#  $a_{1}(1260)$-meson longitudinal twist-2 distribution amplitude and the
$D\to a_{1}(1260)\ell^{+}\nu_{\ell}$ decay processes

Dan-Dan Hu  Hai-Bing Fu<EMAIL_ADDRESS>(Corresponding author) Tao Zhong
<EMAIL_ADDRESS>Zai-Hui Wu  Department of Physics, Guizhou Minzu
University, Guiyang 550025, P.R.China  Xing-Gang Wu<EMAIL_ADDRESS>Department of Physics, Chongqing Key Laboratory for Strongly Coupled Physics,
Chongqing University, Chongqing 401331, P.R. China

###### Abstract

In the paper, we investigate the moments $\langle\xi_{2;a_{1}}^{\|;n}\rangle$
of the axial-vector $a_{1}(1260)$-meson distribution amplitude by using the
QCD sum rules approach under the background field theory. By considering the
vacuum condensates up to dimension-six and the perturbative part up to next-
to-leading order QCD corrections, its first five moments at an initial scale
$\mu_{0}=1~{}{\rm GeV}$ are
$\langle\xi_{2;a_{1}}^{\|;2}\rangle|_{\mu_{0}}=0.223\pm 0.029$,
$\langle\xi_{2;a_{1}}^{\|;4}\rangle|_{\mu_{0}}=0.098\pm 0.008$,
$\langle\xi_{2;a_{1}}^{\|;6}\rangle|_{\mu_{0}}=0.056\pm 0.006$,
$\langle\xi_{2;a_{1}}^{\|;8}\rangle|_{\mu_{0}}=0.039\pm 0.004$ and
$\langle\xi_{2;a_{1}}^{\|;10}\rangle|_{\mu_{0}}=0.028\pm 0.003$, respectively.
We then construct a light-cone harmonic oscillator model for
$a_{1}(1260)$-meson longitudinal twist-2 distribution amplitude
$\phi_{2;a_{1}}^{\|}(x,\mu)$, whose model parameters are fitted by using the
least squares method. As an application of $\phi_{2;a_{1}}^{\|}(x,\mu)$, we
calculate the transition form factors (TFFs) of $D\to a_{1}(1260)$ in large
and intermediate momentum transfers by using the QCD light-cone sum rules
approach. At the largest recoil point ($q^{2}=0$), we obtain
$A(0)=0.130_{-0.013}^{+0.015}$, $V_{1}(0)=1.898_{-0.121}^{+0.128}$,
$V_{2}(0)=0.228_{-0.021}^{+0.020}$, and $V_{0}(0)=0.217_{-0.025}^{+0.023}$. By
applying the extrapolated TFFs to the semi-leptonic decay $D^{0(+)}\to
a_{1}^{-(0)}(1260)\ell^{+}\nu_{\ell}$, we obtain ${\cal B}(D^{0}\to
a_{1}^{-}(1260)e^{+}\nu_{e})=(5.261_{-0.639}^{+0.745})\times 10^{-5}$, ${\cal
B}(D^{+}\to a_{1}^{0}(1260)e^{+}\nu_{e})=(6.673_{-0.811}^{+0.947})\times
10^{-5}$, ${\cal B}(D^{0}\to
a_{1}^{-}(1260)\mu^{+}\nu_{\mu})=(4.732_{-0.590}^{+0.685})\times 10^{-5}$,
${\cal B}(D^{+}\to
a_{1}^{0}(1260)\mu^{+}\nu_{\mu})=(6.002_{-0.748}^{+0.796})\times 10^{-5}$.

###### pacs:

13.25.Hw, 11.55.Hx, 12.38.Aw, 14.40.Be

##  I Introduction

The $D$-meson semileptonic decays into axial-vector mesons are the key
components for understanding the nonperturbative effects in weak interactions,
which have been studied by many theoretical and experimental groups. Based on
the constituent quark model, the quantum number of a meson is determined by
the quantum numbers of all the constituent quarks. There are two types of
axial-vector mesons, ${}^{1}P_{1}$-state with the quantum state
$J^{PC}=1^{+-}$ and ${}^{3}P_{1}$-state with $J^{PC}=1^{++}$. Among the axial-
vector mesons, only the isospin triplet heavy state mesons $b_{1}(1235)$ and
$a_{1}(1260)$ do not have a mixing phenomenon. Their internal structures are
relatively clear. At the hadron level, the $a_{1}(1260)$-meson is a good
subject which is considered as the chiral adjoint state of $\rho$-meson. Thus
$a_{1}(1260)$ and $\rho$-meson are defined as the light-quark pair $q\bar{q}$
with $q=(u,d)$ Dhar:1983fr ; Wingate:1995hy ; Wakayama:2019crb . Since 1986,
the properties of $a_{1}(1260)$-meson have been accurately measured
Weinstein:1993jp . Besides, the observation of the charmless hadronic decay
processes involving $a_{1}(1260)$-meson, such as $B^{0}\to
a_{1}^{\pm}(1260)\pi^{\mp}$, which have been issued by the BABAR and Belle
collaborations Aubert:2004xg ; Aubert:2006dd ; Aubert:2006gb ; Abe:2007jn ,
indicates that $a_{1}(1260)$ is a ${}^{3}P_{1}$-state. Those measurements help
us to investigate the production mechanism of axial-vectors via hadronic decay
processes and to probe the structures of axial-vector meson. Thus it is
important to give a detailed theoretical study on the semileptonic decay $D\to
a_{1}(1260)\ell^{+}\nu_{\ell}$.

The transition form factors (TFFs) of $D\to a_{1}(1260)$ are key components
for investigating the corresponding semileptonic decays. The TFFs for heavy-
to-light decay processes have been calculated under various approaches, such
as the QCD sum rules (QCDSR) Aliev:1999mx , the covariant light-front quark
model (CLFQM) Cheng:2003sm ; Cheng:2017pcq , the constituent quark model (CQM)
Deandrea:1998ww , the light-cone sum rules (LCSR) Wang:2008bw ; Yang:2007zt ;
YanJun:2011rn ; Momeni:2016kjz , the relativistic quark model (RQM)
Faustov:2019mqr , the perturbative QCD (PQCD) Li:2009tx ; Liu:2012jb , the QCD
factorization (QCDF) Lu:2021ttf , the three-point QCD sum rules (3PSR)
Khosravi:2014hqa , and etc. Among them, the LCSR approach provides an
effective way in determining the non-perturbative parameters of hadronic
states. By using the LCSR approach, one can carry out the operator product
expansion (OPE) near the light-cone $x^{2}\approx 0$, and the nonperturbative
hadronic matrix elements can be parameterized as the light-cone distribution
amplitudes (LCDAs) with various twists. In the present paper, we shall adopt
the LCSR approach to deal with the $D\to a_{1}(1260)$ TFFs by using a left-
handed chiral current, which can highlight the longitudinal leading-twist LCDA
contribution.

Within the LCSR approach, the LCDAs not only are basic parameters of hard
exclusive processes, but also can reflect the dynamic information of the
internal structure of hadrons. One can expand the longitudinal twist-2 LCDA
$\phi_{2;a_{1}}^{\|}(x,\mu)$ of $a_{1}(1260)$-meson at an arbitrary scale
$\mu$ into a Gegenbauer series,
$\phi_{2;a_{1}}^{\|}(x,\mu)=6x\bar{x}[1+\sum_{n}a^{\|;n}_{2;a_{1}}(\mu)C_{n}^{3/2}(\xi)]$,
where $\bar{x}=(1-x)$, $\xi=(2x-1)$ and $a^{\|;n}_{2;a_{1}}(\mu)$ are
Gegenbauer moments, whose first nonzero one has been given by Yang at the
initial scale $\mu_{0}=1~{}{\rm GeV}$ Yang:2007zt ,
$a_{2;a_{1}}^{\|;2}(\mu_{0})=-0.02(2)$. This value, along with the higher
order of $a^{\|;n}_{2;a_{1}}(\mu)$, can be calculated within the framework of
QCDSR under the background field theory (BFTSR) Huang:1989gv . It has been
pointed out that the LCDA model based on conformal expansion that is truncated
after its first few terms is not suitable for all cases, since the higher-
order Gegenbauer terms may have sizable contributions, even if they are
generally power suppressed with the increment of $n$ for large momentum
transforms. Thus it is important to know more moments for a precise
determination of LCDA.

The leading-twist LCDA of a meson can be related to its Bethe-Salpeter wave
function. Previous works mainly focused on the wave functions of the
pseudoscalar or the vector mesons, cf. Refs. Ball:2005ei ; Zhong:2018exo ;
Fu:2014cna ; Huang:1994dy ; Vega:2009zb , and there is few research on the
axial-vector mesons. In this work, we will construct a light-cone harmonic
oscillator (LCHO) model for the $a_{1}(1260)$-meson longitudinal twist-2 LCDA.
The parameters of the model shall be fitted by using the newly calculated
moments $\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$. And we will use the least
squares method to do the fitting and to get the optimal solution. For the
purpose, we will use BFTSR to calculate the moments
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$. Within the framework of BFTSR,
the quark and gluon fields are composed by the background fields and their
surrounding quantum fluctuations, and the usual vacuum condensates are
described by the classical background fields, which provides a clear physical
picture for the bound-state internal structures and makes the sum rules
calculation more simplified. The BFTSR have been widely used in calculating
the LCDAs of the heavy/light mesons Zhong:2014jla ; Fu:2018vap ; Zhong:2016kuv
; Huang:2004tp ; Zhong:2011rg ; Hu:2021zmy . Here we will adopt this approach
to investigate the $a_{1}(1260)$-meson moments
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ and then provide a more accuracy
LCDA $\phi_{2;a_{1}}^{\|}(x,\mu)$.

The remaining parts of the paper are organized as follows. In Sec. II, we
present the calculation procedures for the moments of $a_{1}(1260)$-meson
longitudinal twist-2 LCDA, the LCHO model, the TFFs and the branching ratios.
Numerical results and discussions are presented in Sec. III. Section IV is
reserved for a summary.

##  II Calculation Technology

Firstly, the $a_{1}(1260)$-meson longitudinal twist-2 LCDA is defined as
Yang:2007zt

$\displaystyle\langle
0|{\bar{q}_{1}}(z){\gamma_{\mu}}{\gamma_{5}}{q_{2}}(-z)|{a_{1}}(q,\lambda)\rangle$
$\displaystyle\qquad=im_{a_{1}}f^{\|}_{a_{1}}\int_{0}^{1}dxe^{i(xz\cdot
q-\bar{x}z\cdot q)}q_{\mu}\frac{{{e^{*(\lambda)}}\cdot z}}{{q\cdot
z}}\phi_{2;{a_{1}}}^{\parallel}(x,\mu).$ (1)

Here, two light quarks $q_{1}=q_{2}$, which are $(u,d)$-quark for
$a_{1}(1260)$-meson, respectively. This convention shall be followed
throughout the remaining parts of this paper. The $f_{a_{1}}^{\|}$ is
$a_{1}(1260)$-meson decay constant, $q$ and ${e^{*(\lambda)}}$ are momentum
and polarization vector of $a_{1}(1260)$-meson. The polarization vector
satisfies the relationship $(e^{*(\lambda)}\cdot z)/({q\cdot z})\to
1/{m_{a_{1}}}$ Ball:2004rg . By doing the series expansion near $z\to 0$ on
both sides of Eq. (1), one will get:

$\displaystyle\langle
0|{\bar{q}_{1}}(0)/\\!\\!\\!z\gamma_{5}{(iz\cdot\tensor{D})^{n}}{q_{2}}(0)|$
$\displaystyle a_{1}(q,\lambda)\rangle$ $\displaystyle=i{(z\cdot
q)}^{n+1}f_{a_{1}}^{\|}\langle\xi^{n;\|}_{2;a_{1}}\rangle|_{\mu},$ (2)

where the covariant derivative satisfies the relation
$(iz\cdot\tensor{D})^{n}=(iz\cdot\overrightarrow{D}-iz\cdot\overleftarrow{D})^{n}$.
The $n$-th order moments of the $a_{1}(1260)$-meson DA are defined as

$\displaystyle\langle\xi_{2;{a_{1}}}^{\parallel;n}\rangle{|_{\mu}}=\int_{0}^{1}dx{{{(2x-1)}^{n}}}\phi_{2;{a_{1}}}^{\parallel}(x,\mu)$
(3)

One can start from the following correlation function (correlator) to derive
the sum rules, i.e.

$\displaystyle\Pi_{2;a_{1}}^{(n,0)}(z,q)$ $\displaystyle=$ $\displaystyle
i\int d^{4}xe^{iq\cdot x}\langle
0|T\\{J_{n}(x),J_{0}^{\dagger}(0)\\}|0\rangle$ (4) $\displaystyle=$
$\displaystyle(z\cdot q)^{n+2}I_{2;a_{1}}^{(n,0)}(q^{2}),$

with
$J_{n}(x)=\bar{q}_{1}(x)/\\!\\!\\!z\gamma_{5}(iz\cdot\tensor{D})^{n}q_{2}(x)$,
$J_{0}^{\dagger}(0)=\bar{q}_{2}(0)/\\!\\!\\!z\gamma_{5}q_{1}(0)$ and
$z^{2}=0$.111Here, the current $J_{n}(x)$ mainly comes from the basic
definition of $\phi_{2;^{3}P_{1}}^{\|}(x,\mu)$, which is also in agreement
with the current in calculating the Gegenbauer moments of ${}^{3}P_{1}$ axial-
vector meson, i.e. $a_{2;^{3}P_{1}}^{\|,n}(\mu)$ from Ref. Yang:2007zt . The
slightly difference lies in the relationship between two types of moments.
Because of the G-parity, $\phi_{2;a_{1}}^{\|}(x,\mu)$ for ${}^{3}P_{1}$-state
defined by the nonlocal axial-vector current is symmetric, indicating only
even moments are non-zero, i.e. $n=(0,2,4,6,\cdots)$. Based on the idea of
BFTSR and the Feynman rules for one hand, one can apply the OPE for the
correlator (4) in deep Euclidean region $q^{2}\ll 0$. Then, the correlator can
be expanded into three terms including the quark propagators $S_{F}^{d}(0,x)$,
$S_{F}^{u}(x,0)$ and the vertex operators $(iz\cdot\tensor{D})^{n}$, which
have been given in our previous work Hu:2021zmy . In dealing with Lorentz
invariant scalar function $\Pi_{2;a_{1}}^{(n,0)}(z,q^{2})$, the vacuum matrix
element should be used, which can be found in Ref. Zhong:2014jla . On the
other hand, one can insert a complete set of $a_{1}(1260)$-meson intermediated
hadronic states with the same $J^{P}$ quantum number into the correlator and
obtain

$\displaystyle\text{Im}I_{2;a_{1},{\rm Had}}^{(n,0)}(q^{2})$ $\displaystyle=$
$\displaystyle\pi\delta(q^{2}-m_{a_{1}}^{2})f_{a_{1}}^{2}\langle\xi^{\|;n}_{2;a_{1}}\rangle|_{\mu}\langle\xi_{2;a_{1}}^{\|;0}\rangle|_{\mu}$
(5) $\displaystyle+$
$\displaystyle\frac{3}{4\pi(n+1)(n+3)}\theta(q^{2}-s_{a_{1}}),$

where $s_{a_{1}}$ is the continuum threshold. The first (second) terms on the
r.h.s of Eq. (5) are the $a_{1}(1260)$-meson ground state (continuum states)
contribution. Due to not all higher-order contributions and higher-dimensional
operators have been included, the fixed-order prediction of
$\langle\xi_{2;a_{1}}^{\|;0}\rangle|_{\mu}$ is close but not exactly equals to
$1$. So we reserve the term $\langle\xi_{2;a_{1}}^{\|;0}\rangle|_{\mu}$ in the
hadronic expression, i.e. Eq. (5), which is different from other researches in
dealing with the axial-meson DA by using the QCDSR approach. Then, one can
bridge the invariant function and the OPE side by using the dispersion
relation. Furthermore, the Borel transformation are used to suppress the
contribution from the continuum states and high dimension condensates. The sum
rule expression is

$\displaystyle\frac{1}{\pi M^{2}}\int
dse^{-s/M^{2}}\textrm{Im}I_{a_{1},\textrm{had}}^{(n,0)}(s)=\hat{\cal
B}_{M^{2}}I_{2;a_{1},{\rm QCD}}^{(n,0)}(q^{2}),$ (6)

where the Borel parameter $M^{2}$ coming from the Borel transformation with
the operator $\hat{\cal B}_{M^{2}}$. Following the standard SVZ sum rules
procedures, we then obtain the expression of the moment of $a_{1}(1260)$-meson
longitudinal twist-2 LCDA, i.e.,

$\displaystyle~{}~{}\frac{f_{a_{1}}^{\|2}\langle\xi^{\|;n}_{2;a_{1}}\rangle|_{\mu}\langle\xi^{\|;0}_{2;a_{1}}\rangle|_{\mu}}{M^{2}e^{m_{a_{1}}^{2}/M^{2}}}=\frac{3}{4\pi^{2}(n+1)(n+3)}\bigg{(}1+\frac{\alpha_{s}}{\pi}A^{\prime}_{n}\bigg{)}(1-e^{-{s_{a_{1}}}/M^{2}})+\frac{(m_{u}+m_{d})\langle\bar{q}q\rangle}{M^{4}}+\frac{\langle\alpha_{s}G^{2}\rangle}{12\pi
M^{4}}\frac{1+n\theta(n-2)}{n+1}$
$\displaystyle~{}~{}\qquad-\frac{8n+1}{18}~{}\frac{{(m_{u}+m_{d})}\langle
g_{s}\bar{q}\sigma TGq\rangle}{M^{6}}+\frac{\langle
g_{s}\bar{q}q\rangle^{2}}{81{M^{6}}}4(2n+1)-\frac{\langle
g_{s}^{3}f{G^{3}}\rangle}{48\pi^{2}M^{6}}~{}n\theta(n-2)~{}+\,\frac{\langle
g_{s}^{2}\bar{q}q\rangle^{2}}{M^{6}}~{}\frac{2+\kappa^{2}}{486\pi^{2}}\bigg{\\{}3\,(17n+35)$
$\displaystyle~{}~{}\qquad-2(51n+25)\bigg{(}-\ln\frac{M^{2}}{\mu^{2}}\bigg{)}+\theta(n-2)\bigg{[}2n\bigg{(}-\ln\frac{M^{2}}{\mu^{2}}\bigg{)}-25(2n+1)~{}\tilde{\psi}(n)+\frac{1}{n}(49{n^{2}}+100n+56)\bigg{]}\bigg{\\}}.$
(7)

Here, $\tilde{\psi}(n)=\psi((n+1)/2)-\psi(n/2)+\ln 4$. The next-to-leading
(NLO) corrections are $A^{\prime}_{0}=0$, $A^{\prime}_{2}=5/3$,
$A^{\prime}_{4}=59/127$, $A^{\prime}_{6}=353/135$ Ball:1996tb . When taking
$n=0$ for the Eq. (7), one can get the sum rule of zeroth moment, e.g.
$\langle\xi_{2;a_{1}}^{\|;0}\rangle|_{\mu}$. To derive more accurate
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$, we adopt

$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}=\frac{(\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}\langle\xi_{2;a_{1}}^{\|;0}\rangle|_{\mu})\Big{|}_{\text{From
Eq.~{}\eqref{xi2}}}}{\sqrt{(\langle\xi_{2;a_{1}}^{\|;0}\rangle|_{\mu})^{2}}},$
(8)

where, the numerator of Eq. (8) are coming from the Eq. (7), and denominator
are the zeroth moment.

Owning to the fact that the high-order Gegenbauer moment for
$a_{1}(1260)$-meson longitudinal twist-2 DA still have large uncertainties,
one can construct a new LCDA model, i.e. the LCHO model based on the Brodsky-
Huang-Lepage (BHL) prescription Guo:1991eb ; Huang:1994dy . The BHL suggested
a connection between the equal-time wave function in the rest frame and the
light-cone wave function by equating the off-shell propagator $\epsilon$ in
the two frames. For the former
$\epsilon=M^{2}-\left(\sum_{i=1}^{n}q_{i}^{0}\right)^{2}$ with
$\sum_{i=1}^{n}\textbf{q}_{i}=0$, for the latter
$\epsilon=M^{2}-\sum_{i=1}^{n}\left[(\textbf{k}_{\perp
i}^{2}+m_{i}^{2})/x_{i}\right]$ with $\sum_{i=1}^{n}\textbf{k}_{\perp i}=0$
and $\sum_{i=1}^{n}x_{i}=1$. In the two-particle system, one has

$\displaystyle\textbf{q}^{2}\longleftrightarrow\frac{\textbf{k}_{\perp}^{2}+m_{q}^{2}}{4x(1-x)}-m_{q}^{2},$
(9)

with $m_{1}=m_{2}=m_{q}$. Then the possible connection between the rest frame
wave function $\psi_{\rm CM}(\textbf{q}^{2})$ and the light-cone wave function
$\psi_{\rm LC}(x,\textbf{k}_{\perp})$ can be formally represented by

$\displaystyle\psi_{\rm CM}(\textbf{q}^{2})\longleftrightarrow\psi_{\rm
LC}\left(\frac{\textbf{k}_{\perp}^{2}+m_{q}^{2}}{4x(1-x)}-m_{q}^{2}\right).$
(10)

On the other hand, the wave function of the harmonic oscillator model in the
rest frame is

$\displaystyle\psi_{\rm
CM}(\textbf{q}^{2})=A\exp\left(-\frac{\textbf{q}^{2}}{2\beta^{2}}\right),$
(11)

from an approximate bound state solution in the quark models for mesons. By
combining Eqs. (10) and (11), the LCHO model of $a_{1}(1260)$-meson wave
function satisfies

$\displaystyle\Psi_{2;a_{1}}^{\|}(x,{\mathbf{k}_{\bot}})\propto\exp\bigg{[}-\frac{{\mathbf{k}_{\bot}^{2}+m_{q}^{2}}}{8\beta_{2;a_{1}}^{2}x\bar{x}}\bigg{]}.$
(12)

Then the LCHO model for the $a_{1}(1260)$-meson wave function gives

$\displaystyle\Psi_{2;a_{1}}^{\|}(x,{\mathbf{k}_{\bot}})$
$\displaystyle=A_{2;a_{1}}^{\|}\varphi_{2;a_{1}}^{\|}(x)\exp\bigg{[}-\frac{{\mathbf{k}_{\bot}^{2}+m_{q}^{2}}}{8\beta_{2;a_{1}}^{2}x\bar{x}}\bigg{]},$
(13)

where $A_{2;a_{1}}^{\|}$ is the normalization constant, $\beta_{2;a_{1}}$ is a
harmonic parameter, and $m_{q}$ is the mass of the constitute quark $u$ and
$d$ in $a_{1}(1260)$-meson. In addition, the function
$\varphi_{2;a_{1}}^{\|}(x)$ dominates longitudinal distribution and can be
expressed as Zhong:2021epq

$\displaystyle\varphi_{2;a_{1}}^{\|}(x)=(x\bar{x})^{\alpha_{2;a_{1}}^{\|}}[1+B_{2;a_{1}}^{\|}C_{2}^{3/2}(\xi)].$
(14)

Since the meson LCDA is related to its wave function
$\Psi_{2;a_{1}}^{\|}(x,{\mathbf{k}_{\bot}})$ via the following relation:

$\displaystyle\phi_{2;a_{1}}^{\|}(x,\mu)=\frac{2\sqrt{6}}{f_{a_{1}}^{\|}}\int_{|{\bf
k}_{\bot}{|^{2}}\leq{\mu^{2}}}{\frac{{d^{2}}{\bf
k}_{\bot}}{16\pi^{3}}}\Psi_{2;a_{1}}^{\|}(x,{\mathbf{k}_{\bot}}).$ (15)

Then, the twist-2 LCDA of $a_{1}(1260)$-meson can be derived by integrating
the transverse momentum, which have the following form

$\displaystyle\phi_{2;a_{1}}^{\|}(x,\mu)=\frac{\sqrt{6}A_{2;a_{1}}^{\|}\beta_{2;a_{1}}^{2}}{{\pi^{2}}{f_{a_{1}}^{\|}}}x\bar{x}\varphi_{2;a_{1}}^{\|}(x)$
$\displaystyle~{}~{}~{}~{}~{}\times\bigg{\\{}\\!\exp\\!\bigg{[}\\!-\frac{m_{q}^{2}}{8\beta_{2;a_{1}}^{2}x\bar{x}}\bigg{]}-\exp\bigg{[}-\frac{m_{q}^{2}+{\mu^{2}}}{8\beta_{2;a_{1}}^{2}x\bar{x}}\bigg{]}\bigg{\\}}.$
(16)

The two parameters $A_{2;a_{1}}^{\|}$ and $\beta_{2;a_{1}}$ are constrained by
the following two conditions:

  * •

The wave function normalization condition,

$\displaystyle\int_{0}^{1}dx\int\frac{d^{2}{\mathbf{k}_{\bot}}}{16\pi^{3}}\Psi_{2;a_{1}}^{\|}(x,{\bf
k}_{\bot})=\frac{f_{a_{1}}^{\|}}{2\sqrt{6}}.$ (17)

  * •

The probability of $|q\bar{q}\rangle$ Fock state in a $a_{1}(1260)$-meson
should be less than $1$, e.g. $P_{a_{1}}<1$,

$\displaystyle P_{a_{1}}$ $\displaystyle=\int_{0}^{1}dx\int\frac{d^{2}{\bf
k}_{\bot}}{16\pi^{3}}|\Psi_{2;a_{1}}^{\|}(x,{\bf k}_{\bot})|^{2}$
$\displaystyle=\frac{(A_{2;a_{1}}^{\|})^{2}\beta_{2;a_{1}}^{2}}{4{\pi^{2}}}\int_{0}^{1}dx[\varphi_{2;a_{1}}^{\|}(x)]^{2}x\bar{x}$
$\displaystyle\times\exp\bigg{[}-\frac{m_{q}^{2}}{4x\bar{x}\beta_{2;a_{1}}^{2}}\bigg{]}.$
(18)

We shall fit the parameters $\alpha_{2;a_{1}}^{\|}$ and $B_{2;a_{1}}^{\|}$ by
using the least squares method so as to achieve the same moments
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ from the sum rules (7). The
detailed analysis about this point can be found in Refs.Zhong:2021epq ;
Zhong:2022ecl .

Secondly, we adopt the following correlator to derive the LCSRs for the $D\to
a_{1}(1260)$ TFFs,

$\displaystyle\Pi_{\mu}(p,q)=$ $\displaystyle i\int d^{4}xe^{iq\cdot x}\langle
a_{1}(p,\lambda)|T\\{j_{\mu}(x),j_{D}^{\dagger}(0)\\}|0\rangle$
$\displaystyle=$
$\displaystyle-\Pi_{1}e_{\mu}^{*(\lambda)}+\Pi_{2}(e^{*(\lambda)}\cdot
q)(2p+q)_{\mu}$ $\displaystyle+\Pi_{3}(e^{*(\lambda)}\cdot
q){q_{\mu}}+i{\Pi_{V}}\epsilon_{\mu}^{\alpha\beta\gamma}e_{\alpha}^{*(\lambda)}q_{\beta}p_{\gamma},$
(19)

where $j_{\mu}(x)=\bar{q}_{2}(x)\gamma_{\mu}(1-\gamma_{5})c(x)$ and
$j_{D}^{\dagger}(x)=i\bar{c}(x)(1-{\gamma_{5}})q_{1}(x)$. In the time-like
$q^{2}$-region, the long distance quark-gluon interactions are dominant. To
deal with the correlator in the time-like region, one can insert a complete
set of the $D$-meson states, which have the same $J^{P}$ quantum numbers to
obtain the hadronic expression. After separating the $D$-meson pole term, we
obtain

$\displaystyle\Pi_{\mu}^{\rm H}(p,q)=\frac{{\langle
a_{1}|\bar{q}_{2}\gamma_{\mu}(1-\gamma_{5})c|D\rangle\langle
D|\bar{c}i\gamma_{5}q_{1}|0\rangle}}{m_{D}^{2}-(p+q)^{2}}$
$\displaystyle\quad+\sum\limits_{\rm H}\frac{\langle
a_{1}|\bar{q}_{2}\gamma_{\mu}(1-\gamma_{5})c|D^{\rm H}\rangle\langle D^{\rm
H}|\bar{c}i(1-\gamma_{5})q_{1}|0\rangle}{m_{D^{\rm H}}^{2}-(p+q)^{2}},$ (20)

where $\langle D|\bar{c}i\gamma_{5}q|0\rangle=m_{D}^{2}f_{D}/m_{c}$. The $D\to
a_{1}(1260)$ transition matrix elements have the expressions Li:2009tx :

$\displaystyle\langle
a_{1}(p,\lambda)|\bar{q}_{2}\gamma_{\mu}\gamma_{5}c|D(p+q)\rangle\\!=\\!-\epsilon^{\mu\nu\alpha\beta}\\!e_{\nu}^{*(\lambda)}\\!q_{\alpha}p_{\beta}\frac{2iA(q^{2})}{m_{D}-m_{a_{1}}},$
(21) $\displaystyle\langle
a_{1}(p,\lambda)|\bar{q}_{2}\gamma_{\mu}c|D(p+q)\rangle~{}=-e_{\mu}^{*(\lambda)}~{}(m_{D}~{}-m_{a_{1}})~{}V_{1}(q^{2})$
$\displaystyle\qquad+(2p+q)_{\mu}\frac{e^{*(\lambda)}\cdot
q}{m_{D}-m_{a_{1}}}V_{2}(q^{2})+q_{\mu}(e^{*(\lambda)}\cdot
q)\frac{2m_{a_{1}}}{q^{2}}$
$\displaystyle\qquad\times[V_{3}(q^{2})-V_{0}(q^{2})],$

where $p$ is $a_{1}(1260)$-meson momentum and $q=p_{D}-p_{a_{1}}$ is the
momentum transfer, ${e^{*(\lambda)}}$ stands for $a_{1}(1260)$-meson
polarization vector with $\lambda=(\bot,\|)$ being its transverse or
longitudinal component, respectively. There are one linear relationships among
the TFFs Wang:2008bw ; Verma:2011yw :

$\displaystyle
V_{3}(q^{2})=\frac{m_{D}-m_{a_{1}}}{2m_{a_{1}}}V_{1}(q^{2})-\frac{m_{D}+m_{a_{1}}}{2m_{a_{1}}}V_{2}(q^{2}),$
(23)

Following the standard sum rules procedures, one can represent the
contributions of the higher resonances and the continuum states by dispersion
integrations so as to derive the expressions for the hadronic invariant
amplitudes $\Pi_{i}^{\rm H}[q^{2},(p+q)^{2}]$ with $i=(1,2,3,V)$ defined in
Eq. (19). The continuum threshold parameter $s_{0}$ can be set as the value
close to the squared mass of the lowest scalar $D$-meson. Meanwhile, the
conventional quark-hadron duality ansatz,
$\rho_{i}^{\textrm{had}}=\rho_{i}^{\text{QCD}}\theta(s-s_{0})$, can be used to
calculate the hadron spectrum density $\rho_{i}^{\textrm{had}}$. On the other
hand, in the space-like region, one can calculate the correlator via the QCD
theory. In this region, the correlator can be treated by the OPE with the
coefficients being pQCD calculable. The $c$-quark propagator which shall be
used in the calculation can be found in Ref. Hu:2021zmy . After applying the
OPE and using the expressions for the transition matrix elements, one can
arrange the resultant expressions by twist-2, 3, 4 LCDAs Yang:2007zt ;
Momeni:2016kjz . After matching the correlator with the dispersion relation,
and applying the conventional Borel transformation to suppress the less known
continuum contributions, the resultant TFFs under the LCSR approach are

$\displaystyle
V_{1}(q^{2})=\frac{2m_{c}^{2}m_{a_{1}}f_{a_{1}}^{\|}}{m_{D}^{2}f_{D}(m_{D}-m_{a_{1}})}\int_{0}^{1}\frac{du}{u}e^{(m_{D}^{2}-s(u))/{\cal
M}^{2}}\bigg{[}\Theta(c(u,s_{0}))\phi_{3;a_{1}}^{\bot}(u)-\widetilde{\Theta}(c(u,s_{0}))\frac{m_{a_{1}}^{2}}{u{\cal
M}^{2}}\Psi_{4;a_{1}}^{\|}(u)\bigg{]}+\frac{2m_{c}^{2}m_{a_{1}}^{2}}{m_{D}^{2}f_{D}}$
$\displaystyle\qquad\quad\times\frac{(f_{3;a_{1}}^{V}-f_{3;a_{1}}^{A})}{(m_{D}-m_{a_{1}})}\int{{\rm{\cal
D}}\alpha_{i}}\int_{0}^{1}dve^{(m_{D}^{2}-s(X))/{\cal
M}^{2}}\frac{1}{X^{2}{\cal
M}^{2}}\Theta(c(X,s_{0}))\left[\tilde{\Phi}_{3;a_{1}}^{\|}(\alpha_{i})-\Phi_{3;a_{1}}^{\|}(\alpha_{i})\right],$
(24) $\displaystyle
V_{2}(q^{2})=\frac{2m_{c}^{2}m_{a_{1}}f_{a_{1}}^{\|}(m_{D}-m_{a_{1}})}{m_{D}^{2}f_{D}}\int_{0}^{1}\frac{du}{u}e^{(m_{D}^{2}-s(u))/{\cal
M}^{2}}~{}\bigg{[}\frac{1}{u{\cal
M}^{2}}\widetilde{\Theta}(c(u,s_{0}))\Phi_{2;a_{1}}^{\|}(u)+\frac{m_{a_{1}}^{2}}{u{\cal
M}^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))\Psi_{4;a_{1}}^{\|}(u)\bigg{]}$
$\displaystyle\qquad\quad-\frac{m_{c}^{2}m_{a_{1}}^{2}(f_{3;a_{1}}^{V}\\!-\\!f_{3;a_{1}}^{A})(m_{D}\\!-\\!m_{a_{1}})}{m_{D}^{2}f_{D}}\int\\!{\cal
D}\alpha_{i}\\!\int_{0}^{1}\\!dve^{(m_{D}^{2}-s(X))/{\cal
M}^{2}}\frac{1}{X^{3}{\cal
M}^{4}}\Theta(c(X,s_{0}))\bigg{[}\tilde{\Phi}_{3;a_{1}}^{\|}(\alpha_{i})-\Phi_{3;a_{1}}^{\|}(\alpha_{i})\bigg{]},$
(25) $\displaystyle
V_{0}(q^{2})=V_{3}(q^{2})+\bigg{\\{}\frac{q^{2}m_{c}^{2}f_{a_{1}}^{\|}}{m_{D}^{2}f_{D}}\int_{0}^{1}\frac{du}{u}e^{(m_{D}^{2}-s(u))/{\cal
M}^{2}}\bigg{[}\frac{1}{u{\cal
M}^{2}}\widetilde{\Theta}(c(u,s_{0}))\Phi_{2;a_{1}}^{\|}(u)\\!-\\!\frac{m_{a_{1}}^{2}(2-u)}{u^{2}{\cal
M}^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))\Psi_{4;a_{1}}^{\|}(u)\bigg{]}$
$\displaystyle\qquad\quad-\frac{q^{2}\,m_{c}^{2}\,{\color[rgb]{0,0,1}m_{a_{1}}}\,(f_{3;a_{1}}^{V}-f_{3;a_{1}}^{A})}{m_{D}^{2}{f_{D}}}\int{\cal
D}\alpha_{i}\int_{0}^{1}dv~{}e^{(m_{D}^{2}-s(X))/{\cal
M}^{2}}~{}\frac{1}{X^{3}{\cal
M}^{4}}~{}\Theta(c(X,s_{0}))~{}\bigg{[}\,\tilde{\Phi}_{3;a_{1}}^{\|}(\alpha_{i})\,-\,\Phi_{3;a_{1}}^{\|}(\alpha_{i})\,\bigg{]}\bigg{\\}},$
(26) $\displaystyle
A(q^{2})=\frac{m_{c}^{2}m_{a_{1}}f_{a_{1}}^{\|}(m_{D}-m_{a_{1}})}{2m_{D}^{2}f_{D}}\int_{0}^{1}due^{(m_{D}^{2}-s(u))/{\cal
M}^{2}}\frac{1}{u^{2}{\cal
M}^{2}}\widetilde{\Theta}(c(u,s_{0}))\psi_{3;a_{1}}^{\bot}(u),$ (27)

where the $q^{2}$-dependence factors $s(u)$ and $s(X)$ are defined as

$\displaystyle s(\zeta)$ $\displaystyle=$
$\displaystyle\frac{m_{c}^{2}-(1-\zeta)({q^{2}}-\zeta
m_{a_{1}}^{2})}{\zeta}\text{ with }\zeta=(u,X),$ (28)

where $X=\alpha_{1}-\alpha_{2}+v\alpha_{3}$, $\alpha_{1},\alpha_{2}$ and
$\alpha_{3}$ are the respective momentum fractions carried by
${\bar{q}}_{1},q_{2}$ quarks and gluon in the $a_{1}(1260)$-meson Yang:2007zt
. $\Theta(c(u,s_{0}))$ is the conventional step function,
$\widetilde{\Theta}(c(u,s_{0}))$ and
$\widetilde{\widetilde{\Theta}}(c(u,s_{0}))$ are defined as

$\displaystyle\int_{0}^{1}\frac{du}{u^{2}{\cal M}^{2}}e^{-s(u)/{\cal
M}^{2}}\widetilde{\Theta}(c(u,s_{0}))f(u)$
$\displaystyle\qquad=\int_{u_{0}}^{1}\frac{du}{u^{2}{\cal
M}^{2}}e^{-s(u)/{\cal M}^{2}}f(u)+\delta(c(u_{0},s_{0})),$ (29)
$\displaystyle\int_{0}^{1}\frac{du}{2u^{3}{\cal M}^{4}}e^{-s(u)/{\cal
M}^{2}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))f(u)$
$\displaystyle\qquad=\int_{u_{0}}^{1}\frac{du}{2u^{3}{\cal
M}^{4}}e^{-s(u)/{\cal M}^{2}}f(u)+\Delta(c(u_{0},s_{0})),$ (30)

where $c(u,s_{0})=us_{0}-m_{b}^{2}+\bar{u}q^{2}-u\bar{u}m_{a_{1}}^{2}$ and

$\displaystyle\delta(c(u,s_{0}))=e^{-s_{0}/{\cal
M}^{2}}\frac{f(u_{0})}{m_{c}^{2}+u_{0}^{2}m_{a_{1}}^{2}-q^{2}},$ (31)
$\displaystyle\Delta(c(u,s_{0}))=e^{-s_{0}/{\cal
M}^{2}}\bigg{[}\frac{1}{2u_{0}{\cal
M}^{2}}\frac{f(u_{0})}{m_{c}^{2}+u_{0}^{2}m_{a_{1}}^{2}-q^{2}}$
$\displaystyle\left.-\frac{u_{0}^{2}}{2(m_{c}^{2}+u_{0}^{2}m_{a_{1}}^{2}-q^{2})}\frac{d}{du}\left(\frac{f(u)}{u(m_{c}^{2}+u^{2}m_{a_{1}}^{2}-q^{2})}\right)\right|_{u={u_{0}}}\bigg{]}.$
(32)

Here $u_{0}$ is the solution of $c(u_{0},s_{0})=0$ with $0\leq u_{0}\leq 1$.
The simplified LCDAs are defined as

$\displaystyle\Phi_{2;a_{1}}^{\|}(u)=\int_{0}^{u}dv\phi_{2;a_{1}}^{\|}(v),$
(33)
$\displaystyle\Phi_{3;a_{1}}^{\bot}(u)=\int_{0}^{u}{dv}\phi_{3;a_{1}}^{\bot}(v),$
(34)
$\displaystyle\Psi_{4;a_{1}}^{\|}(u)=\int_{0}^{u}{dv}\int_{0}^{v}{dw}\psi_{4;a_{1}}^{\|}(w).$
(35)

The coupling constants $f_{3;a_{1}}^{V}$ and $f_{3;a_{1}}^{A}$ for
$a_{1}(1260)$-meson are defined as the following matrix elements:

$\displaystyle\langle
a_{1}(q,\lambda)|J_{3,\mu}^{3,A}(0)|0\rangle=-f_{3;a_{1}}^{A}(z\cdot
q)^{3}e^{(\lambda)}_{\bot,\mu}+{\cal O}(z_{\mu}),$ $\displaystyle\langle
a_{1}(q,\lambda)|J_{3,\mu}^{1,V}(0)|0\rangle=-if_{3;a_{1}}^{V}(z\cdot
q)^{2}e^{(\lambda)}_{\bot,\mu}+{\cal O}(z_{\mu}),$ (36)

where the interpolating currents,
$J_{3,\mu}^{3,A}(0)=z^{\alpha}z^{\beta}\bar{q}_{2}(0)\gamma_{\alpha}\gamma_{5}[G_{\beta\mu}(0)i(z\cdot\overrightarrow{D})-i(z\cdot\overleftarrow{D})G_{\beta\mu}(0)]q_{1}(0)$,
$J_{3,\mu}^{1,V}(0)=z^{\alpha}z^{\beta}\bar{q}_{2}(0)\gamma_{\alpha}g_{s}{\tilde{G}_{\beta\mu}}(0)q_{1}(0)$,
and ${\cal O}(z_{\mu})$ involve the twist-4 corrections Yang:2007zt .

Then, the longitudinal and transverse differential decay widths for
semileptonic decay $D\to a_{1}(1260)\ell^{+}\nu_{\ell}$ can be expressed as

$\displaystyle\frac{d\Gamma_{\rm L}(D\to
a_{1}(1260)\ell^{+}\nu_{\ell})}{dq^{2}}=\bigg{(}\frac{q^{2}-m_{\ell}^{2}}{q^{2}}\bigg{)}^{2}\frac{\sqrt{\lambda}G_{F}^{2}|V_{cd}|^{2}}{384\pi^{3}m_{D}^{3}}$
$\displaystyle\times\frac{1}{q^{2}}\bigg{[}3m_{\ell}^{2}\,\lambda\,V_{0}^{2}(q^{2})+\frac{m_{\ell}^{2}+2q^{2}}{2m_{a_{1}}}\Big{|}(m_{D}^{2}-m_{a_{1}}^{2}-q^{2})$
$\displaystyle\times(m_{D}-m_{a_{1}})V_{1}(q^{2})-\frac{\lambda}{m_{D}-m_{a_{1}}}{V_{2}}(q^{2})\Big{|}^{2}\bigg{]},$
$\displaystyle\frac{d\Gamma_{\pm}(D\\!\to\\!a_{1}(1260)\ell^{+}\nu_{\ell})}{dq^{2}}\\!=\\!\bigg{(}\frac{q^{2}-m_{\ell}^{2}}{q^{2}}\bigg{)}^{2}\frac{\lambda^{3/2}G_{F}^{2}|{V_{cd}}|^{2}}{384\pi^{3}m_{D}^{3}}$
$\displaystyle\times(m_{\ell}^{2}+2q^{2})\bigg{|}\frac{A(q^{2})}{m_{D}-m_{a_{1}}}\,\mp\,\frac{(m_{D}-m_{a_{1}})\,V_{1}(q^{2})}{\sqrt{\lambda}}\bigg{|}^{2},$

where $G_{F}$ is the Fermi coupling constant, $|V_{cd}|$ is the CKM matrix
element, and
$\lambda=(m_{D}^{4}+m_{a_{1}}^{4}+q^{4}-2m_{a_{1}}^{2}m_{D}^{2}-2q^{2}m_{D}^{2}-2q^{2}m_{a_{1}}^{2})$.
The total differential decay width of the semileptonic decay is $d\Gamma_{\rm
L}+d\Gamma_{\rm T}$, where $d\Gamma_{\rm L}$ and $d\Gamma_{\rm
T}=d\Gamma_{+}+d\Gamma_{-}$ corresponds to longitudinal and transverse parts,
respectively.

##  III Numerical results and discussions

To do the numerical calculation, the input parameters are taken as follows.
The current charm-quark mass, ${\bar{m}_{c}}({\bar{m}_{c}})=1.27(2)~{}{\rm
GeV}$, and the current quark-masses for the light quarks are
${\bar{m}_{u}}(2~{}{\rm GeV})=2.16_{-0.26}^{+0.49}~{}{\rm MeV}$ and
${\bar{m}_{d}}(2~{}{\rm GeV})=4.67_{-0.17}^{+0.48}~{}{\rm MeV}$. The $D$-meson
and the $a_{1}(1260)$-meson masses are, $m_{D^{0}}=1.865~{}{\rm
GeV},m_{D^{+}}=1.870~{}{\rm GeV}$ and $m_{a_{1}}\approx 1.230~{}{\rm GeV}$,
accordingly, which are taken from the Particle Data Group (PDG) Zyla:2020zbs .
The $D$-meson and $a_{1}(1260)$-meson decay constants
$f_{D^{0}}=f_{D^{+}}=0.210(12)~{}{\rm GeV}$, $f_{a_{1}}^{\|}=0.238(10)~{}{\rm
GeV}$ Mutuk:2018lki . The $D\to a_{1}(1260)$ decay processes typical scale in
this paper is $\mu_{k}=(m_{D}^{2}-m_{c}^{2})^{1/2}\approx 1.4~{}{\rm GeV}$.
Furthermore, the non-perturbative vacuum condensates are the significant
parameters to the sum rule, and we take Colangelo:2000dp ; Narison:2014wqa ;
Zhong:2021epq

$\displaystyle\langle\bar{q}q\rangle|_{2~{}{\rm GeV}}$ $\displaystyle=$
$\displaystyle(-289.14^{+9.34}_{-4.47})^{3}~{}{\rm MeV}^{3},$
$\displaystyle\langle g_{s}\bar{q}\sigma TGq\rangle|_{2~{}{\rm GeV}}$
$\displaystyle=$ $\displaystyle(-1.934_{-0.103}^{+0.188})\times 10^{-2}~{}{\rm
GeV}^{5},$ $\displaystyle\langle g_{s}\bar{q}q\rangle^{2}|_{2~{}{\rm GeV}}$
$\displaystyle=$ $\displaystyle(2.082_{-0.697}^{+0.734})\times 10^{-3}~{}{\rm
GeV}^{6},$ $\displaystyle\langle g_{s}^{2}\bar{q}q\rangle^{2}|_{\mu}$
$\displaystyle=$ $\displaystyle(7.420_{-2.483}^{+2.614})\times 10^{-3}~{}{\rm
GeV}^{6},$ $\displaystyle\langle\alpha_{s}G^{2}\rangle|_{\mu}$
$\displaystyle=$ $\displaystyle 0.038(11)~{}{\rm GeV}^{4},$
$\displaystyle\langle g_{s}^{3}fG^{3}\rangle|_{\mu}$ $\displaystyle\approx$
$\displaystyle 0.045~{}{\rm GeV}^{6},$ $\displaystyle\kappa$ $\displaystyle=$
$\displaystyle 0.74(3).$ (39)

It should be noted that, the values of the gluon condensates are the most
commonly used in QCD sum rules. The value of the double-gluon condensate
$\langle\alpha_{s}G^{2}\rangle$ is determined by the sum rule of the
charmonium, and the one for triple-gluon condensate $\langle
g_{s}^{3}fG^{3}\rangle$ is based on the instanton model222For more detailed
discussion, one can refer to Refs. Colangelo:2000dp ; Shifman:1978bx ;
Shifman:1978by . The double-quark condensate $\langle\bar{q}q\rangle$ and the
quark-gluon mixed condensate $\langle g_{s}\bar{q}\sigma TGq\rangle$ were
updated in our previous work Zhong:2021epq  based on the GellMann-Oakes-Renner
relation and the relationship $\langle g_{s}\bar{q}\sigma
TGq\rangle=m_{0}^{2}\langle q\bar{q}\rangle$ with $m_{0}^{2}=0.80(2)~{}{\rm
GeV}^{2}$ Narison:2014ska . One can calculate the four-quark condensate
$\langle g_{s}\bar{q}q\rangle^{2}$ by using
$\rho\alpha_{s}\langle\bar{q}q\rangle^{2}=(5.8\pm 1.8)\times 10^{-4}{\rm
GeV}^{6}$ with $\rho\simeq 3-4$ Narison:2014ska , and determine the value of
$\langle g_{s}^{2}\bar{q}q\rangle^{2}$ by combining with the new value of
$\langle\bar{q}q\rangle$. All those scale-dependent parameters, such as the
quark masses and the vacuum condensates, shall be run from an initial scale
$\mu_{0}$ to a special choice of scale such as $\mu_{k}$ by applying the
renormalization group equations (RGE) given by Refs. Yang:1993bp ;
Hwang:1994vp ; Lu:2006fr ; Zhang:2021wnv .

Table 1: The ratios of the continuum states’ and the dimension-six condensates’ contributions over the total moments of $a_{1}(1260)$-meson longitudinal twist-2 LCDA $\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ with $n=(2,4,6,8,10)$ within the determined Borel windows. The abbreviations “Con.” and “Six.” stand for the continuum and dimension-six contributions, respectively. $n$ |  Con. |  Six. |  $M^{2}$ |  $\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$
---|---|---|---|---
2 | $<35\%$ | $<5\%$ | $[1.782,2.886]$ | $[0.240,0.198]$
4 | $<35\%$ | $<5\%$ | $[2.740,3.385]$ | $[0.101,0.091]$
6 | $<40\%$ | $<5\%$ | $[3.669,4.567]$ | $[0.057,0.051]$
8 | $<40\%$ | $<5\%$ | $[4.585,5.269]$ | $[0.037,0.034]$
10 | $<45\%$ | $<5\%$ | $[5.494,6.790]$ | $[0.027,0.023]$

Two important parameters for the BFTSR approach for the moments are continuum
threshold $s_{0}$ and Borel parameter $M^{2}$, whose range is called as the
Borel Window. To fix their values and make the sum rules predictions reliable,
the contributions from the continuum states and the contributions from the
dimension-six condensates should be small enough. For the purpose, we
determine the Borel window by allowing the contribution of continuum states to
be less than $45\%$ and the contribution of dimension-six condensates to be
less than $5\%$. When determining the threshold parameter $s_{a_{1}}$, one can
normalized the $0_{\rm th}$-order $a_{1}(1260)$-meson longitudinal DA in the
appropriate Borel window. Followed by this approach, we can get
$s_{a_{1}}=1.4(4)~{}{\rm GeV}$. We present the determined Borel windows and
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ at the scale $\mu={\sqrt{M^{2}}}$
in Table 1. Here, we have set the continuum contributions to be no more than
$(35\%,35\%,40\%,40\%,45\%)$ for $n=(2,4,6,8,10)$, respectively, and the
dimension-six contributions for all
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ to be less than $5\%$.

Principally, the moments including all the condensates should be independent
to the Borel parameter $M^{2}$, and for a fixed-order OPE expansion, it may
change with different choices of $M^{2}$ within the allowable Borel window.
Such change depends heavily on the convergence of the OPE expansion over
$1/M^{2}$. For the present LCSR up to dimension-6 condensates, e.g the series
(7), as a conservative prediction, we require the variations of
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ within the Borel window to be less
than $10\%$. We present the first five moments for $a_{1}(1260)$-meson twist-2
LCDA versus the Borel parameter $M^{2}$ in Fig. 1. The shaded bands indicate
the corresponding Borel windows, which are in the region of $[1.0,7.0]~{}{\rm
GeV}^{2}$, respectively.

Figure 1: Moments $\langle\xi^{\|;n}_{2;a_{1}}\rangle|_{\mu}$ up to
$n=(2,4,6,8,10)$ order level versus the Borel parameter $M^{2}$. The shaded
bands stand for the corresponding Borel windows.

By taking all uncertainty sources into consideration and applying the RGE of
the moments, $\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ at the initial scale
$\mu_{0}$ and special scale $\mu_{k}$ are

$\displaystyle\langle\xi_{2;a_{1}}^{\|;2}\rangle|_{\mu_{0}}=0.223(29),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;2}\rangle|_{\mu_{k}}=0.219(25),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;4}\rangle|_{\mu_{0}}=0.098(8),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;4}\rangle|_{\mu_{k}}=0.097(8),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;6}\rangle|_{\mu_{0}}=0.056(6),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;6}\rangle|_{\mu_{k}}=0.055(5),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;8}\rangle|_{\mu_{0}}=0.039(4),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;8}\rangle|_{\mu_{k}}=0.037(3),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;10}\rangle|_{\mu_{0}}=0.028(3),$
$\displaystyle\langle\xi_{2;a_{1}}^{\|;10}\rangle|_{\mu_{k}}=0.027(3).$ (40)

In order to determine the two LCHO model parameters $\alpha_{2;a_{1}}^{\|}$
and $B_{2;a_{1}}^{\|}$, one can use the specific fitting by taking the two
parameters as the fitting parameters, e.g.
$\theta=(\alpha_{2;a_{1}}^{\|},B_{2;a_{1}}^{\|})$. The moments
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ from Eq. (16) with the definition
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}=\int_{0}^{1}\xi^{n}\phi_{2;a_{1}}^{\|}(x,\mu)$
been regarded as the mean function $\mu(x_{i};\theta)(x_{i}\to n)$, where the
moments calculated with BFTSR, i.e. Eq. (40) are considered as the independent
measurements $y_{i}$ with the known variance $\sigma_{i}$. To obtain the best
values of fitting parameters $\theta$, one can minimize the function

$\displaystyle\chi^{2}(\theta)=\sum\limits_{i=1}^{5}\frac{(y_{i}-\mu(x_{i},\theta))^{2}}{\sigma_{i}^{2}}.$
(41)

The goodness of fit is judged by the magnitude of the probability

$\displaystyle P_{\chi^{2}}=\int^{\infty}_{\chi^{2}}f(y;n_{d})dy.$ (42)

Here $f(y;n_{d})$ with the number of degrees of freedom $n_{d}$ is the
probability density function of $\chi^{2}(\theta)$, and

$\displaystyle
f(y;n_{d})=\frac{1}{\Gamma\left(\dfrac{n_{d}}{2}\right)2^{\frac{n_{d}}{2}}}y^{\frac{n_{d}}{2}-1}e^{-\frac{y}{2}}.$
(43)

Figure 2: The relationship curve between parameters $\alpha_{2;a_{1}}^{\|}$, $B_{2;a_{1}}^{\|}$ and goodness of fit $P_{\chi_{\min}^{2}}$. Table 2: Fitting parameters $A_{2;a_{1}}^{\|}({\rm GeV^{-1}})$, $\beta_{2;a_{1}}({\rm GeV})$, $\alpha_{2;a_{1}}^{\|}$ and $B_{2;a_{1}}^{\|}$ with different constituent quark mass $m_{q}$ under initial scale $\mu_{0}$. Meanwhile, the goodness of fit $P_{\chi_{\min}^{2}}$ and the probability $P_{a_{1}}$ are also given. $m_{q}$ |  200MeV |  250MeV |  300MeV |  330MeV |  350MeV
---|---|---|---|---|---
$A_{2;a_{1}}^{\|}$ | +6.157 | +5.382 | +4.775 | +4.464 | +4.336
$\beta_{2;a_{1}}$ | +0.522 | +0.549 | +0.582 | +0.603 | +0.618
$\alpha_{2;a_{1}}^{\|}$ | $-0.884$ | $-0.946$ | $-0.996$ | $-1.022$ | $-1.029$
$B_{2;a_{1}}^{\|}$ | $-0.116$ | $-0.115$ | $-0.114$ | $-0.114$ | $-0.113$
$P_{\chi_{\min}^{2}}$ | +0.977 | +0.954 | +0.923 | +0.903 | +0.889
$P_{a_{1}}$ | +0.641 | +0.611 | +0.584 | +0.570 | +0.562

Then, we obtain the fitting parameters at the initial scale $\mu_{0}$. Due to
the quark component of $a_{1}(1260)$-meson here is $\bar{u}u$ or $\bar{d}d$,
so the treatment for light-quark mass in this paper is the same with usual
constituent quark mass. There are different values for constituent quark mass
$m_{q}$, which is taken to be $250~{}{\rm MeV}$ in the invariant meson mass
scheme Jaus:1989au ; Chung:1988mu ; Choi:1997qh ; Schlumpf:1994bc ,
$330~{}{\rm MeV}$ in the spin-averaged meson mass scheme Dziembowski:1986dr ;
Dziembowski:1987zp ; Ji:1992yf ; Choi:1996mq . In addition, $m_{q}=300~{}{\rm
MeV}$ and $m_{q}=200~{}{\rm MeV}$ Zhong:2021epq  of the simplest in Refs.
Wu:2008yr ; Wu:2011gf . In this work, we present the results for different
choices of the constituent quark mass, e.g.
$m_{q}=(200,250,300,330,350)~{}{\rm MeV}$, respectively. The fitting results
are given in Table 2. As a default value, we shall take $m_{q}=250~{}{\rm
MeV}$ to do our calculation, whose corresponding goodness of fit is $95.4\%$.
This value is also agree with the usual pion and kaon cases Zhong:2021epq ;
Zhong:2022ecl . One can find that the parameter $A_{2;a_{1}}^{\|}$ and
$B_{2;a_{1}}^{\|}$ gradually decrease with the increment of $m_{q}$, and the
goodness of fit $P_{\chi_{\min}^{2}}$ is also decreasing with the increment of
$m_{q}$. To show more clearly the relationship between the magnitudes of
$\alpha_{2;a_{1}}^{\|}$ and $B_{2;a_{1}}^{\|}$ and the goodness of fit
$P_{\chi_{\min}^{2}}$, we present the relationship curve between them in Fig.
2. The darker shaded band of Fig. 2 represents the higher goodness of fit.
When the range of goodness of fit is $80\%\leq P_{\chi^{2}_{\min}}\leq 96\%$,
the allowable ranges for the parameters $\alpha_{2;a_{1}}^{\|}$ and
$B_{2;a_{1}}^{\|}$ are quite small.

Figure 3: The $a_{1}(1260)$-meson longitudinal twist-2 LCDA $\phi_{2;a_{1}}^{\|}(x,\mu_{0})$. As a comparison, the QCDSR Yang:2008xw ; Yang:2007zt  and the asymptotic curves are also given. Table 3: The $D\to a_{1}(1260)$ TFFs at the large recoil region, i.e. $A(0),V_{1}(0),V_{2}(0),V_{0}(0)$. Predictions under various approaches are also listed. References | $A(0)$ | $V_{1}(0)$ | $V_{2}(0)$ | $V_{0}(0)$
---|---|---|---|---
This work | $0.130_{-0.013}^{+0.015}$ | $1.898_{-0.121}^{+0.128}$ | $0.228_{-0.021}^{+0.020}$ | $0.217_{-0.025}^{+0.023}$
LCSR-I Momeni:2019uag  | $0.07\pm 0.05$ | $0.37\pm 0.01$ | $-0.03\pm 0.02$ | $0.15\pm 0.05$
LCSR-II Huang:2021owr  | $0.34_{-0.04}^{+0.03}$ | $2.63_{-0.21}^{+0.20}$ | $0.34_{-0.04}^{+0.03}$ | $0.24_{-0.01}^{+0.00}$
3PSR Zuo:2016msr  | $0.314_{-0.046}^{+0.048}$ | $0.039_{-0.010}^{+0.012}$ | $0.112_{-0.032}^{+0.037}$ | $-0.114_{-0.019}^{+0.018}$
CLFQM Cheng:2003sm  | $0.20$ | $1.54$ | $0.06$ | $0.31$
CLFQM Verma:2011yw  | $0.19_{-0.01}^{+0.00}$ | $1.51_{-0.04}^{+0.00}$ | $0.05_{-0.00}^{+0.01}$ | $0.32_{-0.00}^{+0.00}$

We present the $a_{1}(1260)$-meson twist-2 LCDA in Fig. 3. As a comparison, we
have also presented the QCDSR prediction Yang:2008xw ; Yang:2007zt  and the
asymmetry behavior $\phi_{2;a_{1}}^{\|}(x)=6x\bar{x}$ in Fig. 3. Our
prediction prefers a single-peak behavior, which is as the same as the QCDSR
and asymptotic ones. In order to deal with the $a_{1}(1260)$-meson twist-3
distribution amplitudes, one can decompose the $\phi_{3;a_{1}}^{\bot}(x)$ and
$\psi_{3;a_{1}}^{\bot}(x)$ into several terms according to various source
terms Ball:1998sk :

$\displaystyle\phi_{3;a_{1}}^{\bot}(x)=\phi_{3;a_{1}}^{\bot
WW}(x)+\phi_{3;a_{1}}^{\bot g}(x)+\phi_{3;a_{1}}^{\bot m}(x)$ (44)
$\displaystyle\psi_{3;a_{1}}^{\bot}(x)=\psi_{3;a_{1}}^{\bot
WW}(x)+\psi_{3;a_{1}}^{\bot g}(x)+\psi_{3;a_{1}}^{\bot m}(x)$ (45)

where $\phi_{3;a_{1}}^{\bot g}(x)$ and $\psi_{3;a_{1}}^{\bot g}(x)$ are
contributions from the three-particle distribution amplitudes, which can be
neglected due to their smallest and negligible contributions.
$\phi_{3;a_{1}}^{\bot m}(x)$ and $\psi_{3;a_{1}}^{\bot m}(x)$ are related to
the coefficients $\tilde{\delta}_{\pm}$, whose magnitudes tend to be zero in
the $q\bar{q}$ with $q=(u,d)$ meson’s components system 333The detailed
analysis can be found in Section 4 in Ref. Ball:1998sk . The
$\phi_{3;a_{1}}^{\bot WW}(x)$ and $\psi_{3;a_{1}}^{\bot WW}(x)$ denotes the
contribution from the twist-2 longitudinal distribution amplitudes, which is
also called Wandzura-Wilczek approximation. Thus, one can get the following
relationship

$\displaystyle\phi_{3;a_{1}}^{\bot}(x)=\frac{1}{2}\bigg{[}\int_{0}^{x}\frac{dv}{\bar{v}}\phi_{2;a_{1}}^{\|}(v)+\int_{x}^{1}\frac{dv}{v}\phi_{2;a_{1}}^{\|}(v)\bigg{]},$
(46)
$\displaystyle\psi_{3;a_{1}}^{\bot}(x)=2\bar{x}\int_{0}^{x}\frac{dv}{\bar{v}}\phi_{2;a_{1}}^{\|}(v)+2x\int_{x}^{1}\frac{dv}{v}\phi_{2;a_{1}}^{\|}(v).$
(47)

The twist-3,4 LCDAs for $a_{1}(1260)$-meson is taken from Ref. Yang:2007zt

$\displaystyle\psi_{4;a_{1}}^{\|}(x)=6x\bar{x}+(1-3\xi^{2})\bigg{[}\frac{1}{7}a_{2;a_{1}}^{\|;2}-\frac{20}{3}\frac{f_{3;a_{1}}^{A}}{f_{a_{1}}^{\|}m_{a_{1}}}\bigg{]},$
(48)
$\displaystyle\Phi_{3;a_{1}}^{\|}(\alpha_{i})=5040(\alpha_{1}-\alpha_{2})\alpha_{1}\alpha_{2}\alpha_{3}^{2},$
(49)
$\displaystyle\tilde{\Phi}_{3;a_{1}}^{\|}(\alpha_{i})=360\alpha_{1}\alpha_{2}\alpha_{3}^{2}\bigg{[}1+\frac{1}{2}(7\alpha_{3}-3)\omega_{a_{1}}^{V}\bigg{]}$
(50)

where $a_{2;a_{1}}^{\|;2}=0.056\pm 0.072$ is the result obtained by using the
BFTSR, the coupling constants $f_{3;a_{1}}^{A}=0.0012~{}{\rm GeV^{2}}$ and
$f_{3;a_{1}}^{V}=0.0036~{}{\rm GeV^{2}}$ Yang:2008xw , the coefficient
$\omega_{a_{1}}^{V}=-2.9$ Yang:2008xw . $\Phi_{3;a_{1}}^{\|}(\alpha_{i})$ and
$\tilde{\Phi}_{3;a_{1}}^{\|}(\alpha_{i})$ represent the twist-3 LCDAs of the
three-particle part Yang:2007zt ; Cheng:2007mx , accordingly. Here the twist-3
LCDAs $\phi_{3;a_{1}}^{\bot}(x)$ and $\psi_{3;a_{1}}^{\bot}(x)$ are related to
the twist-2 LCDAs by using the Wandzura-Wliczek approximation. In order to
evolve the hadronic parameters in $a_{1}(1260)$-meson twist-2, 3, 4 LCDAs from
the initial factorization scale $\mu_{0}$ to the special scale $\mu_{k}$, one
can use the RGE with the form
$c_{i}(\mu_{k})=L^{\gamma_{c_{i}}/\beta_{0}}c_{i}(\mu_{0})$, where
$L=\alpha_{s}(\mu_{k})/\alpha_{s}(\mu_{0})$, $\beta_{0}=11-2/3n_{f}$. The one-
loop anomalous dimensions $\gamma_{c_{i}}$ satisfy the following equation
Fu:2020uzy .

$\displaystyle\gamma_{a_{n}}=C_{F}\bigg{(}1-\frac{2}{(n+1)(n+2)}-\sum\limits_{m=2}^{n+1}\frac{1}{m}\bigg{)}.$
(51)

The TFFs of $D\to a_{1}(1260)$ are key elements for investigating the
$D$-meson semileptonic decay. To derive the exact value for the TFFs
(24)-(27), we need to fix the continuum threshold $s_{0}$ and Borel parameter
$M^{2}$. Normally, the continuum threshold $s_{0}$ should be taken near the
squared mass of the $D$-meson’s first excited state with the same $J^{P}$
number, e.g. $D_{0}(2550)^{0}$. And we take $s_{0}^{A}=6.5(3)~{}{\rm
GeV}^{2}$, $s_{0}^{V_{1}}=5.7(3)~{}{\rm GeV}^{2}$,
$s_{0}^{V_{2}}=6.0(3)~{}{\rm GeV}^{2}$ and $s_{0}^{V_{0}}=6.0(3)~{}{\rm
GeV}^{2}$. One can use four criteria of the LCSR approach listed in Ref.
Hu:2021zmy  to determine the Borel parameters for the four TFFs. And the
determined Borel parameters are ${\cal M}^{2}_{A}=4.0(3)~{}{\rm GeV}^{2}$ ,
${\cal M}^{2}_{V_{1}}=5.4(3)~{}{\rm GeV}^{2}$, ${\cal
M}^{2}_{V_{2}}=5.0(3)~{}{\rm GeV}^{2}$ and ${\cal M}^{2}_{V_{0}}=5.0(3)~{}{\rm
GeV}^{2}$, respectively. We present the $D\to a_{1}(1260)$ TFFs at the large
recoil region, i.e. $q^{2}\to 0~{}{\rm GeV}^{2}$ within errors in Table 3. To
make a comparison, the predictions from various approaches are presented, i.e.
the LCSR-I, II Huang:2021owr ; Momeni:2019uag , 3PSR Zuo:2016msr  and CLFQM
Cheng:2003sm ; Verma:2011yw , respectively. Our predictions are close to the
LCSR-II ones. To have a clear look at the uncertainties caused by different
input parameters, we present the TFFs as follows

$\displaystyle A(0)$ $\displaystyle=$ $\displaystyle
0.130+(_{-0.005}^{+0.005})_{s_{0}}+(_{-0.008}^{+0.009})_{{\cal
M}^{2}}+(_{-0.009}^{+0.010})_{m_{c}f_{D}}$ (52) $\displaystyle=$
$\displaystyle 0.130_{-0.013}^{+0.015}$ $\displaystyle V_{1}(0)$
$\displaystyle=$ $\displaystyle
1.898+(_{-0.065}^{+0.060})_{s_{0}}+(_{-0.002}^{+0.002})_{{\cal
M}^{2}}+(_{-0.103}^{+0.114})_{m_{c}f_{D}}$ (53) $\displaystyle+$
$\displaystyle(_{-0.001}^{+0.001})_{a_{2;a_{1}}^{\|;2}}=1.898_{-0.121}^{+0.128}$
$\displaystyle V_{2}(0)$ $\displaystyle=$ $\displaystyle
0.228+(_{-0.006}^{+0.006})_{s_{0}}+(_{-0.013}^{+0.015})_{{\cal
M}^{2}}+(_{-0.012}^{+0.014})_{m_{c}f_{D}}$ (54) $\displaystyle+$
$\displaystyle(_{-0.001}^{+0.001})_{a_{2;a_{1}}^{\|;2}}=0.228_{-0.021}^{+0.020}$
$\displaystyle V_{0}(0)$ $\displaystyle=$ $\displaystyle
0.217+(_{-0.007}^{+0.006})_{s_{0}}+(_{-0.020}^{+0.018})_{{\cal
M}^{2}}+(_{-0.012}^{+0.013})_{m_{c}f_{D}}$ (55) $\displaystyle=$
$\displaystyle 0.217_{-0.025}^{+0.023}$

Table 4: The $D\to a_{1}(1260)$ TFFs at the large recoil region $q^{2}=0$, in which the twsit-2,3,4 LCDAs’ contributions are presented separately. |  $A(q^{2})$ |  $V_{1}(q^{2})$ |  $V_{2}(q^{2})$ |  $V_{0}(q^{2})$
---|---|---|---|---
$\Phi_{2;a_{1}}^{\|}(u)$ | $/$ | $/$ | $0.191$ | $-0.240$
$\phi_{3;a_{1}}^{\bot}(u)$ | $/$ | $2.061$ | $/$ | $0.550$
$\psi_{3;a_{1}}^{\bot}(u)$ | $0.130$ | $/$ | $/$ | $/$
$\Psi_{4;a_{1}}^{\|}(u)$ | $/$ | $-0.180$ | $0.019$ | $-0.075$
${\cal H}(\alpha_{i})$ | $/$ | $0.017$ | $0.018$ | $-0.018$
Total | $0.130$ | $1.898$ | $0.228$ | $0.217$

We present the $D\to a_{1}(1260)$ TFFs at the large recoil region $q^{2}=0$ in
Table 4, in which the contributions from the DAs with various twist structures
are presented. Ref. Lu:2021mri  indicates that the corrections from the two-
particle higher-twist contributions are ($27\%$-$36\%$), and our twist-4
contribution falls within this margin of error. As for the twist-4
contribution to the form factor $V_{0}(0)$, its magnitude is about $34.56\%$
of the total result. For $V_{1}(0)$ and $V_{2}(0)$, their twist-4
contributions change to $9.48\%$ and $8.33\%$, respectively. For convenience,
we use ${\cal
H}(\alpha_{i})=\tilde{\Phi}_{3;a_{1}}^{\|}(\alpha_{i})-\Phi_{3;a_{1}}^{\|}(\alpha_{i})$
to represent the net contribution of the three-particle twist-3 LCDAs, whose
contribution to the TFFs $V_{1}(0)$, $V_{2}(0)$ and $V_{0}(0)$ are $0.90\%$,
$7.90\%$ and $8.30\%$, respectively.

Table 5: The masses of low-lying $D$ resonances, coefficients $\alpha_{1,2}$ and $\Delta$ for the TFFs $A(q^{2}),V_{1}(q^{2}),V_{2}(q^{2}),V_{0}(q^{2})$, in which all the input parameters are set to be their central values. |  $A(q^{2})$ |  $V_{1}(q^{2})$ |  $V_{2}(q^{2})$ |  $V_{0}(q^{2})$
---|---|---|---|---
$m_{R,i}$ | 2.007 | 2.420 | 2.420 | 1.865
$\alpha_{1}$ | $-1.640$ | $-5.178$ | $-0.703$ | $-4.384$
$\alpha_{2}$ | $31.085$ | $231.525$ | $18.574$ | $38.942$
$\Delta$ | $0.004\%$ | $0.002\%$ | $0.020\%$ | $0.020\%$

Theoretically, the LCSR approach for $D\to a_{1}(1260)$ TFFs are reliable in
low and intermediate $q^{2}$-regions, which can be extrapolated to all the
physically allowable region $m_{\ell}^{2}\leq(m_{D}-m_{a_{1}})^{2}\approx
0.4~{}{\rm GeV}^{2}$. In this paper, we mainly consider the simplified series
expansion (SSE), which has the following form

$\displaystyle
F_{i}(q^{2})=\frac{1}{P_{i}(q^{2})}\sum\limits_{k}\alpha_{k}[z(q^{2})-z(0)]^{k},$
(56)

with the function $z(t)$ including $t_{\pm}$, $t_{0}$ and $t$, whose
definition can be found Ref. Hu:2021zmy . $F_{i}(q^{2})$ are the TFFs
$A(q^{2})$ and $V_{0,1,2}(q^{2})$, respectively. In this approach, the simple
pole $P_{i}(q^{2})=(1-q^{2}/m^{2}_{R,i})$ accounting for low-lying resonances,
instead of Blaschke factor $B(t)$ is more applicable in many processes. Here,
the masses of low-lying $D$ resonances are mainly determined by the $J^{P}$
states. Followed by the Ref. Momeni:2020zrb  and PDG values Zyla:2020zbs , we
listed the $m_{R,i}$ in Table 5. Meanwhile, the fitting quality should
satisfied the relationship $\Delta<1\%$, which is defined as

$\displaystyle\Delta=\frac{\sum\nolimits_{t}|F_{i}(t)-F_{i}^{\rm
fit}(t)|}{\sum\nolimits_{t}|F_{i}(t)|}\times 100$ (57)

where $t\in[0,1/40,\cdots,40/40]\times 0.28~{}{\rm GeV}^{2}$. The fitting
parameters $\alpha_{i}$ for every TFFs and the quality of fit $\Delta$ are
also listed in Table 5. From which, the $\Delta$ of $D\to a_{1}(1260)$ TFFs
are less than 0.020$\%$.

Table 6: The $D\to a_{1}(1260)\ell^{+}\nu_{\ell}$ branching fractions with error. Other predictions are given as a comparison. Method | ${\cal B}(D^{0}\to a_{1}^{-}(1260)e^{+}\nu_{e})$ | ${\cal B}(D^{+}\to a_{1}^{0}(1260)e^{+}\nu_{e})$ | ${\cal B}(D^{0}\to a_{1}^{-}(1260)\mu^{+}\nu_{\mu})$ | ${\cal B}(D^{+}\to a_{1}^{0}(1260)\mu^{+}\nu_{\mu})$
---|---|---|---|---
This work | $(5.261_{-0.639}^{+0.745})\times 10^{-5}$ | $(6.673_{-0.811}^{+0.947})\times 10^{-5}$ | $(4.732_{-0.590}^{+0.685})\times 10^{-5}$ | $(6.002_{-0.748}^{+0.796})\times 10^{-5}$
LCSR-I Momeni:2019uag  | $(3.58\pm 0.52)\times 10^{-5}$ | $(4.73\pm 0.63)\times{10^{-5}}$ | - | -
LCSR-II Huang:2021owr  | $6.90\times 10^{-5}$ | $9.38\times 10^{-5}$ | $6.27\times 10^{-5}$ | $8.52\times 10^{-5}$
3PSR Zuo:2016msr  | $(1.11_{-0.34}^{+0.41})\times 10^{-5}$ | $(1.47_{-0.44}^{+0.55})\times 10^{-5}$ | - | -
CLFQM Wang:2015cis  | $4.1\times 10^{-5}$ | - | $3.6\times{10^{-5}}$ | -

Furthermore, the $|V_{cd}|$-independence longitudinal and transverse
differential decay widths $d\Gamma_{\rm L,T}$ and total width $d\Gamma_{\rm
total}=d\Gamma_{\rm L}+d\Gamma_{\rm T}$ of $D\to
a_{1}(1260)\ell^{+}\nu_{\ell}$ can be obtained according to Eqs.
(LABEL:Eq:dgamma1) and (LABEL:Eq:dgamma2), which are shown in Fig. 4. As a
comparison, we also present the LCSR-II predictions Huang:2021owr . Fig. 4
shows that the contributions of the decay widths mainly come from the
longitudinal parts in small $q^{2}$-region, and transverse parts contribute
sizably in intermediate and large $q^{2}$-regions. In different to LCSR-II
prediction, our predictions tend to $0$ when
$q^{2}\rightsquigarrow(m_{D}-m_{a_{1}})^{2}\approx 0.4~{}{\rm GeV}^{2}$, which
are similar to most of the other semileptonic decay processes such as final
state involving $\pi,K,\rho,K^{*},D,...$.

Figure 4: The $D\to a_{1}(1260)\ell^{+}\nu_{\ell}$ decay width with $q^{2}$ by
using the chiral LCSR for the TFFs, where the upper and lower ones are for
$\ell=(e,\mu)$, respectively. As a comparison, we also present the other LCSR
results from Ref. Huang:2021owr .

Finally, by using the lifetimes of $D^{0},D^{+}$-mesons
${\tau_{D^{0}}}=(0.410\pm 0.015)~{}\rm ps$ and $\tau_{D^{+}}=(1.040\pm
0.007)~{}\rm ps$ issued by the PDG Zyla:2020zbs , we get the branching
fractions for the two different semileptonic decay channels $D^{0}\to
a_{1}^{-}(1260)\ell^{+}\nu_{\ell}$ and $D^{+}\to
a_{1}^{0}(1260)\ell^{+}\nu_{\ell}$, which are listed in Table 6. Meanwhile,
other theoretical predictions such as LCSR-I, II Momeni:2019uag ;
Huang:2021owr , 3PSR Zuo:2016msr  and CLFQM Wang:2015cis  are also given as a
comparison. Our results are in agreement with other predictions which are in
order of $10^{-5}$. At present, $D^{0(+)}\to
a_{1}^{-(0)}(1260)\ell^{+}\nu_{\ell}$ has not been measured. In year 2020, the
BESIII collaboration measured $D^{0(+)}\to b_{1}^{-(0)}e^{+}\nu_{e}$ decay
processes and provided the upper limits for the product branching fractions,
which are ${\cal B}(D^{0}\to b_{1}^{-}(1235)e^{+}\nu_{e})\cdot{\cal
B}(b_{1}(1235)^{-}\to\omega\pi^{-})<1.12\times 10^{-4}$ and ${\cal B}(D^{+}\to
b_{1}^{0}(1235)e^{+}\nu_{e})\cdot{\cal
B}(b_{1}(1235)^{0}\to\omega\pi^{0})<1.75\times 10^{-4}$ Ablikim:2020agq . Here
${\cal B}(b_{1}(1235)^{0(-)}\to\omega\pi^{0(-)})=1$ Huang:2021owr  and one can
get the branching fraction for $D\to b_{1}(1235)e^{+}\nu_{e}$ directly.
Furthermore, the branching fractions of $D\to a_{1}(1260)\ell^{+}\nu_{\ell}$
of this work satisfies this upper limit.

##  IV Summary

In the present paper, we have calculated the $a_{1}(1260)$-meson moments of
LCDA $\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$ by using the BFTSR approach
up to NLO QCD corrections for the perturbative part and up to dimension-six
condensates for the non-perturbative part. The moments of LCDA up to $10_{\rm
th}$-order have been given in Eq.(40). Then, by combining the two constraints
(17) and (18) with the least squares fitting approach for
$\langle\xi_{2;a_{1}}^{\|;n}\rangle|_{\mu}$, we get the $a_{1}(1260)$-meson
longitudinal LCDA $\phi_{2;a_{1}}^{\|}(x,\mu_{0})$. Figure 3 shows that
$\phi_{2;a_{1}}^{\|}(x,\mu_{0})$ tends to a single-peak behavior. Moreover, by
using the derived twist-2 LCDA, we have calculated the $D\to a_{1}(1260)$ TFFs
$A(q^{2})$ and $V_{0,1,2}(q^{2})$ by using the LCSR approach up to twist-4
accuracy. Furthermore, the $|V_{cd}|$-independence differential decay width of
semileptonic decay $D\to a_{1}(1260)\ell^{+}\nu_{\ell}$ with $\ell=(e,\mu)$
have been given in Fig. 4, and the branching fractions for $D^{0(+)}\to
a_{1}^{-(0)}\ell^{+}\nu_{\ell}$ are given in Table 6. The branching fractions
are of order $10^{-5}$, which is close to the present experimental upper
limit. It is hoped that the decays $D\to a_{1}(1260)\ell^{+}\nu_{\ell}$ can be
observed in near future, which inversely could provide a (potential) helpful
test for QCD sum rules approach.

##  V Acknowledgments

We are grateful for the referee’s valuable comments and suggestions. This work
was supported in part by the National Natural Science Foundation of China
under Grant No.11765007, No.11875122, No.11947406, No.12147102 and
No.12175025, the Project of Guizhou Provincial Department of Science and
Technology under Grant No.KY[2019]1171, and No.ZK[2021]024, the Project of
Guizhou Provincial Department of Education under Grant No.KY[2021]030 and
No.KY[2021]003, the Fundamental Research Funds for the Central Universities
under Grant No.2020CQJQY-Z003, the graduate research and innovation foundation
of Chongqing, china (No.ydstd1912), and the Project of Guizhou Minzu
University under Grant No. GZMU[2019]YB19.

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