int\frac{d^{4}xd^{4}k}{(2\pi)^{4}}\frac{e^{i(q-k)x}}{m_{c}^{2}-k^{2}}Tr{[\sigma_{\mu\nu}(1\pm\gamma_{5})q^{\nu}(\not{k}+m_{c})(1\pm\gamma_{5})]_{\delta\alpha}\langle
M(p)|\bar{q}_{2\delta}(x)q_{1\alpha}(0)}|0\rangle.$ (20)

In the LCSR method, the non-vanishing matrix elements are defined in terms of
LCDAs. Generally, the matrix element $\langle
S(p)|\bar{q}_{\delta}(x)q_{\alpha}(0)|0\rangle$ is written as Cheng:2005nb

$\displaystyle\langle S(p)|\bar{q}_{\delta}(x)q_{\alpha}(0)|0\rangle$
$\displaystyle=\frac{1}{4}\int_{0}^{1}due^{iup\cdot
x}\\{\not{p}\phi_{S}(u)+m_{S}(\phi^{s}_{S}(u)-\sigma_{\mu\nu}p^{\mu}x^{\nu}\frac{\phi^{\sigma}_{s}(u)}{6})\\}_{\delta\alpha},$
(21)

where $\phi_{S}(u)$ is the twist-2 LCDA, $\phi^{s}_{S}(u)$ and
$\phi^{\sigma}_{s}(u)$ are twist-3 LCDAs of scalar meson. Similarly, the
matrix element $\langle A(p)|\bar{q}_{\delta}(x)q_{\alpha}(0)|0\rangle$ is
expressed as Yang:2008xw

$\displaystyle\langle
A(p,\varepsilon^{*})|\bar{q}_{\delta}(x)q_{\alpha}(0)|0\rangle=$
$\displaystyle-\frac{1}{4}\int^{1}_{0}due^{i(up\cdot{x})}\times\\{f_{A}m_{A}[\not{p}\gamma_{5}\frac{\varepsilon^{*}x}{px}(\phi_{\parallel}(u)+\frac{m_{A}^{2}x^{2}}{16}A_{\parallel}^{2})$
$\displaystyle+(\not{\varepsilon}^{*}-\not{p}{\varepsilon^{*}x}{px})\gamma_{5}g^{(a)}_{\bot}(u)-\not{x}\gamma_{5}\frac{\varepsilon^{*}x}{2(px)^{2}}m_{A}^{2}\bar{g}_{3}(u)+\epsilon_{\mu\nu\rho\sigma}\varepsilon^{*\nu}p^{\rho}x^{\sigma}\gamma^{\mu}\frac{g^{(v)}_{\bot}(u)}{4}]$
$\displaystyle+f_{\bot}^{A}[\frac{1}{2}(\not{p}\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\not{p})\gamma_{5}(\phi_{\bot}(u)+\frac{m_{A}^{2}x^{2}}{16}A_{\bot}^{2})-\frac{1}{2}(\not{p}\not{x}-\not{x}\not{p})\gamma_{5}\frac{\varepsilon^{*}x}{2(px)^{2}}m_{A}^{2}\bar{h}_{s}(u)$
$\displaystyle-\frac{1}{4}(\not{\varepsilon}^{*}\not{x}-\not{x}\not{\varepsilon}^{*})\gamma_{5}\frac{m_{A}^{2}}{px}m_{A}^{2}\bar{h}_{3}(u)+i(\varepsilon^{*}x)m_{A}^{2}\gamma_{5}\frac{h_{\parallel}^{(p)}(u)}{2}]\big{\\}}_{\delta\alpha},$
(22)

where $\phi_{\parallel}(u)$ and $\phi_{\bot}(u)$ are twist-2 LCDAs,
$g^{(a)}_{\bot}$, $g^{(v)}_{\bot}$, $h_{\parallel}^{(p)}$ and
$h_{\parallel}^{(t)}$ are twist-3 LCDAs, and $g_{3}$ and $h_{3}$ are twist-4
LCDAs of axial-vector meson,

$\displaystyle\bar{g}_{3}(u)$
$\displaystyle=g_{3}(u)+\phi_{\parallel}(u)-2g_{\bot}^{a}(u),$
$\displaystyle\bar{h}_{\parallel}^{t}$
$\displaystyle=h_{\parallel}^{h}-\frac{1}{2}\phi_{\bot}(u)-\frac{1}{2}h_{3}(u),$
$\displaystyle\bar{h}_{3}(u)$ $\displaystyle=h_{3}(u)-\phi_{\bot}(u).$ (23)

For $D\rightarrow S$, substituting Eq. (21) into correlation functions (19)
and (20), and making a trace , we obtain the correlator on the theoretical
side as

$\displaystyle\Pi_{1\mu}$
$\displaystyle=2im_{c}p_{\mu}\int^{1}_{0}du\frac{\phi_{S}(u)}{m^{2}_{c}-(q+up)^{2}},$
(24) $\displaystyle\Pi_{2\mu}$ $\displaystyle=-[2p_{\mu}q^{2}-2q_{\mu}(p\cdot
q)]\int^{1}_{0}du\frac{\phi_{S}(u)}{m^{2}_{c}-(q+up)^{2}}.$ (25)

Different from the process $D\rightarrow S$, for $D\rightarrow A$ process,
axial-vector meson LCDAs are usually expressed as longitudinal and transverse
projection operator Yang:2008xw . The transverse projection of the correlation
functions are

$\displaystyle\Pi_{1\mu}(p,q)=$ $\displaystyle
i\int\frac{d^{4}xd^{4}k}{(2\pi)^{4}}\frac{e^{i(q-k)x}}{m_{c}^{2}-k^{2}}Tr{[\gamma_{\mu}(1-\gamma_{5})(\not{k}+m_{c})(1+\gamma_{5})]}_{\delta\alpha}M^{A}_{\bot\delta\alpha},$
(26) $\displaystyle\Pi_{2\mu}(p,q)=$ $\displaystyle
i\int\frac{d^{4}xd^{4}k}{(2\pi)^{4}}\frac{e^{i(q-k)x}}{m_{c}^{2}-k^{2}}Tr{[\sigma_{\mu\nu}(1+\gamma_{5})q^{\nu}(\not{k}+m_{c})(1+\gamma_{5})]}_{\delta\alpha}M^{A}_{\bot\delta\alpha},$
(27)

where $M^{A}_{\bot\delta\alpha}$ is transverse projection operator Yang:2008xw
,

$\displaystyle M^{A}_{\bot\delta\alpha}=$ $\displaystyle
i\frac{f_{\bot}}{4}E\\{\not{\varepsilon}^{*}_{\bot}\not{n}_{-}\gamma_{5}\phi_{\bot}(u)-\frac{f_{V}}{f_{A}^{\bot}}\frac{m_{A}}{E}[\not{\varepsilon}^{*}_{\bot}\gamma_{5}g^{(a)}_{\bot}(u)$
$\displaystyle-E\int^{u}_{0}dv(\phi_{\parallel}(v)-g^{(a)}_{\bot}(v))\not{n}_{-}\gamma_{5}\varepsilon^{*}_{\bot\mu}\frac{\partial}{\partial{k_{\bot\mu}}}$
$\displaystyle+i\epsilon_{\mu\nu\rho\sigma}\gamma_{\mu}\varepsilon^{*\nu}n_{-}^{\rho}n_{+}^{\sigma}\frac{g^{(v)^{\prime}}_{\bot}(u)}{8}-E\frac{g^{(v)}_{\bot}(u)}{4}\frac{\partial}{\partial{k_{\bot\sigma}}}]\\}\mid_{k=up}+\mathcal{O}(\frac{m_{A}^{2}}{E^{2}}).$
(28)

Substituting Eq. (28) into correlation functions (26) and (27), and performing
the trace operation for Eq. (26) and Eq. (27), respectively, we obtain the
correlator of $D\rightarrow A$ on the theoretical side as

$\displaystyle\Pi_{1\mu}=$
$\displaystyle-2f^{A}_{\bot}\int^{1}_{0}du[-p\cdot{(q+up)}\varepsilon^{*}_{\bot\mu}-(\varepsilon^{*}_{\bot}\cdot{q})p_{\mu}-i\epsilon_{\mu\nu\beta\tau}\varepsilon^{*\beta}_{\bot}q^{\mu}p^{\tau}]\frac{\phi_{\bot}(u)}{m^{2}_{c}-(q+up)^{2}},$
(29) $\displaystyle\Pi_{2\mu}=$ $\displaystyle
2f^{A}_{\bot}m_{c}\int^{1}_{0}du[i(q\cdot{p})\varepsilon^{*}_{\bot\mu}-i(\varepsilon^{*}_{\bot}\cdot{q})p_{\mu}-\epsilon_{\mu\nu\rho\tau}\varepsilon^{*\rho}_{\bot}q^{\nu}p^{\tau}]\frac{\phi_{\bot}(u)}{m^{2}_{c}-(q+up)^{2}}.$
(30)

Base on the conformal symmetry hidden in the QCD Lagrangian, $\phi_{S}(u)$ and
$\phi_{\bot}(u)$ can be expanded in a series of Gegenbauer polynomials
$C_{m}^{3/2}$ with increasing conformal spin as Braun:2003rp

$\displaystyle\phi_{S}(u,\mu)=\bar{f_{S}}(\mu)6u(1-u)[B_{0}(\mu)+\sum_{m=1}B_{m}(\mu)C_{m}^{3/2}(2u-1)]$
(31)

for scalar mesons, where $B_{m}(\mu)$ is Gegenbauer coefficient,

$\displaystyle
B_{m}(\mu)=\frac{1}{\bar{f_{S}}}\frac{2(2m+1)}{3(m+1)(m+2)}\int_{0}^{1}C_{m}^{3/2}(2u-1)\phi_{S}(u,\mu)du$
(32)

and

$\displaystyle\bar{f_{S}}(\mu)$
$\displaystyle=\bar{f_{S}}(\mu_{0})(\frac{\alpha_{s}(\mu_{0})}{\alpha_{s}(\mu)})^{4/b}.$
(33)

In the SU(3) limit, $B_{0}=0$, and the twist-2 LCDAs of all scalar mesons are
antisymmetric in the $u\rightarrow 1-u$ transformation, thus, only odd
Gegenbauer coefficients ($B_{1}(\mu),B_{3}(\mu)$) are considered in the
following discussion. For axial-vector meson, the twist-2 LCDAs are
Yang:2007zt

$\displaystyle\phi_{\bot}^{A}(u)=6u(1-u)[a_{0}^{\bot}+3a_{1}^{\bot}\xiup+a_{2}^{\bot}\frac{3}{2}(5\xiup^{2}-1)]$
(34)

for $1^{3}P_{1}$ meson, and

$\displaystyle\phi_{\bot}^{A}(u)=6u(1-u)[1+3a_{1}^{\bot}\xiup+a_{2}^{\bot}\frac{3}{2}(5\xiup^{2}-1)]$
(35)

for $1^{1}P_{1}$ meson, where $\xiup=2u-1$, and $a_{0}^{\bot}$, $a_{1}^{\bot}$
and $a_{2}^{\bot}$ are Gegenbauer coefficients.

Matching the expression on phenomenological side with that on theoretical
side, we get

$-\frac{2if_{+}^{D\rightarrow S}(q^{2})p_{\mu}+i[f_{+}^{D\rightarrow
S}(q^{2})+f_{-}^{D\rightarrow
S}(q^{2})]q_{\mu}}{m_{D}^{2}-(p+q)^{2}}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}=2im_{c}p_{\mu}\int^{1}_{\Delta}du\frac{\phi(u)}{m^{2}_{c}-(q+up)^{2}},\\\
$ (36) $-\frac{[q^{2}2p_{\mu}-2q_{\mu}(p\cdot
q)]}{m_{D}^{2}-(p+q)^{2}}\frac{if_{T}^{D\rightarrow
S}(q^{2})}{m_{D}+m_{P}}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}=-[2p_{\mu}q^{2}-2q_{\mu}(p\cdot
q)]\int^{1}_{\Delta}du\frac{\phi(u)}{m^{2}_{c}-(q+up)^{2}}\\\ $ (37)

for $D\rightarrow S$, and

$\displaystyle\\{\frac{2iA^{D\rightarrow
A}(q^{2})}{m_{D}-m_{A}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}(q+p)^{\alpha}p^{\beta}-(m_{D}-m_{A})V_{1}^{D\rightarrow
A}(q^{2})\varepsilon^{*}_{\mu}+\frac{V_{2}^{D\rightarrow
A}(q^{2})}{m_{D}-m_{A}}(\varepsilon^{*}\cdot{q})$
$\displaystyle\times(2p+q)_{\mu}+\frac{2(\varepsilon^{*}\cdot{q})m_{A}}{q^{2}}q_{\mu}[V_{3}^{D\rightarrow
A}(q^{2})-V_{0}^{D\rightarrow
A}(q^{2})]\\}\frac{1}{m_{D}^{2}-(p+q)^{2}}\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}$
$\displaystyle=-2f^{A}_{\bot}\int^{1}_{\Delta}du[-p\cdot{(q+up)}\varepsilon^{*}_{\bot\mu}-(\varepsilon^{*}_{\bot}\cdot{q})p_{\mu}-i\epsilon_{\mu\nu\beta\tau}\varepsilon^{*\beta}_{\bot}q^{\mu}p^{\tau}]\frac{\phi_{\bot}(u)}{m^{2}_{c}-(q+up)^{2}},$
(38)
$\displaystyle\\{-2\epsilon_{\mu\nu\rho\sigma}\varepsilon^{*\nu}q^{\rho}p^{\sigma}T_{1}^{D\rightarrow
A}(q^{2})+i(\varepsilon^{*}\cdot{q})p_{\mu}[T_{2}^{D\rightarrow
A}(q^{2})+(\frac{q^{2}}{m_{D}^{2}-m_{A}^{2}}-1)T_{3}^{D\rightarrow
A}(q^{2})]\\}$
$\displaystyle+2i(\varepsilon^{*}\cdot{q})p_{\mu}[T_{2}^{D\rightarrow
A}(q^{2})+\frac{q^{2}}{m_{D}^{2}-m_{A}^{2}}T_{3}^{D\rightarrow
A}(q^{2})]-i\varepsilon^{*}_{\mu}(m_{D}^{2}-m_{A}^{2})T_{2}^{D\rightarrow
A}(q^{2})\\}\frac{1}{m_{D}^{2}-(p+q)^{2}}$
$\displaystyle\times\frac{m^{2}_{D}f_{D}}{m_{c}+m_{q_{1}}}=2f^{A}_{\bot}m_{c}\int^{1}_{\Delta}du[i(q\cdot{p})\varepsilon^{*}_{\bot\mu}-i(\varepsilon^{*}_{\bot}\cdot{q})p_{\mu}-\epsilon_{\mu\nu\rho\tau}\varepsilon^{*\rho}_{\bot}q^{\nu}p^{\tau}]\frac{\phi_{\bot}(u)}{m^{2}_{c}-(q+up)^{2}}$
(39)

for $D\rightarrow A$, where $\Delta$ is the solution to the equation $us-
m_{c}^{2}-u(1-u)m_{M}^{2}+(1-u)q^{2}=0$,

$\displaystyle\Delta$
$\displaystyle=\frac{\sqrt{(s_{0}-m_{M}^{2}-q^{2})^{2}+4(m_{c}^{2}-q^{2})m_{M}^{2}}-(s_{0}-m_{M}^{2}-q^{2})}{2m_{M}^{2}}$
(40)

and $u\in{[0,1]}$.

In order to suppress the contribution of higher excited states and continuum,
we perform Borel transformation Colangelo:2000dp  on both sides of Eps.
(36)-(39),

$\displaystyle B_{M^{2}}\frac{1}{m_{D}^{2}-(q+p)^{2}}$
$\displaystyle=\frac{1}{M^{2}}e^{-\frac{m_{D}^{2}}{M^{2}}},$ $\displaystyle
B_{M^{2}}\frac{1}{m_{c}^{2}-(q+up)^{2}}$
$\displaystyle=\frac{1}{uM^{2}}e^{-\frac{m_{c}^{2}+u(1-u)p^{2}-(1-u)q^{2}}{uM^{2}}},$
(44)

where $M^{2}$ is the Borel parameter. Up to this point, we get the analytical
results of the form factors

$\displaystyle f_{+}^{D\rightarrow S}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}m_{c}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle f_{-}^{D\rightarrow S}(q^{2})$
$\displaystyle=\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}m_{c}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle f_{T}^{D\rightarrow S}(q^{2})$
$\displaystyle=(m_{D}+m_{S})\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF}$
(42)

for $D\rightarrow S$, and

$\displaystyle A^{D\rightarrow A}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}(m_{D}-m_{A})f^{A}_{\bot}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle V_{1}^{D\rightarrow A}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}\frac{f^{A}_{\bot}}{m_{D}-m_{A}}\int^{1}_{\Delta}\frac{\phi(u)}{u}\frac{m_{c}^{2}-q^{2}+u^{2}p^{2}}{u}due^{FF},$
$\displaystyle V_{2}^{D\rightarrow A}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}(m_{D}-m_{A})f^{A}_{\bot}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle T_{1}^{D\rightarrow A}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}m_{c}f^{A}_{\bot}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle T_{2}^{D\rightarrow A}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}m_{c}f^{A}_{\bot}(1-\frac{q^{2}}{m_{D}^{2}-m_{A}^{2}})\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle T_{3}^{D\rightarrow A}(q^{2})$
$\displaystyle=-\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}m_{c}f^{A}_{\bot}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle V_{0}^{D\rightarrow A}(q^{2})=$
$\displaystyle-\frac{f^{A}_{\bot}}{2m_{A}}\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}\int^{1}_{\Delta}\frac{\phi(u)}{u}\frac{m_{c}^{2}-q^{2}+u^{2}p^{2}}{u}due^{FF}$
$\displaystyle+\frac{f^{A}_{\bot}}{2m_{A}}\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}(m_{D}^{2}-m_{A}^{2})f^{A}_{\bot}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF}$
$\displaystyle+\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}\frac{q^{2}f^{A}_{\bot}}{2m_{A}}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF},$
$\displaystyle V_{3}^{D\rightarrow A}(q^{2})=$
$\displaystyle-\frac{f^{A}_{\bot}}{2m_{A}}\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}\int^{1}_{\Delta}\frac{\phi(u)}{u}\frac{m_{c}^{2}-q^{2}+u^{2}p^{2}}{u}due^{FF}$
$\displaystyle+\frac{f^{A}_{\bot}}{2m_{A}}\frac{m_{c}+m_{q_{1}}}{m_{D}^{2}f_{D}}(m_{D}^{2}-m_{A}^{2})f^{A}_{\bot}\int^{1}_{\Delta}\frac{\phi(u)}{u}due^{FF}$
(43)

for $D\rightarrow A$, where

$\displaystyle FF$
$\displaystyle=-\frac{1}{uM^{2}}(m^{2}_{c}+u\bar{u}p^{2}-\bar{u}q^{2})+\frac{m^{2}_{D}}{M^{2}}.$
(44)

Obviously, there exist some relations between form factors, which are similar
to that obtained from $B\rightarrow Sl\bar{\nu}_{l}$ process Sun:2010nv ;
YanJun:2011rn ,

$\displaystyle f_{-}^{D\rightarrow S}(q^{2})$
$\displaystyle=-f_{+}^{D\rightarrow S}(q^{2}),$ $\displaystyle
f_{T}^{D\rightarrow S}(q^{2})$
$\displaystyle=\frac{m_{D}+m_{S}}{m_{c}}f_{+}^{D\rightarrow S}(q^{2}),$
$\displaystyle A^{D\rightarrow A}(q^{2})$ $\displaystyle=V_{2}^{D\rightarrow
A}(q^{2}),$ $\displaystyle V_{0}^{D\rightarrow A}(0)$
$\displaystyle=V_{3}^{D\rightarrow A}(0),$ $\displaystyle T_{1}^{D\rightarrow
A}(q^{2})$ $\displaystyle=T_{3}^{D\rightarrow A}(q^{2}),$ $\displaystyle
T_{2}^{D\rightarrow A}(q^{2})$
$\displaystyle=(1-\frac{q^{2}}{m_{D}^{2}-m_{V}^{2}})T_{3}^{D\rightarrow
A}(q^{2}).$ (45)

Branching ratios

Using the above form factors, we can further calculate the differential decay
widths of these decay processes. For the semileptonic decay $D\rightarrow
Sl\bar{\nu}_{l}$, the differential decay width can be written as Cheng:2017fkw

$\displaystyle\frac{d\Gamma(D\rightarrow Sl\bar{\nu}_{l})}{dq^{2}}=$
$\displaystyle\frac{G_{F}^{2}|V_{cd}|^{2}}{768\pi^{3}m_{D}^{3}}\frac{(q^{2}-m_{l}^{2})^{2}}{q^{6}}\sqrt{(m_{D}^{2}+m_{S}^{2}-q^{2})^{2}-4m_{D}^{2}m_{S}^{2}}$
$\displaystyle\times\\{(f_{+}(q^{2}))^{2}[(q^{2}+m_{S}^{2}-m_{D}^{2})^{2}(q^{2}+2m_{l}^{2})-q^{2}m_{S}^{2}(4q^{2}+2m_{l}^{2})]$
$\displaystyle+6f_{+}(q^{2})f_{-}(q^{2})q^{2}m_{l}^{2}(m_{D}^{2}-m_{S}^{2}-q^{2})+6(f_{-}(q^{2}))^{2}q^{4}m_{l}^{2}\\}.$
(46)

For the semileptonic decay $D\rightarrow Al\bar{\nu}_{l}$, the differential
decay widths can be written as Momeni:2019uag

$\displaystyle\frac{d\Gamma_{L}(D\rightarrow Al\bar{\nu}_{l})}{dq^{2}}=$
$\displaystyle(\frac{q^{2}-m_{l}^{2}}{q^{2}})^{2}\frac{\sqrt{\lambdaup}G_{F}^{2}|V_{cd}|^{2}}{384\pi^{3}m_{D}^{3}}\times\frac{1}{q^{2}}\\{3m_{l}^{2}\lambdaup
V_{0}^{2}(q^{2})+(m_{l}^{2}+2q^{2})$
$\displaystyle\times|\frac{1}{2m_{A}^{2}}[(m_{D}^{2}-m_{A}^{2}-q^{2})(m_{D}-m_{A})V_{1}(q^{2})-\frac{\lambdaup}{m_{D}-m_{A}}V_{2}(q^{2})]|^{2}\\}$
(47)

and Momeni:2019uag

$\displaystyle\frac{d\Gamma_{\pm}(D\rightarrow Al\bar{\nu}_{l})}{dq^{2}}=$
$\displaystyle(\frac{q^{2}-m_{l}^{2}}{q^{2}})^{2}\frac{\sqrt{\lambdaup}G_{F}^{2}|V_{cd}|^{2}}{384\pi^{3}m_{D}^{3}}\times\\{(m_{l}^{2}+2q^{2})\lambdaup|\frac{A(q^{2})}{m_{D}-m_{A}}\mp\frac{(m_{D}-m_{A})V_{1}(q^{2})}{\lambdaup}|^{2}\\},$
(48)

where
$\lambdaup=m_{D}^{4}+m_{A}^{4}+q^{4}-2m_{A}^{2}m_{D}^{2}-2q^{2}m_{A}^{2}$,
$\frac{d\Gamma_{L}}{dq^{2}}$ and $\frac{d\Gamma_{\pm}}{dq^{2}}$ are the
longitudinal and transverse components of the differential decay width,
respectively. The total differential decay width can be written as
Momeni:2019uag

$\displaystyle\frac{d\Gamma(D\rightarrow
Al\bar{\nu}_{l})}{dq^{2}}=\frac{d\Gamma_{L}(D\rightarrow
Al\bar{\nu}_{l})}{dq^{2}}+\frac{d\Gamma_{\pm}(D\rightarrow
Al\bar{\nu}_{l})}{dq^{2}},$ (49)

where $G_{F}$ is Fermi coupling constant, and $|V_{cd}|$ is CKM matrix
element.

It should be noted that when calculating $\frac{d\Gamma}{dq^{2}}$, due to the
quark contents of scalar and axial-vector meson,

$\displaystyle|a_{0}^{0}(980)\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|u\bar{u}\rangle-|d\bar{d}\rangle),|a_{0}^{-}(980)\rangle=|d\bar{u}\rangle,$
$\displaystyle|a_{0}^{0}(1450)\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|u\bar{u}\rangle-|d\bar{d}\rangle),|a_{0}^{-}(1450)\rangle=|d\bar{u}\rangle,$
$\displaystyle|a_{1}^{0}(1260)\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|u\bar{u}\rangle-|d\bar{d}\rangle),|a_{1}^{-}(1260)\rangle=|d\bar{u}\rangle,$
$\displaystyle|b_{1}^{0}(1235)\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|u\bar{u}\rangle-|d\bar{d}\rangle),|b_{1}^{-}(1235)\rangle=|d\bar{u}\rangle,$
(50)

the semileptonic decay widths for $D^{+}\rightarrow S(A)l\bar{\nu}_{l}$
corresponding to Eqs. (46)-(49) should have an additional factor
$\frac{1}{2}$.

##  III Numerical analyses and discussion

###  III.1 Choices of input parameters

To calculate the numerical results of form factors and branching ratios, we
briefly discuss the parameters involved in these semileptonic decay processes.
For the initial state, we take the masses as $m_{D^{0}}=1.865$ GeV and
$m_{D^{+}}=1.870$ GeV Zyla:2020zbs , decay constant as
$f_{D^{0}}=f_{D^{+}}=205.4$ MeV Amhis:2019ckw . The quark masses are taken as
$m_{u}=2.16$ MeV, $m_{d}=4.67$ MeV and $m_{c}=1.27$ GeV Zyla:2020zbs . Masses,
decay constants Zyla:2020zbs  and Gegenbauer coefficients Cheng:2005nb ;
Yang:2008xw  for LCDAs of the final state mesons are shown in Table II. And
the parameters Zyla:2020zbs  involved in the calculation of the branching
ratios are shown in Table III.

Table 2: Masses, decay constants and Gegenbauer coefficients of distribution amplitudes of final state mesons at the scale $\mu=1$ GeV. Process |  $M$ [GeV] |  $\bar{f}_{M}$ [MeV] |  $f^{M}_{\bot}$ [MeV] |  $B_{1}$ |  $B_{3}$ |  $a_{0}^{\bot}$ |  $a_{1}^{\bot}$ |  $a_{2}^{\bot}$
---|---|---|---|---|---|---|---|---
$D\rightarrow a_{0}(980)$ |  0.980 |  0.365 |  - |  -0.93 |  0.14 |  - |  - |  -
$D\rightarrow a_{0}(1450)$ |  1.474 |  -0.280 |  - |  0.89 |  -1.38 |  - |  - |  -
$D\rightarrow a_{1}(1260)$ |  1.230 |  - |  238 |  - |  - |  0 |  -1.04 |  0
$D\rightarrow b_{1}(1235)$ |  1.230 |  - |  180 |  - |  - |  1 |  0 |  0.03
Table 3: Parameters in branching ratios. $G_{F}$ [GeV-2] |  $m_{e}$ $[\rm{GeV}]$ |  $m_{\mu}$ $[\rm{GeV}]$ |  $|V_{cd}|$ |  $\tau_{D^{+}}$ [s] |  $\tau_{D^{0}}$ [s]
---|---|---|---|---|---
$1.1664\times 10^{-5}$ |  $0.511\times 10^{-3}$ |  $0.106$ |  0.220 |  $1.040\pm 0.007\times 10^{-12}$ |  $4.101\pm 0.015\times 10^{-13}$

###  III.2 Dependence of form factors on threshold and Borel parameter

The choice of the continuum threshold and the Borel parameter is also crucial.
The threshold is generally chosen as the mass square of the first excited
state of $D$ meson, but this value is not universal and has to be determined
individually according to the sum rule of different processes. The choice of
the Borel parameter should satisfy:

(1) the contributions from the continuum and higher excited states are less
than 30%;

(2) the dependence of form factors on the Borel parameter is weak.
Based on the above conditions, the corresponding thresholds and the Borel
windows for different semileptonic decay processes are ascertained, as is
shown in Fig. 1-4.

For the convenience of discussion in the following, we take the central value
of threshold as $s_{0}=5.18$ GeV2 for $D\rightarrow a_{0}(980)$, $D\rightarrow
a_{0}(1450)$, $D\rightarrow a_{1}(1260)$ and $D\rightarrow b_{1}(1235)$
respectively.

Figure 1: Dependence of $D\rightarrow a_{0}(980)$ transition form factors and
penguin form factors on the Borel parameter $M^{2}$ at $q^{2}=0$ GeV2. The
dash, solid and dot lines correspond to thresholds $s_{0}=5.03$, $5.18$ and
$5.33$ GeV2 respectively.

Figure 2: Dependence of $D\rightarrow a_{0}(1450)$ transition form factors and
penguin form factors on the Borel parameter $M^{2}$ at $q^{2}=0$ GeV2. The
dash, solid and dot lines correspond to thresholds $s_{0}=5.03$, $5.18$ and
$5.33$ GeV2 respectively.

Figure 3: Dependence of $D\rightarrow a_{1}(1260)$ transition form factors and
penguin form factors on the Borel parameter $M^{2}$ at the transfer momentum
$q^{2}=0$ GeV2. The dash, solid and dot lines correspond to thresholds
$s_{0}=5.03$, $5.18$ and $5.33$ GeV2 respectively.

Figure 4: Dependence of $D\rightarrow b_{1}(1235)$ transition form factors and
penguin form factors on the Borel parameter $M^{2}$ at the transfer momentum
$q^{2}=0$ GeV2. The dash, solid and dot lines correspond to thresholds
$s_{0}=5.03$, $5.18$ and $5.33$ GeV2 respectively.

###  III.3 Results and discussion

The form factors at $q^{2}=0$ GeV2

Based on the above parameters, we get the numerical results of form factors in
case of $q^{2}=0$ GeV2, shown in Table IV and Table V .

Table 4: Comparison of numerical results of form factors for $D\rightarrow S$ with that of Covariant Confined Quark Model(CCQM) and Covariant Light-Front Quark Model(CLFQM). Process | Method | $f_{+}(0)$ | $f_{-}(0)$ | $f_{T}(0)$
---|---|---|---|---
$D^{0}\rightarrow a_{0}^{-}(980)$ | this work | $0.85^{+0.10}_{-0.11}$ | $-0.85^{+0.10}_{-0.11}$ | $2.29^{+0.29}_{-0.30}$
LCSRCheng:2017fkw  | $1.75^{+0.26}_{-0.27}$ | $0.31\pm 0.13$ | -
CCQMSoni:2020sgn  | $0.55\pm 0.02$ | $0.03\pm 0.01$ | -
$D^{+}\rightarrow a_{0}^{0}(980)$ | this work | $0.85^{+0.10}_{-0.11}$ | $-0.85^{+0.10}_{-0.11}$ | $2.29^{+0.29}_{-0.30}$
LCSRCheng:2017fkw  | $1.76\pm 0.26$ | $0.31\pm 0.13$ | -
CCQMSoni:2020sgn  | $0.55\pm 0.02$ | $0.03\pm 0.01$ | -
$D^{0}\rightarrow a_{0}^{-}(1450)$ | this work | $0.94^{+0.02}_{-0.03}$ | $-0.94^{+0.02}_{-0.03}$ | $2.48^{+0.07}_{-0.08}$
CLFQMVerma:2011yw  | $0.51^{-0.01+0.01}_{+0.01-0.02}$ | - | -
$D^{+}\rightarrow a_{0}^{0}(1450)$ | this work | $0.94^{+0.02}_{-0.03}$ | $-0.94^{+0.02}_{-0.03}$ | $2.48^{+0.07}_{-0.08}$
CLFQMVerma:2011yw  | $0.51^{-0.01+0.01}_{+0.01-0.02}$ | - | -

For $D\rightarrow a_{0}(980)$, our results for $f_{+}(0)$ are $51\%$ smaller
than those of Ref. Cheng:2017fkw , mainly due to the different input parameter
$\bar{f}_{s}$ and the inclusion of twist-3 LCDAs in Ref. Cheng:2017fkw .

Table 5: Comparison of numerical results of form factors for $D\rightarrow A$ with that of LCSR, 3-point QCD sum rules (3QR) and CLFQM (n=1,2,3).  Process | Method | $A(0)$ | $V_{1}(0)$ | $V_{2}(0)$ | $V_{0}(0)$ | $T_{n}(0)$
---|---|---|---|---|---|---
$D^{0}\rightarrow a_{1}^{-}(1260)$ | this work | $0.34^{+0.03}_{-0.04}$ | $2.63^{+0.20}_{-0.21}$ | $0.34^{+0.03}_{-0.04}$ | $0.24^{+0.00}_{-0.01}$ | $0.67^{+0.07}_{-0.07}$
LCSR Momeni:2019uag  | $0.07\pm 0.05$ | $0.37\pm 0.01$ | $-0.03\pm 0.02$ | $0.15\pm 0.05$ | -
3SR Zuo:2016msr  | $0.09$ | $0.77$ | $-0.01$ | $0.19$ | -
CLFQM Verma:2011yw  | $0.19^{-0.01+0.00}_{+0.01-0.00}$ | $1.51^{-0.04+0.00}_{+0.04-0.01}$ | $0.05^{-0.01+0.00}_{+0.00-0.00}$ | $0.32^{-0.00-0.00}_{+0.00-0.00}$ | -
$D^{+}\rightarrow a_{1}^{0}(1260)$ | this work | $0.34^{+0.03}_{-0.04}$ | $2.63^{+0.20}_{-0.21}$ | $0.34^{+0.03}_{-0.04}$ | $0.24^{+0.00}_{-0.01}$ | $0.67^{+0.07}_{-0.07}$
LCSR Momeni:2019uag  | $0.04\pm 0.04$ | $0.26\pm 0.08$ | $-0.02\pm 0.01$ | $0.10\pm 0.03$ | -
3SR Zuo:2016msr  | $0.08$ | $0.54$ | $-0.00$ | $0.12$ | -
CLFQM Verma:2011yw  | $0.19^{-0.01+0.00}_{+0.01-0.00}$ | $1.51^{-0.04+0.00}_{+0.04-0.01}$ | $0.05^{-0.01+0.00}_{+0.00-0.00}$ | $0.32^{-0.00-0.00}_{+0.00-0.00}$ | -
$D^{0}\rightarrow b_{1}^{-}(1235)$ | this work | $-0.24^{+0.01}_{-0.00}$ | $-1.78^{+0.03}_{-0.02}$ | $-0.24^{+0.01}_{-0.00}$ | $-0.16^{+0.00}_{-0.00}$ | $-0.47^{+0.01}_{-0.01}$
LCSR Momeni:2019uag  | $-0.41$ | $-0.22$ | 0.21 | $-0.32$ | -
CLFQM Verma:2011yw  | $0.12^{+0.00+0.00}_{-0.00-0.02}$ | $1.39^{+0.02+0.03}_{-0.02-0.04}$ | $-0.10^{+0.02-0.01}_{-0.02+0.01}$ | $0.50^{-0.01+0.02}_{+0.01-0.02}$ | -
$D^{+}\rightarrow b_{1}^{0}(1235)$ | this work | $-0.24^{+0.01}_{-0.00}$ | $-1.78^{+0.03}_{-0.02}$ | $-0.24^{+0.01}_{-0.00}$ | $-0.16^{+0.00}_{-0.00}$ | $-0.47^{+0.01}_{-0.01}$
LCSR Momeni:2019uag  | $-0.28$ | $-0.16$ | 0.15 | $-0.23$ | -
CLFQM Verma:2011yw  | $0.12^{+0.00+0.00}_{-0.00-0.02}$ | $1.39^{+0.02+0.03}_{-0.02-0.04}$ | $-0.10^{+0.02-0.01}_{-0.02+0.01}$ | $0.50^{-0.01+0.02}_{+0.01-0.02}$ | -

In Table V we give the form factors for $D\rightarrow a_{1}(1260)$ and
$D\rightarrow b_{1}(1235)$, which is larger compared to the results of Ref.
Momeni:2019uag , since contribution from the high twist LCDAs is included.

The form factors at $q^{2}\neq 0$ GeV2

Fig. 3 shows the dependence of the form factors on $m_{l}^{2}\leq
q^{2}\leq(m_{D}-m_{S})^{2}$ for $D\rightarrow S$. For $D\rightarrow
a_{0}(980)$, we take the threshold as $s_{0}=5.18$ GeV2 and the Borel
parameter as $M^{2}=1.72$ GeV2. For $D\rightarrow a_{0}(1450)$, we take the
threshold as $s_{0}=5.18$ GeV2 and the Borel parameter as $M^{2}=1.74$ GeV2.

Figure 5: Dependence of the form factors on the transfer momentum $q^{2}$ for
$D\rightarrow a_{0}(980)$ and $D\rightarrow a_{0}(1450)$ decay. The solid and
dash lines depict the form factors $f_{+}(q^{2})$, $f_{T}(q^{2})$ ,
respectively.

Fig. 4 shows the dependence of the form factors on $m_{l}^{2}\leq
q^{2}\leq(m_{D}-m_{A})^{2}$ for $D\rightarrow A$. For $D\rightarrow
a_{1}(1260)$, we take the threshold $s_{0}=5.18$ GeV2 and the Borel parameter
$M^{2}=4$ GeV2, and for $D\rightarrow b_{1}(1235)$, we take the threshold
$s_{0}=5.18$ GeV2 and the Borel parameter $M^{2}=6$ GeV2.

Figure 6: Dependence of the form factors decay on the transfer momentum
$q^{2}$ for $D\rightarrow a_{1}(1260)$ and $D\rightarrow b_{1}(1235)$ . The
solid, dash, dash-dot, dash-dot-dot, and dot lines depict the form factors
$V_{1}(q^{2})$, $A(q^{2})(V_{2}(q^{2}))$, $V_{0}(q^{2})$,
$T_{1}(q^{2})(T_{3}(q^{2}))$, $T_{2}(q^{2})$, respectively.

Branching ratios

Fig. 5 and Fig. 6 show the dependence of the differential decay widths on the
transfer momentum $q^{2}$ for $D\rightarrow Sl\bar{\nu}_{l}$ and $D\rightarrow
Al\bar{\nu}_{l}$, respectively.

Figure 7: $D\rightarrow a_{0}(980)l\bar{\nu}_{l}$ differential decay widths at
$m_{e}^{2}\leq q^{2}\leq(m_{D}-m_{a_{0}(980)})^{2}$ , and $D\rightarrow
a_{0}(1450)l\bar{\nu}_{l}$ differential decay widths at $m_{e}^{2}\leq
q^{2}\leq(m_{D}-m_{a_{0}(1450)})^{2}$.

Figure 8: $D\rightarrow a_{1}(1260)l\bar{\nu}_{l}$ differential decay widths
at $m_{e}^{2}\leq q^{2}\leq(m_{D}-m_{a_{1}(1260)})^{2}$, and $D\rightarrow
b_{1}(1235)l\bar{\nu}_{l}$ differential decay widths at $m_{e}^{2}\leq
q^{2}\leq(m_{D}-m_{b_{1}(1235)})^{2}$ .

For differential decay widths, integrating over a range of $m_{e}^{2}\leq
q^{2}\leq(m_{D}-m_{M})^{2}$, we obtain the decay widths and hence the decay
branching ratios, the numerical results of which are shown in Tables VI and
VII.

Table 6: Comparison of numerical results of branching ratios for $D\rightarrow Sl\bar{\nu}_{l}$ with other theoretical methods and experiments.  Process | Method | Branching ratios | $\mathcal{B}(D^{0(+)}\rightarrow Sl\bar{\nu}_{e})\times\mathcal{B}(S\rightarrow\etaup\pi)$
---|---|---|---
$D^{0}\rightarrow a_{0}^{-}(980)e^{+}\nu_{e}$ | this work | $1.36\times 10^{-4}$ | $1.15\times 10^{-4}$
CCQM Soni:2020sgn  | $1.68\pm 0.15\times 10^{-4}$ | $1.84\times 10^{-4}$
LCSR Cheng:2017fkw  | $4.08^{+1.37}_{-1.22}\times 10^{-4}$ | $3.45\times 10^{-4}$
BES III Ablikim:2018ffp  | - | $1.33^{+0.33}_{-0.29}\times 10^{-4}$
$D^{0}\rightarrow a_{0}^{-}(980)\mu^{+}\nu_{\mu}$ | this work | $1.21\times 10^{-4}$ | -
CCQM Soni:2020sgn  | $1.63\pm 0.14\times 10^{-4}$ | -
$D^{+}\rightarrow a_{0}^{0}(980)e^{+}\nu_{e}$ | this work | $1.79\times 10^{-4}$ | $1.51\times 10^{-4}$
CCQM Soni:2020sgn  | $2.18\pm 0.38\times 10^{-4}$ | $1.84\times 10^{-4}$
LCSR Cheng:2017fkw  | $5.40^{+1.78}_{-1.59}\times 10^{-4}$ | $4.56\times 10^{-4}$
BES III Ablikim:2018ffp  | - | $1.66^{+0.81}_{-0.66}\times 10^{-4}$
$D^{+}\rightarrow a_{0}^{0}(980)\mu^{+}\nu_{\mu}$ | this work | $1.59\times 10^{-4}$ | -
CCQM Soni:2020sgn  | $2.12\pm 0.37\times 10^{-4}$ | -
$D^{0}\rightarrow a_{0}^{-}(1450)e^{+}\nu_{e}$ | this work | $3.14\times 10^{-6}$ | $0.29\times 10^{-6}$
$D^{0}\rightarrow a_{0}^{-}(1450)\mu^{+}\nu_{\mu}$ | this work | $2.01\times 10^{-6}$ | -
$D^{+}\rightarrow a_{0}^{0}(1450)e^{+}\nu_{e}$ | this work | $4.28\times 10^{-6}$ | $0.40\times 10^{-6}$
CLFQM Cheng:2017pcq  | $5.4\pm 0.5\times 10^{-6}$ | -
$D^{+}\rightarrow a_{0}^{0}(1450)\mu^{+}\nu_{\mu}$ | this work | $2.76\times 10^{-6}$ | -
CLFQM Cheng:2017pcq  | $3.8\pm 0.3\times 10^{-6}$ | -

In Table VI, for $D^{0(+)}\rightarrow a_{0}^{-(0)}(980)e^{+}\nu_{e}$ , it is
easy to see that our results are well consistent with those of BES III
Ablikim:2018ffp  within the error, where Cheng:2013fba

$\displaystyle\mathcal{B}(a_{0}(980)\rightarrow\etaup\pi)$
$\displaystyle=0.845\pm 0.017.$ (51)

The difference between our results and the results of LCSR Cheng:2017fkw
methods is mainly caused by form factors. It is easy to see in Eq (46) that
$(f_{+}(q^{2}))^{2}$ makes the main contribution in the differential decay
width, with $(f_{+}(q^{2})f_{-}(q^{2}))$ and $f_{-}(q^{2})$ being suppressed
by $m_{l}^{2}$. We also calculate the ratio of the partial width

$\displaystyle\frac{\Gamma(D^{0}\rightarrow
a_{0}^{-0}(980)e^{+}\nu_{e})}{\Gamma(D^{+}\rightarrow
a_{0}^{0}(980)e^{+}\nu_{e})}=2.17,$ (52)

which is consistent with the prediction $2.03\pm 0.95\pm 0.06$ by BES III
Ablikim:2018ffp  within the error.

For $D\rightarrow a_{0}(1450)l\bar{\nu}_{l}$, we predict the results of the
branching ratios at $\mathcal{B}(a_{0}(1450)\rightarrow\etaup\pi)=0.093$
Zyla:2020zbs  and also calculate the partial width ratios,

$\displaystyle\frac{\Gamma(D^{0}\rightarrow
a_{0}^{-0}(1450)e^{+}\nu_{e})}{\Gamma(D^{+}\rightarrow
a_{0}^{0}(1450)e^{+}\nu_{e})}=2.03,$ (53)

expecting them to be tested experimentally in the future.

Table 7: Comparison of numerical results of branching ratios of $D\rightarrow Al\bar{\nu}_{l}$ with the results of other theoretical methods and experiments.  Process | Method | Branching ratios | $\mathcal{B}(D\rightarrow b_{1}(1235)l\bar{\nu}_{l})\times\mathcal{B}(b_{1}(1235)\rightarrow\omegaup\pi)$
---|---|---|---
$D^{0}\rightarrow a_{1}^{-}(1260)e^{+}\nu_{e}$ | this work | $6.90\times 10^{-5}$ | -
$D^{0}\rightarrow a_{1}^{-}(1260)\mu^{+}\nu_{\mu}$ | this work | $6.27\times 10^{-5}$ | -
$D^{+}\rightarrow a_{1}^{0}(1260)e^{+}\nu_{e}$ | this work | $9.38\times 10^{-5}$ | -
$D^{+}\rightarrow a_{1}^{0}(1260)\mu^{+}\nu_{\mu}$ | this work | $8.52\times 10^{-5}$ | -
$D^{0}\rightarrow b_{1}^{-}(1235)e^{+}\nu_{e}$ | this work | $4.85\times 10^{-5}$ | $4.85\times 10^{-5}$
BES III Ablikim:2020agq  | - | $<1.12\times 10^{-4}$
$D^{0}\rightarrow b_{1}^{-}(1235)\mu^{+}\nu_{\mu}$ | this work | $4.40\times 10^{-5}$ | -
$D^{+}\rightarrow b_{1}^{0}(1235)e^{+}\nu_{e}$ | this work | $6.58\times 10^{-5}$ | $6.58\times 10^{-5}$
CLFQM Cheng:2017pcq  | $7.4\pm 0.7\times 10^{-5}$ | $7.4\pm 0.7\times 10^{-5}$
BES III Ablikim:2020agq  | - | $<1.75\times 10^{-4}$
$D^{+}\rightarrow b_{1}^{0}(1235)\mu^{+}\nu_{\mu}$ | this work | $6.00\times 10^{-5}$ | -
CLFQM Cheng:2017pcq  | $6.4\pm 0.6\times 10^{-5}$ | -

For $D\rightarrow a_{1}(1260)l\bar{\nu}_{l}$ and $D^{0(+)}\rightarrow
b^{-(0)}_{1}(1235)le^{+}\nu_{e}$, BES III gave an upper limit for
$\mathcal{B}(D^{0(+)}\rightarrow
b_{1}^{-(0)}(1235)e^{+}\nu_{e})\times\mathcal{B}(b_{1}^{-(0)}(1235)\rightarrow\omegaup\pi^{-(0)})$
Ablikim:2020agq . When taking
$\mathcal{B}(b_{1}(1235)\rightarrow\omegaup\pi$)=1 Zyla:2020zbs , our results
for $D^{0(+)}\rightarrow b^{-(0)}_{1}(1235)le^{+}\nu_{e}$ are consistent with
the experiment within the error, while that for $D\rightarrow
a_{1}(1260)l\bar{\nu}_{l}$ are expected to be tested by experiment in the
future.

##  IV Summary

We systematically study the semileptonic decay process of $D\rightarrow
S,Al\bar{\nu_{l}}$ by LCSR with chiral currents. For $D\rightarrow
Sl\bar{\nu_{l}}$ we take $a_{0}(980)$ and $a_{0}(1450)$ meson as $q\bar{q}$
states, expecting our results to be helpful in determining the hadron
structures of $a_{0}(980)$ and $a_{0}(1450)$. For $D\rightarrow
Al\bar{\nu_{l}}$, we study $a_{1}(1260)(1^{3}p^{1})$ and
$b_{1}(1235)(1^{1}p^{1})$ mesons. Due to the chiral currents, our methods
eliminate the contributions of the higher twist LCDAs, which avoid the
uncertainty from the high twist LCDAs. Simple relations between form factors
for $D\rightarrow S,Al\bar{\nu_{l}}$ are also obtained, which are similar to
that obtained from the $B\rightarrow S,Al\bar{\nu_{l}}$ process Sun:2010nv ;
YanJun:2011rn .

We also systematically analyze the dependence of form factors and differential
decay widths on the transfer momentum $q^{2}$. Applying the results of form
factors and differential decay widths, we present the branching ratios of
these semileptonic decay processes. Our numerical results for $D\rightarrow
a_{0}(980),b_{1}(1235)l\bar{\nu_{l}}$ are in good agreement with experiments
and that for the $D\rightarrow a_{0}(1450),a_{1}(1260)l\bar{\nu_{l}}$
processes are expected to be tested experimentally in the future.

## ACKNOWLEDGEMENTS

Y.J. Sun would like to thank Y.M. Wang for helpful discussions. This work was
supported in part by Natural Science Foundation of China under Grant
Nos.11365018, 11375240 and 11565023.

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