#  $K_{S}^{0}-K_{L}^{0}$ asymmetries in $D$-meson decays

Di Wang1, Fu-Sheng Yu1,2, Peng-Fei Guo1, and Hua-Yu Jiang1 1 School of Nuclear
Science and Technology, Lanzhou University, Lanzhou 730000, People’s Republic
of China
2Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, People’s Republic of China

###### Abstract

The $K_{S}^{0}-K_{L}^{0}$ asymmetries in the $D$ meson decays, induced by the
interference between the Cabibbo-favored and the doubly Cabibbo-suppressed
amplitudes, can help to understand the dynamics of charm decays. All possible
processes of two-body non-leptonic $D$ decays into one neutral kaon and
another pseudoscalar or vector meson are considered. We study the
$K_{S}^{0}-K_{L}^{0}$ asymmetries and the branching fractions of corresponding
processes in the factorization-assisted topological-amplitude approach in
which significant flavor $SU(3)$ symmetry breaking effects are included. The
branching fractions of $K_{L}^{0}$ modes are predicted. It is first found that
the $K_{S}^{0}-K_{L}^{0}$ asymmetries in the $D^{0}$-meson decays are shifted
by the $D^{0}-\overline{D}^{0}$ mixing parameter $y_{D}\simeq 0.006$, to be
$0.113\pm 0.001$ for all the relevant $D^{0}$ decay modes. Our results on
$K_{S}^{0}-K_{L}^{0}$ asymmetries are consistent with the current data and
could be tested by experiments in the future.

##  I Introduction

The study of $D$-meson decays and mixing can provide some useful information
with respect to flavor mixing and $CP$ asymmetries lab01 . The two-body
nonleptonic decays of $D$ mesons can be classified into three types: Cabibbo-
favored (CF), singly Cabibbo-suppressed (SCS), and doubly Cabibbo-suppressed
(DCS) processes. In the Standard Model (SM), they are, respectively,
corresponding to the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements,
$|V_{cs}^{*}V_{ud}|\sim 1$,
$|V_{cd}^{*}V_{ud}|\sim|V_{cs}^{*}V_{us}|\sim\lambda$, and
$|V_{cd}^{*}V_{us}|\sim\lambda^{2}$ with the Wolfenstein parameter
$\lambda=\sin\theta_{C}\approx 0.225$ and $\theta_{C}$ as the Cabibbo angle.
Unlike the CF and SCS processes mostly observed in experiments, only a few DCS
modes are well measured due to the relatively small branching fractions PDG .
However, the studies on DCS processes have great interests for us. Because of
the relative smallness in the SM, the DCS processes can be significantly
affected by new physics beyond the Standard Model. For example, there would be
new $CP$ violating effects in the DCS processes in some new physics models,
thereby affecting the determination of mixing parameters and indirect $CP$
phases in the $D^{0}-\overline{D}^{0}$ system HFAG ; Kagan:2009gb . Besides,
the studies on DCS processes can help to test the flavor $SU(3)$ symmetry and
understand the dynamics of charmed hadron decays and the mechanism of
$D^{0}-\overline{D}^{0}$ mixing FAT1 ; FAT2 ; Cheng:2010ry ; Cheng:2016ejf ;
Cheng:2012xb ; Muller:2015lua ; Bhattacharya:2008ss ; Bhattacharya:2008ke ;
Bhattacharya:2009ps ; Donoghue:1985hh ; Buccella:1996uy ; lab3 ; Falk:2001hx ;
Cheng:2010rv .

Among the DCS modes, the decaying of $D$ mesons into $K^{0}$ in the final
states is actually involved in the processes with $K_{S}^{0}$, which are
dominated, however, by CF modes of $D$ decaying into $\overline{K}^{0}$. We
cannot distinguish the effects of the CF and DCS amplitudes in the individual
data of $D\to K_{S}^{0}f$. In some of the literatures, the decays with
$K_{S}^{0}$ in the final states are always approximately considered as
saturated by CF contributions, and hence the DCS information is neglected in
such processes Cheng:2010ry ; Cheng:2016ejf ; Cheng:2012xb ; Rosner:1999xd .

The difference between the $K^{0}_{S}$ and $K^{0}_{L}$ modes induced by
interference between the CF and DCS amplitudes was first pointed out by Bigi
and Yamamoto Bigi . They proposed the observable of the $K_{S}^{0}-K_{L}^{0}$
asymmetry to describe the difference between modes with $K_{S}^{0}$ and
$K_{L}^{0}$. In the two-body decays of $D\to K^{0}_{S,L}f$ with $f$ as the
other meson in the final state except for the neutral kaons, the
$K_{S}^{0}-K_{L}^{0}$ asymmetries are defined by

$R(f)\equiv\frac{\Gamma(D\rightarrow K_{S}^{0}f)-\Gamma(D\rightarrow
K_{L}^{0}f)}{\Gamma(D\rightarrow K_{S}^{0}f)+\Gamma(D\rightarrow
K_{L}^{0}f)}.$ (1)

The nonvanishing values of $R(f)$ would be induced by the interference between
the decay amplitudes of $D\to\overline{K}^{0}f$ (CF transitions) and $D\to
K^{0}f$ (DCS transitions). Therefore, the determination on the
$K_{S}^{0}-K_{L}^{0}$ asymmetries in $D$ decays can be useful to study the DCS
processes. The asymmetries in $D\to K_{S,L}^{0}\pi$ decays have been measured
by the CLEO Collaboration lab10 111The BESIII collaboration has reported their
preliminary result with only statistical error that RBES  $R(D^{0}\to
K_{S,L}^{0}\pi^{0})=0.1077\pm 0.0125.$

$\begin{split}R(D^{0}\to K_{S,L}^{0}\pi^{0})=0.108\pm 0.025\pm 0.024,\\\
R(D^{+}\to K_{S,L}^{0}\pi^{+})=0.022\pm 0.016\pm 0.018.\end{split}$ (2)

The $K_{S}^{0}-K_{L}^{0}$ asymmetries have been studied in the QCD
factorization approach Gao:2006nb ; lab4 . However, since the charm quark mass
is not heavy enough, the QCD-inspired methods, such as the QCD factorization
approach lab5 , the perturbative QCD approach lab6 , and the soft-collinear
effective theory lab02 , are not suitable for charmed hadron decays. The
asymmetries are also predicted in the conventional topological diagrammatic
approach under the $SU(3)$ flavor symmetry Bhattacharya:2008ss ;
Bhattacharya:2009ps ; Cheng:2010ry , but it is known that the $SU(3)$ breaking
effects can be as large as $30\%$ in charm decays and, thus have to be
considered. In Muller:2015lua , the authors studied the $K_{S}^{0}-K_{L}^{0}$
asymmetries in the topological approach including linear $SU(3)$ breaking
effects. Since there are too many parameters in this method, the predictive
power is limited.

In this work, we study the $K_{S}^{0}-K_{L}^{0}$ asymmetries in the
factorization-assisted topological-amplitude (FAT) approach FAT1 ; FAT2 , in
which nonperturbative contributions are included and significant flavor
$SU(3)$ symmetry breaking effects are well expressed. It has been shown that
the FAT approach works well in $D$ meson decays. The results on branching
fractions are consistent with experimental data in the $D$ decays into two
pseudoscalar mesons ($PP$), or one pseudoscalar meson and one vector meson
$(PV)$. Furthermore, the prediction on the $CP$ asymmetry difference $\Delta
A_{CP}=A_{CP}(D^{0}\to K^{+}K^{-})-A_{CP}(D^{0}\to\pi^{+}\pi^{-})$ in the FAT
approach FAT1  is verified by recent LHCb collaboration Aaij:2016cfh . The FAT
approach will be introduced in details in the following sections.

In this paper, we will study the $K_{S}^{0}-K_{L}^{0}$ asymmetries in the SM
in the $D\to PP$ decay modes of

$D^{0}\rightarrow K_{S,L}^{0}\pi^{0},\quad D^{0}\rightarrow
K_{S,L}^{0}\eta,\quad D^{0}\rightarrow K_{S,L}^{0}\eta^{\prime},\quad
D^{+}\rightarrow K_{S,L}^{0}\pi^{+},\quad D^{+}_{s}\rightarrow
K_{S,L}^{0}K^{+},$

and firstly in the $D\rightarrow PV$ decay modes of

$\begin{split}D^{0}\rightarrow K_{S,L}^{0}\rho^{0},\quad D^{0}\rightarrow
K_{S,L}^{0}\omega,\quad D^{0}\rightarrow K_{S,L}^{0}\phi,\quad
D^{+}\rightarrow K_{S,L}^{0}\rho^{+},\quad D^{+}_{s}\rightarrow
K_{S,L}^{0}K^{*+}.\end{split}$

Furthermore, the $D^{0}-\overline{D}^{0}$ mixing effects will first be
considered in $K_{S}^{0}-K_{L}^{0}$ asymmetries in the neutral $D$ meson decay
modes, which we find to be non-negligible.

The plan of this paper is as follows. In Sec. II, we will show the general
formulas of the $K_{S}^{0}-K_{L}^{0}$ asymmetries, $R(f)$, and the
$D^{0}-\overline{D}^{0}$ mixing effects on it. In Sec. III, we shall present
the amplitude decompositions of $D\to PP$ and $D\to PV$ decays in the FAT
approach. The numerical results on branching fractions and
$K_{S}^{0}-K^{0}_{L}$ asymmetries will be given in Sec. IV. Sec. V is the
conclusion.

##  II The Formalism of the $K_{S}^{0}-K_{L}^{0}$ asymmetries

###  II.1 The $K_{S}^{0}-K_{L}^{0}$ asymmetries in charged $D$ decays

The $K_{S}^{0}$ and $K_{L}^{0}$ states are linear combinations of $K^{0}$ and
$\overline{K}^{0}$, under the convention of
$\mathcal{CP}|K^{0}\rangle=-|\overline{K}^{0}\rangle$, as

$\begin{split}|K_{S}^{0}\rangle&=\frac{1}{\sqrt{2(1+|\epsilon|^{2})}}\left[(1+\epsilon)|K^{0}\rangle-(1-\epsilon)|\overline{K}^{0}\rangle\right],\\\
|K_{L}^{0}\rangle&=\frac{1}{\sqrt{2(1+|\epsilon|^{2})}}\left[(1+\epsilon)|K^{0}\rangle+(1-\epsilon)|\overline{K}^{0}\rangle\right],\end{split}$
(3)

where $\epsilon=|\epsilon|e^{i\phi_{\epsilon}}$ is a small complex parameter
indicating the indirect $CP$ violating effect, with the value of
$|\epsilon|=(2.228\pm 0.011)\times 10^{-3}$ and
$\phi_{\epsilon}=43.5^{\circ}\pm 0.5^{\circ}$ PDG . We start with this more
general formula of $K^{0}-\overline{K}^{0}$ mixing, and will find later that
the parameter $\epsilon$ is negligible in the $K_{S}^{0}-K_{L}^{0}$
asymmetries.

In order to study the $K_{S}^{0}-K_{L}^{0}$ asymmetries in Eq.(1), we express
the amplitudes of $D\to\overline{K}^{0}f$ and $D\to K^{0}f$ decays as

$\displaystyle\mathcal{A}(D\to\overline{K}^{0}f)=\mathcal{T}_{\rm
CF}e^{i(\phi_{\rm CF}+\delta_{\rm CF})},~{}~{}~{}~{}~{}\mathcal{A}(D\to
K^{0}f)=\mathcal{T}_{\rm DCS}\,e^{i(\phi_{\rm DCS}+\delta_{\rm DCS})},$ (4)

where $\mathcal{T}_{\rm CF,\,DCS}$ are real, $\phi_{\rm CF,\,DCS}$ and
$\delta_{\rm CF,\,DCS}$ are the weak and strong phases for the CF and DCS
amplitudes, respectively. The CKM matrix elements have been involved in
$\mathcal{T}_{\rm CF,\,DCS}$. From Eq.(3), the amplitudes of the $D\rightarrow
K_{S}^{0}f$ and $D\rightarrow K_{L}^{0}f$ decays are x1

$\begin{split}\mathcal{A}(D\rightarrow
K_{S}^{0}f)&=\frac{1}{\sqrt{2(1+|\epsilon|^{2})}}\left[(1+\epsilon^{*})\,\mathcal{T}_{\rm
DCS}\,e^{i(\phi_{\rm DCS}+\delta_{\rm
DCS})}-(1-\epsilon^{*})\,\mathcal{T}_{\rm CF}\,e^{i(\phi_{\rm CF}+\delta_{\rm
CF})}\right],\\\ \mathcal{A}(D\rightarrow
K_{L}^{0}f)&=\frac{1}{\sqrt{2(1+|\epsilon|^{2})}}\left[(1+\epsilon^{*})\,\mathcal{T}_{\rm
DCS}\,e^{i(\phi_{\rm DCS}+\delta_{\rm
DCS})}+(1-\epsilon^{*})\,\mathcal{T}_{\rm CF}\,e^{i(\phi_{\rm CF}+\delta_{\rm
CF})}\right].\end{split}$ (5)

For convenience, we define the ratio between the DCS and CF amplitudes as

$\frac{\mathcal{A}(D\rightarrow
K^{0}f)}{\mathcal{A}(D\rightarrow\overline{K}^{0}f)}=r_{f}\,e^{i(\phi_{f}+\delta_{f})},$
(6)

where $r_{f}=\mathcal{T}_{\rm DCS}/\mathcal{T}_{\rm CF}$, $\phi_{f}=\phi_{\rm
DCS}-\phi_{\rm CF}$ and $\delta_{f}=\delta_{\rm DCS}-\delta_{\rm CF}$. Note
that $r_{f}$ is small,
$r_{f}\propto|V_{cd}^{*}V_{us}/V_{cs}^{*}V_{ud}|\approx\lambda^{2}=\mathcal{O}(10^{-2})$.
The parameters $r_{f}$ and $\delta_{f}$ depend on the individual processes and
$\phi_{f}$ is mode independent in the SM. Then the $K_{S}^{0}-K_{L}^{0}$
asymmetries can be written as

$\displaystyle R(f)$
$\displaystyle={|(1-\epsilon^{*})-(1+\epsilon^{*})\,r_{f}\,e^{i(\phi_{f}+\delta_{f})}|^{2}-|(1-\epsilon^{*})+(1+\epsilon^{*})\,r_{f}\,e^{i(\phi_{f}+\delta_{f})}|^{2}\over|(1-\epsilon^{*})-(1+\epsilon^{*})\,r_{f}\,e^{i(\phi_{f}+\delta_{f})}|^{2}+|(1-\epsilon^{*})+(1+\epsilon^{*})\,r_{f}\,e^{i(\phi_{f}+\delta_{f})}|^{2}}$
$\displaystyle=-2\,r_{f}\frac{\cos(\phi_{f}+\delta_{f})(1-|\epsilon|^{2})+2\sin(\phi_{f}+\delta_{f})Im(\epsilon)}{|1-\epsilon^{*}|^{2}+|1+\epsilon^{*}|^{2}\,r_{f}^{2}}$
$\displaystyle\simeq-2r_{f}\cos(\phi_{f}+\delta_{f})-4r_{f}[Re(\epsilon)\cos(\phi_{f}+\delta_{f})+Im(\epsilon)\sin(\phi_{f}+\delta_{f})].$
(7)

The second term in the last line is sub-leading, at the order of $10^{-4}$,
and hence can be safely neglected compared to the first term which is
$\mathcal{O}(10^{-2})$. Thus the $K_{S}^{0}-K_{L}^{0}$ asymmetries are not
sensitive to the $CP$ violating effect in the $K^{0}-\overline{K}^{0}$ mixing
system. Besides, the weak phase difference $\phi_{f}$ is tiny in the SM,
$\phi_{f}=Arg\left[{V_{cd}^{*}V_{us}/V_{cs}^{*}V_{ud}}\right]$,
$\sin\phi_{f}=\mathcal{O}(\lambda^{4})=\mathcal{O}(10^{-3})$. Hence as a good
approximation, the $K_{S}^{0}-K_{L}^{0}$ asymmetries can be expressed as

$\displaystyle R(f)=-2\,r_{f}\cos\delta_{f}.$ (8)

It can be expected that $R(f)=\mathcal{O}(10^{-2})$. Therefore, the
determination of the $K_{S}^{0}-K_{L}^{0}$ asymmetries is useful to understand
the dynamics of the DCS decays, especially the relative strong phases between
DCS and CF amplitudes.

Note that all the above formulas are in general for $D$ decays. In $D^{0}$
decay modes, the $D^{0}-\overline{D}^{0}$ mixing effects have to be
considered, which will be discussed in the next subsection. Besides, as seen
above, the $CP$ violating effect in $K^{0}-\overline{K}^{0}$ mixing is
negligible in the discussion of $K_{S}^{0}-K_{L}^{0}$ asymmetries. Thereby
$K_{S}^{0}$ and $K_{L}^{0}$ are the $CP$ even and $CP$ odd states,
respectively,

$|K_{S}^{0}\rangle=\frac{1}{\sqrt{2}}\left(|K^{0}\rangle-|\overline{K}^{0}\rangle\right),~{}~{}~{}~{}~{}|K_{L}^{0}\rangle=\frac{1}{\sqrt{2}}\left(|K^{0}\rangle+|\overline{K}^{0}\rangle\right).$
(9)

In the following discussions, we will use the above formulas for $K_{S}^{0}$
and $K_{L}^{0}$ states, and the decay amplitudes of

$\begin{split}\mathcal{A}(D\rightarrow
K_{S}^{0}f)&=-\frac{1}{\sqrt{2}}\mathcal{T}_{\rm CF}\,e^{i\delta_{\rm
CF}}\left(1-r_{f}e^{i\delta_{f}}\right),\\\ \mathcal{A}(D\rightarrow
K_{L}^{0}f)&=\frac{1}{\sqrt{2}}\mathcal{T}_{\rm CF}\,e^{i\delta_{\rm
CF}}\left(1+r_{f}e^{i\delta_{f}}\right).\end{split}$ (10)

In the SM, there is a minus sign in $r_{f}=-\tan^{2}\theta_{c}\hat{r}_{f}$,
with $\hat{r}_{f}=|(\mathcal{T}_{\rm DCS}/\mathcal{T}_{\rm
CF})(V_{cs}^{*}V_{ud}/V_{cd}^{*}V_{us})|$. Then the CF and DCS amplitudes
would contribute constructively (destructively) for the
$K_{S}^{0}~{}(K_{L}^{0})$ modes in the case of $\cos\delta_{f}>0$, and
conversely for $\cos\delta_{f}<0$. $R(f)$ can also be expressed as

$\displaystyle R(f)=2\tan^{2}\theta_{C}~{}\hat{r}_{f}\cos\delta_{f}.$ (11)

Physics does not depend on the phase conventions. If
$\mathcal{CP}|K^{0}\rangle=+|\overline{K}^{0}\rangle$, Eq. (8) would become
$R(f)=2r_{f}\cos\delta_{f}$. But as shown in Cheng:2010ry  that the decay
constants of $K^{0}$ and $\overline{K}^{0}$ are opposite in sign in this case,
there would be additional opposite sign between $\mathcal{A}(D\to K^{0}f)$ and
$\mathcal{A}(D\to\overline{K}^{0}f)$, so then
$r_{f}=\tan^{2}\theta_{c}\hat{r}_{f}$. Thus (11) still holds under different
phase conventions.

###  II.2 The effect of the $D^{0}-\overline{D}^{0}$ mixing

We will study the $D^{0}-\overline{D}^{0}$ mixing effect in the $D^{0}\to
K_{S,L}^{0}f_{CP}^{0}$ decays, where $f_{CP}^{0}$ is a $CP$ eigenstate such as
$\pi^{0}$, $\eta^{(\prime)}$, $\rho^{0}$, $\omega$ and $\phi$. Under the
convention of $\mathcal{CP}|D^{0}\rangle=-|\overline{D}^{0}\rangle$, the mass
eigenstates of the neutral $D$ mesons can be written as
$|D^{0}_{1,2}\rangle=p|D^{0}\rangle\mp q|\overline{D}^{0}\rangle$ with
$q/p=|q/p|e^{i\phi_{D}}$. Some standard notations are used in neutral
$D$-meson mixing:

$\begin{split}\mathcal{A}_{K_{S}^{0}}&\equiv\mathcal{A}(D^{0}\to
K_{S}^{0}f_{CP}^{0}),\quad\overline{\mathcal{A}}_{K_{S}^{0}}\equiv\mathcal{A}(\overline{D}^{0}\to
K_{S}^{0}f_{CP}^{0}),\\\ \mathcal{A}_{K_{L}^{0}}&\equiv\mathcal{A}(D^{0}\to
K_{L}^{0}f_{CP}^{0}),\quad\overline{\mathcal{A}}_{K_{L}^{0}}\equiv\mathcal{A}(\overline{D}^{0}\to
K_{L}^{0}f_{CP}^{0}),\\\
\lambda_{K_{S}^{0}}&\equiv\frac{q}{p}\frac{\overline{\mathcal{A}}_{K_{S}^{0}}}{\mathcal{A}_{K_{S}^{0}}},\quad\lambda_{K_{L}^{0}}\equiv\frac{q}{p}\frac{\overline{\mathcal{A}}_{K_{L}^{0}}}{\mathcal{A}_{K_{L}^{0}}},\quad\Gamma_{D^{0}}\equiv\frac{\Gamma_{D^{0}_{1}}+\Gamma_{D^{0}_{2}}}{2},\\\
x_{D}\equiv&\frac{\Delta
m_{D^{0}}}{\Gamma_{D^{0}}}=\frac{m_{D^{0}_{1}}-m_{D^{0}_{2}}}{\Gamma_{D^{0}}},\quad
y_{D}\equiv\frac{\Delta\Gamma_{D^{0}}}{2\Gamma_{D^{0}}}=\frac{\Gamma_{D^{0}_{1}}-\Gamma_{D^{0}_{2}}}{2\Gamma_{D^{0}}},\end{split}$
(12)

where the amplitudes of $\overline{D}^{0}\rightarrow K_{S}^{0}f^{0}_{CP}$ and
$\overline{D}^{0}\rightarrow K_{L}^{0}f^{0}_{CP}$ decays are expressed as

$\begin{split}\mathcal{A}(\overline{D}^{0}\rightarrow
K_{S}^{0}f^{0}_{CP})&=-\eta_{CP}\,\eta_{K_{S}^{0}}\left[\mathcal{T}_{\rm
DCS}e^{i(-\phi_{\rm DCS}+\delta_{\rm DCS})}-\mathcal{T}_{\rm
CF}e^{i(-\phi_{\rm CF}+\delta_{\rm CF})}\right]/\sqrt{2},\\\
\mathcal{A}(\overline{D}^{0}\rightarrow
K_{L}^{0}f^{0}_{CP})&=-\eta_{CP}\,\eta_{K_{L}^{0}}\left[\mathcal{T}_{\rm
DCS}e^{i(-\phi_{\rm DCS}+\delta_{\rm DCS})}+\mathcal{T}_{\rm
CF}e^{i(-\phi_{\rm CF}+\delta_{\rm CF})}\right]/\sqrt{2},\end{split}$ (13)

where the minus sign is from
$\mathcal{CP}|D^{0}\rangle=-|\overline{D}^{0}\rangle$ and
$\eta_{K_{S}^{0}}(\eta_{K_{L}^{0}})=+(-)$ from (9), and
$\eta_{CP}=\mathcal{PC}(-1)^{\mathcal{J}}$ with the quantum numbers
$\mathcal{J}^{\mathcal{PC}}$ of $f_{CP}^{0}$. For the decay modes with
pseudoscalar mesons of $\pi^{0}$, $\eta^{(^{\prime})}$ or vector mesons of
$\rho^{0}$, $\omega$ or $\phi$, the values of $\eta_{CP}=-1$. In the absence
of $CP$ asymmetry, we get the decay amplitudes of

$\begin{split}\mathcal{A}(\overline{D}^{0}\rightarrow
K_{S}^{0}f^{0}_{CP})&=-\frac{1}{\sqrt{2}}\mathcal{T}_{\rm CF}\,e^{i\delta_{\rm
CF}}\left(1-r_{f}e^{i\delta_{f}}\right),\\\
\mathcal{A}(\overline{D}^{0}\rightarrow
K_{L}^{0}f^{0}_{CP})&=-\frac{1}{\sqrt{2}}\mathcal{T}_{\rm CF}\,e^{i\delta_{\rm
CF}}\left(1+r_{f}e^{i\delta_{f}}\right).\end{split}$ (14)

In the neutral $D$ meson system, $CP$ is conserved at the level of $10^{-4}$
so far HFAG . It is a good approximation that $|q/p|=1$ and $\phi_{D}=0$. So
then, with Eqs (10) and (14), we get

$\lambda_{K_{S}^{0}}=1\qquad\text{and}\qquad\lambda_{K_{L}^{0}}=-1.$ (15)

The time-integrated decay rates of $D^{0}\to K_{S,L}^{0}f_{CP}^{0}$ decays can
be expressed as Grossman:2006jg

$\begin{split}\Gamma(D^{0}\to
K_{S,L}^{0}f_{CP}^{0})=&\int_{0}^{\infty}\Gamma(D^{0}(t)\to
K_{S,L}^{0}f_{CP}^{0})dt\\\
=&\left|\mathcal{A}_{K_{S,L}^{0}}\right|^{2}\bigg{[}1+{1+|\lambda_{K_{S,L}^{0}}|^{2}\over
2}\frac{y_{D}^{2}}{1-y_{D}^{2}}-{1-|\lambda_{K_{S,L}^{0}}|^{2}\over
2}\frac{x_{D}^{2}}{1+x_{D}^{2}}\\\
&\quad\quad\quad\quad\quad+Re(\lambda_{K_{S,L}^{0}})\frac{y_{D}}{1-y_{D}^{2}}-Im(\lambda_{K_{S,L}^{0}})\frac{x_{D}}{1+x_{D}^{2}}\bigg{]}.\end{split}$
(16)

Notice that the first term is independent from neutral $D$-meson mixing. The
$K_{S}^{0}-K_{L}^{0}$ asymmetries in the $D^{0}$ decays are

$\displaystyle R(f_{CP}^{0})=-2r_{f}\cos\delta_{f}+y_{D}.$ (17)

Compared to (8), the mixing parameter $y_{D}$ contributes to $R(f_{CP}^{0})$.
The $y_{D}$ term in (17) is attributed by the terms of
$Re(\lambda_{K_{S,L}^{0}})$ in the rates $\Gamma(D^{0}\to
K_{{S,L}}^{0}f_{CP}^{0})$ in (16). The current world averaging result of
$y_{D}$ is $(0.62\pm 0.08)\%$ in the case of $CP$ conservation HFAG . If the
precision of the measurements of $R(f^{0}_{CP})$ could reach up to $10^{-3}$,
the neutral $D$-meson mixing effects have to be considered in the analysis of
$D^{0}$ decays.

Similarly to the case of $K^{0}$ and $\overline{K}^{0}$, the phase conventions
of $D^{0}$ and $\overline{D}^{0}$ would not affect the physical observables.
If $\mathcal{CP}|D^{0}\rangle=+|\overline{D}^{0}\rangle$, an additional minus
sign would exist in each equation of (13) and (15), and the terms of
$Re(\lambda_{K_{S,L}^{0}})$ and $Im(\lambda_{K_{S,L}^{0}})$ in (16). So then
(17) still holds in this convention.

##  III Amplitude decompositions in the FAT approach

The factorization-assisted topological-amplitude (FAT) approach works well for
the charm decays FAT1 ; FAT2 . It is based on the topological amplitudes
according to the weak currents. There are four types of topological diagrams
for the two-body nonleptonic $D$ meson decays at the tree level qaz : the
color-favored tree emission amplitude $T$, the color-suppressed tree emission
amplitude $C$, the $W$-exchange amplitude $E$, and the $W$-annihilation
amplitude $A$, as shown in Fig. $1$. Then the hypothesis of factorization is
used, to calculate each topological amplitude which is factorized into two
parts: the short-distance Wilson coefficients and the long-distance hadronic
matrix elements. The large nonperturbative and nonfactorizable contributions
are parametrized to be determined by the experimental data. In this way, most
$SU(3)$ flavor symmetry breaking effects are included.

Figure 1:  Four types of the topological diagrams contributing to two-body
nonleptonic $D$ meson decays in the Standard Model: the color-favored tree
amplitude $T$, the color-suppressed tree amplitude $C$, the $W$-exchange
amplitude $E$ and the $W$-annihilation amplitude $A$.

The effective Hamiltonian of the charm decays in the SM can be written as
lab22

$\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{CKM}\left[C_{1}(\mu)Q_{1}(\mu)+C_{2}(\mu)Q_{2}(\mu)\right]+H.c.,$
(18)

where $G_{F}$ denotes the Fermi coupling constant, $V_{CKM}$ is the products
of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, $C_{1,2}$ are the
Wilson coefficients. The current-current operators are written as

$\displaystyle
Q_{1}=\bar{u}_{\alpha}\gamma_{\mu}(1-\gamma_{5})q_{2\beta}\bar{q}_{1\beta}\gamma^{\mu}(1-\gamma_{5})c_{\alpha},~{}~{}~{}~{}~{}Q_{2}=\bar{u}_{\alpha}\gamma_{\mu}(1-\gamma_{5})q_{2\alpha}\bar{q}_{1\beta}\gamma^{\mu}(1-\gamma_{5})c_{\beta},$
(19)

with $\alpha,\beta$ being the color indices, $q_{1,2}$ being the $d$ or $s$
quarks.

In the factorization hypothesis, the topological amplitudes in the
$D\rightarrow PP$ modes can be written as FAT1

$\displaystyle T[C]$
$\displaystyle=\frac{G_{f}}{\sqrt{2}}V_{CKM}a_{1}(\mu)[a_{2}(\mu)]f_{P_{2}}(m^{2}_{D}-m^{2}_{P_{1}})F_{0}^{D\rightarrow
P_{1}}(m^{2}_{P_{2}}),$ (20) $\displaystyle E$
$\displaystyle=\frac{G_{f}}{\sqrt{2}}V_{CKM}C_{2}(\mu)\chi^{E}_{q,s}e^{i\phi^{E}_{q,s}}f_{D}m^{2}_{D}\Big{(}\frac{f_{P_{1}}f_{P_{2}}}{f_{\pi}^{2}}\Big{)},$
(21) $\displaystyle A$
$\displaystyle=\frac{G_{f}}{\sqrt{2}}V_{CKM}C_{1}(\mu)\chi^{A}_{q,s}e^{i\phi^{A}_{q,s}}f_{D}m^{2}_{D}\Big{(}\frac{f_{P_{1}}f_{P_{2}}}{f_{\pi}^{2}}\Big{)},$
(22)

with

$\displaystyle a_{1}(\mu)$
$\displaystyle=C_{2}(\mu)+\frac{C_{1}(\mu)}{N_{c}},\quad
a_{2}(\mu)=C_{1}(\mu)+C_{2}(\mu)[\frac{1}{N_{c}}+\chi^{C}e^{i\phi^{C}}],$ (23)

and $N_{c}=3$. Here $P_{1}$ represents the pseudoscalar meson transited from
the $D$ decays, and $P_{2}$ the emitted meson, in the $T$ and $C$ diagrams.
$C_{1,2}(\mu)$ are the Wilson coefficients at the scale of $\mu=\sqrt{\Lambda
m_{D}(1-r^{2}_{2})}$ for $T$ and $C$ diagrams, and $\mu=\sqrt{\Lambda
m_{D}(1-r^{2}_{1})(1-r^{2}_{2})}$ for $E$ and $A$ diagrams, with
$r_{i}=m_{P_{i}}/m_{D}$, to describe the $SU(3)$ breaking effect relating to
the energy release of the final states. $\Lambda$ represents the momentum of
the soft degree of freedom in the $D$ decays, fixed to be $\Lambda=0.5$GeV in
this work. It has been shown that large nonfactorizable contributions exist in
the $C$ diagram, resulting from the final-state interactions, which are
parametrized as $\chi^{C}e^{i\phi^{C}}$. $f_{i}$ and $F_{0}$ are the decay
constants and transition form factors, respectively, whose values are used as
in FAT1 ; Fusheng:2011tw . The $E$ and $A$ diagrams are dominated by the
nonfactorizable contributions, parametrized as
$\chi_{q,s}^{E,A}e^{i\phi_{q,s}^{E,A}}$, while the factorizable ones are
neglected due to the helicity suppression. The subscripts $q$ and $s$ stand
for the quark pairs produced from the vacuum as the $u$, $d$ quarks or the $s$
quark. Due to the fact that the pion boson is a Nambu-Goldstone boson and
quark-antiquark bound state simultaneously lab26 ; lab9 , a strong phase
factor $e^{iS_{\pi}}$ is introduced for each pion involved in the non-
factorizable contributions of $E$ and $A$ amplitudes. In the end, all the non-
factorizable parameters, $\chi^{C}$, $\phi^{C}$, $\chi_{q,s}^{E,A}$,
$\phi_{q,s}^{E,A}$ and $S_{\pi}$ are universal parameters to be fit from the
data.

Similarly, the topological amplitudes of the $D\rightarrow PV$ modes can be
parametrized as FAT2

$\displaystyle T_{P}[C_{P}]$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{1}^{P}(\mu)[a_{2}^{P}(\mu)]f_{V}m_{V}F_{1}^{D\rightarrow
P}(m^{2}_{V})2(\varepsilon_{V}\cdot p_{D}),$ (24) $\displaystyle T_{V}[C_{V}]$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{1}^{V}(\mu)[a_{2}^{V}(\mu)]f_{P}m_{V}A_{0}^{D\rightarrow
V}(m^{2}_{P})2(\varepsilon_{V}\cdot p_{D}),$ (25) $\displaystyle E_{P,V}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{CKM}C_{2}(\mu)\chi^{E}_{q,s}e^{i\phi^{E}_{q,s}}f_{D}m_{D}\frac{f_{P}f_{V}}{f_{\pi}f_{\rho}}(\varepsilon_{V}\cdot
p_{D}),$ (26) $\displaystyle A_{P,V}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{CKM}C_{1}(\mu)\chi^{A}_{q,s}e^{i\phi^{A}_{q,s}}f_{D}m_{D}\frac{f_{P}f_{V}}{f_{\pi}f_{\rho}}(\varepsilon_{V}\cdot
p_{D}),$ (27)

where the subscript $P$ in $T_{P}$ and $C_{P}$ represents the topologies with
a transited pseudoscalar meson and an emitted vector boson, while the
subscript $V$ in $T_{V}$ and $C_{V}$ stands for the transited vector meson and
emitted pseudoscalar meson diagrams. The effective Wilson coefficients
$a_{1}^{P(V)}$ and $a_{2}^{P(V)}$ are

$\begin{split}a_{1}^{P(V)}(\mu)&=C_{2}(\mu)+\frac{C_{1}(\mu)}{N_{C}},\quad
a_{2}^{P(V)}(\mu)=C_{1}(\mu)+C_{2}(\mu)[\frac{1}{N_{c}}+\chi^{C}_{P(V)}e^{i\phi^{C}_{P(V)}}].\end{split}$
(28)

The nonfactorizable parameters $\chi_{P,V}^{C}$ and $\phi_{P,V}^{C}$ are also
free to be determined by the data. For the annihilation-type diagrams, the
subscripts of $E_{P,V}$ and $A_{P,V}$ stand for the anti-quark from weak
decays entering in the pseudoscalar meson or the vector meson. It is assumed
that $E_{P}=E_{V}$ and $A_{P}=A_{V}$ in the FAT approach, due to the almost
vanishing branching fraction of $D_{s}^{+}\to\pi^{+}\rho^{0}$ FAT2 , but
$\chi_{q}^{E,A}\neq\chi_{s}^{E,A}$ and $\phi_{q}^{E,A}\neq\phi_{s}^{E,A}$ to
describe large $SU(3)$ breaking effects.

In the end, the major nonperturbative and nonfactorizable contributions are
involved in these universal parameters, and most $SU(3)$ breaking effects are
considered in the FAT approach. Besides, the penguin contributions are not
included in the CF and DCS decays, are smaller than the tree diagrams, and
hence are neglected in this paper. In the following discussions, the CKM
matrix elements are specified out of each topological diagram to denote the CF
and DCS amplitudes. So we will use the same symbols for the topological
diagrams with and without CKM matrix elements, so there will be no ambiguity.

##  IV Numerical results

Table 1: Branching fractions and representations of topological amplitudes for the $D\to PP$ decays with $K_{S}^{0}$ or $K_{L}^{0}$ in the final states. Our results are given in the last column, compared to the experimental data PDG .  Modes |  Representation |  $\mathcal{B}_{\rm exp}(\%)$ |  $\mathcal{B}_{\rm FAT}(\%)$
---|---|---|---
$D^{0}\to K_{S}^{0}\pi^{0}$ | ${1\over 2}V_{cd}^{*}V_{us}(C-E)-{1\over 2}V_{cs}^{*}V_{ud}(C-E)$ |  1.20$\pm$0.04 |  1.31$\pm$0.06
$D^{0}\to K_{L}^{0}\pi^{0}$ | ${1\over 2}V_{cd}^{*}V_{us}(C-E)+{1\over 2}V_{cs}^{*}V_{ud}(C-E)$ |  1.00$\pm$0.07 |  1.05$\pm$0.04
$D^{0}\to K_{S}^{0}\eta$ |  | $V_{cd}^{*}V_{us}[{1\over 2}(C+E)\cos\phi_{\eta}-{1\over\sqrt{2}}E\sin\phi_{\eta}]$
---
$~{}~{}~{}~{}~{}~{}~{}~{}~{}-V_{cs}^{*}V_{ud}[{1\over
2}(C+E)\cos\phi_{\eta}-{1\over\sqrt{2}}E\sin\phi_{\eta}]$
0.485$\pm$0.030 |  0.50$\pm$0.09
$D^{0}\to K_{L}^{0}\eta$ |  | $V_{cd}^{*}V_{us}[{1\over 2}(C+E)\cos\phi_{\eta}-{1\over\sqrt{2}}E\sin\phi_{\eta}]$
---
$~{}~{}~{}~{}~{}~{}~{}~{}~{}+V_{cs}^{*}V_{ud}[{1\over
2}(C+E)\cos\phi_{\eta}-{1\over\sqrt{2}}E\sin\phi_{\eta}]$
|  0.40$\pm$0.07
$D^{0}\to K_{S}^{0}\eta^{\prime}$ |  | $V_{cd}^{*}V_{us}[{1\over 2}(C+E)\sin\phi_{\eta}+{1\over\sqrt{2}}E\cos\phi_{\eta}]$
---
$~{}~{}~{}~{}~{}~{}~{}~{}~{}-V_{cs}^{*}V_{ud}[{1\over
2}(C+E)\sin\phi_{\eta}+{1\over\sqrt{2}}E\cos\phi_{\eta}]$
0.95$\pm$0.05 |  0.95$\pm$0.09
$D^{0}\to K_{L}^{0}\eta^{\prime}$ |  | $V_{cd}^{*}V_{us}[{1\over 2}(C+E)\sin\phi_{\eta}+{1\over\sqrt{2}}E\cos\phi_{\eta}]$
---
$~{}~{}~{}~{}~{}~{}~{}~{}~{}+V_{cs}^{*}V_{ud}[{1\over
2}(C+E)\sin\phi_{\eta}+{1\over\sqrt{2}}E\cos\phi_{\eta}]$
|  0.77$\pm$0.07
$D^{+}\to K_{S}^{0}\pi^{+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(C+A)-{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(T+C)$ |  1.53$\pm$0.06 |  1.61$\pm$0.13
$D^{+}\to K_{L}^{0}\pi^{+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(C+A)+{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(T+C)$ |  1.46$\pm$0.05 |  1.47$\pm$0.14
$D_{s}^{+}\to K_{S}^{0}K^{+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(T+C)-{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(C+A)$ |  1.50$\pm$0.05 |  1.50$\pm$0.16
$D_{s}^{+}\to K_{L}^{0}K^{+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(T+C)+{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(C+A)$ |  |  1.46$\pm$0.16
Table 2: Same as Table 1 but for the $D\to PV$ decays.  Modes |  Representation |  $\mathcal{B}_{\rm exp}(\%)$ |  $\mathcal{B}_{\rm FAT}(\%)$
---|---|---|---
$D^{0}\to K_{S}^{0}\rho^{0}$ | ${1\over 2}V_{cd}^{*}V_{us}(C_{V}-E_{P})-{1\over 2}V_{cs}^{*}V_{ud}(C_{V}-E_{V})$ | $0.64^{+0.07}_{-0.08}$ | $0.50\pm 0.11$
$D^{0}\to K_{L}^{0}\rho^{0}$ | ${1\over 2}V_{cd}^{*}V_{us}(C_{V}-E_{P})+{1\over 2}V_{cs}^{*}V_{ud}(C_{V}-E_{V})$ |  | $0.40\pm 0.09$
$D^{0}\to K_{S}^{0}\omega$ | ${1\over 2}V_{cd}^{*}V_{us}(C_{V}+E_{P})-{1\over 2}V_{cs}^{*}V_{ud}(C_{V}+E_{V})$ | $1.11\pm 0.06$ | $1.18\pm 0.19$
$D^{0}\to K_{L}^{0}\omega$ | ${1\over 2}V_{cd}^{*}V_{us}(C_{V}+E_{P})+{1\over 2}V_{cs}^{*}V_{ud}(C_{V}+E_{V})$ |  | $0.95\pm 0.15$
$D^{0}\to K_{S}^{0}\phi$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}E_{V}-{1\over\sqrt{2}}V_{cs}^{*}V_{ud}E_{P}$ | $0.424^{+0.033}_{-0.017}$ | $0.40\pm 0.04$
$D^{0}\to K_{L}^{0}\phi$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}E_{V}+{1\over\sqrt{2}}V_{cs}^{*}V_{ud}E_{P}$ |  | $0.33\pm 0.03$
$D^{+}\to K_{S}^{0}\rho^{+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(C_{V}+A_{P})-{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(T_{P}+C_{V})$ | $6.04^{+0.60}_{-0.34}$ | $4.99\pm 0.50$
$D^{+}\to K_{L}^{0}\rho^{+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(C_{V}+A_{P})+{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(T_{P}+C_{V})$ |  | $5.37\pm 0.50$
$D_{s}^{+}\to K_{S}^{0}K^{*+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(T_{P}+C_{V})-{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(C_{V}+A_{P})$ | $2.7\pm 0.6$ | $1.20\pm 0.36$
$D_{s}^{+}\to K_{L}^{0}K^{*+}$ | ${1\over\sqrt{2}}V_{cd}^{*}V_{us}(T_{P}+C_{V})+{1\over\sqrt{2}}V_{cs}^{*}V_{ud}(C_{V}+A_{P})$ |  | $1.37\pm 0.33$

In order to obtain the reasonable results of $K_{S}^{0}-K_{L}^{0}$ asymmetries
$R(f)$, we do a global $\chi^{2}$ fit on the nonperturbative parameters in the
FAT approach using the latest experimental data. The fittings are separately
for the $D\to PP$ and $PV$ modes with 30 and 37 data, respectively. We use the
$\mathcal{B}(D\to K^{0}_{S}f)$ and $\mathcal{B}(D\to K^{0}_{L}f)$ instead of
the $\mathcal{B}(D\to\overline{K}^{0}f)$ so as to include the interference
effects between the CF and DCS decays. The associated best-fit values and
uncertainties are obtained as

$\begin{split}\chi^{C}=-0.406\pm 0.011,&~{}~{}~{}\phi^{C}=0.636\pm 0.011,\\\
\chi_{q}^{E}=0.226\pm 0.006,&~{}~{}~{}\chi_{s}^{E}=0.138\pm 0.005,\\\
\chi_{q}^{A}=0.259\pm 0.013,&~{}~{}~{}\chi_{s}^{A}=0.218\pm 0.015,\\\
\phi^{E}_{q}=-4.44\pm 0.02,&~{}~{}~{}\phi_{s}^{E}=-4.81\pm 0.04,\\\
\phi^{A}_{q}=-4.21\pm 0.03,&~{}~{}~{}\phi_{s}^{A}=-3.93\pm 0.04,\\\
S_{\pi}=0.192\pm 0.010,&\end{split}$

in the $D\to PP$ modes, and

$\begin{split}\chi_{P}^{C}=-0.443\pm 0.007,&~{}~{}~{}\phi_{P}^{C}=0.497\pm
0.027,\\\ \chi_{V}^{C}=-0.694\pm 0.024,&~{}~{}~{}\phi_{V}^{C}=0.828\pm
0.065,\\\ \chi_{q}^{E}=0.194\pm 0.013,&~{}~{}~{}\chi_{s}^{E}=0.283\pm
0.011,\\\ \chi^{A}_{q}=0.147\pm 0.021,&~{}~{}~{}\chi^{A}_{s}=0.135\pm
0.032,\\\ \phi_{q}^{E}=-1.40\pm 0.07,&~{}~{}~{}\phi_{s}^{E}=-3.09\pm 0.13,\\\
\phi_{q}^{A}=-0.584\pm 0.211,&~{}~{}~{}\phi^{A}_{s}=-1.71\pm 0.14,\\\
S_{\pi}=1.28\pm 0.14,&\end{split}$

in the $D\to PV$ modes.

The topological diagrammatic representations and our results of branching
fractions in the $D\to K^{0}_{S}f$ and $D\to K^{0}_{L}f$ decays are presented
in Tables 1 and 2 for the $D\to PP$ and $PV$ decay modes, respectively. The
predictions are given in the last columns, compared to the experimental data
PDG . The additional data and results in the global fitting are listed in
Appendix. In order to obtain a reasonable error estimation, we consider the
uncertainties of those universal parameters as well as the decay constants and
form factors involved. The errors of decay constants of $\pi$, $K$, $D$ and
$D_{s}$ are taken from PDG PDG , those of $\eta$ and $\eta^{\prime}$ are from
Feldmann:1998vh , and those of vector mesons are from Straub:2015ica . The
form factors and their errors of $D\to P$ are taken from Koponen:2012di . The
errors of all the other decay constants and form factors are taken as $10\%$
of the center value due to the theoretical uncertainties. It can be found that
our results are well consistent with the data within the uncertainties.
Besides, the predictions on the branching fractions of $D\to K_{L}^{0}f$ are
to be tested by experiments.

From Tables. 1 and 2, the branching fractions of the $D\to K_{S}^{0}f$ modes
are obviously different from those of the $D\to K_{L}^{0}f$ modes, due to the
effect of interference between the CF and DCS amplitudes. For example,
$\mathcal{B}(D^{0}\to K_{S}^{0}f^{0}_{CP})$ are all larger than
$\mathcal{B}(D^{0}\to K_{L}^{0}f^{0}_{CP})$. As shown in Muller:2015lua ,
$\mathcal{B}(D^{0}\to K_{S}^{0}\pi^{0})>\mathcal{B}(D^{0}\to
K_{L}^{0}\pi^{0})$ holds with a significance of more than $4\sigma$. From (5),
$\mathcal{B}(D\to K_{S}^{0}f)+\mathcal{B}(D\to
K_{L}^{0}f)=\mathcal{B}(D\to\overline{K}^{0}f)+\mathcal{B}(D\to
K^{0}f)\approx\mathcal{B}(D\to\overline{K}^{0}f)$ as a good approximation with
neglected branching fractions of DCS processes.

The difference between $\mathcal{B}(D\to K_{S}^{0}f)$ and $\mathcal{B}(D\to
K_{L}^{0}f)$ is induced by the effect of interference between the CF and DCS
amplitudes, defined by the $K^{0}_{S}-K^{0}_{L}$ asymmetries, $R(f)$. With the
fitting results, the $K_{S}^{0}-K_{L}^{0}$ asymmetries in the $D^{+}$ and
$D_{s}^{+}$ decays are predicted to be

$\begin{split}R(D^{+}\to K_{S,L}^{0}\pi^{+})&=0.025\pm 0.008,\\\
R(D^{+}_{s}\to K_{S,L}^{0}K^{+})&=0.012\pm 0.006,\\\ R(D^{+}\to
K_{S,L}^{0}\rho^{+})&=-0.037\pm 0.011,\\\ R(D^{+}_{s}\to
K_{S,L}^{0}K^{*+})&=-0.070\pm 0.032.\end{split}$ (29)

Our result is consistent with the experimental data of $R(D^{+}\to
K_{S,L}^{0}\pi^{+})_{\rm exp}=0.022\pm 0.016\pm 0.018$ lab10 .

For the $D^{0}$ decays, the amplitudes of
$D^{0}\rightarrow\overline{K}^{0}f^{0}_{CP}$ and $D^{0}\rightarrow
K^{0}f^{0}_{CP}$ are the same except for the CKM matrix elements. For example,

$\frac{\mathcal{A}(D^{0}\to
K^{0}\pi^{0})}{\mathcal{A}(D^{0}\to\overline{K}^{0}\pi^{0})}=\frac{V^{*}_{cd}V_{us}}{V^{*}_{cs}V_{ud}}\frac{C_{K^{0}}+E_{K^{0}}}{C_{\overline{K}^{0}}+E_{\overline{K}^{0}}}=-\tan^{2}\theta_{C}.$
(30)

In the FAT approach, as showed in Eq. (20) and (21),
$C_{K^{0}}=C_{\overline{K}^{0}}$, $E_{K^{0}}=E_{\overline{K}^{0}}$, due to
$f_{K^{0}}=f_{\overline{K}^{0}}$. The above ratio is only related to the CKM
matrix elements. This relation also holds for the $D^{0}\to
K^{0}(\overline{K}^{0})\eta^{(\prime)}$ decays. In the case of the $D\to PV$
modes, due to the assumption of $E_{P}=E_{V}$ in the FAT approach as discussed
in Sec. III and shown in (26), the ratios between the DCS and CF amplitudes in
the modes of $D^{0}$ decaying into $\rho^{0}$, $\omega$ and $\phi$ also only
depend on the CKM matrix elements. Then the $K_{S}^{0}-K_{L}^{0}$ asymmetries
in the $D^{0}$ decays are identical to each other, and according to Eq. (17)

$R(D^{0}\to K_{S,L}^{0}f_{CP}^{0})=2\tan^{2}\theta_{C}+y_{D}.$ (31)

With the current world averaging result of the $D^{0}-\overline{D}^{0}$ mixing
parameter $y_{D}=(0.62\pm 0.08)\%$ assuming no $CP$ violation HFAG , we have

$\begin{split}&R(D^{0}\to K^{0}_{S,L}\pi^{0})=R(D^{0}\to
K^{0}_{S,L}\eta)=R(D^{0}\to K^{0}_{S,L}\eta^{\prime})\\\
&~{}~{}~{}~{}=R(D^{0}\to K^{0}_{S,L}\rho^{0})=R(D^{0}\to
K^{0}_{S,L}\omega)=R(D^{0}\to K^{0}_{S,L}\phi)=0.113\pm 0.001,\end{split}$
(32)

with the error from those of the CKM matrix elements and $y_{D}$. Our result
is consistent with experimental result of $R(D^{0}\to
K_{S,L}^{0}\pi^{0})=0.108\pm 0.025\pm 0.024$ lab10  with large errors. Without
the effect of $D^{0}-\overline{D}^{0}$ mixing, $R(D^{0}\to
K^{0}_{S,L}f^{0}_{CP})\approx 0.107$ which is in agreement with predictions in
other methods as seen in Table.3. The improvement on the precision of
measurements is called for to test the neutral $D$ mixing effect in the
$K_{S}^{0}-K_{L}^{0}$ asymmetries. In experiment, at the current stage with
limited data to determine the effect of $D^{0}-\overline{D}^{0}$ mixing, it is
suggested to measure all the above two-body decays of the $D^{0}$ and combine
the results to decrease the errors.

It is found that the amplitudes of $\mathcal{A}(D^{0}\to K^{0}f_{CP}^{0})$ and
$\mathcal{A}(D^{0}\to\overline{K}^{0}f_{CP}^{0})$ are reflected under the
$U$-spin symmetry and the $K^{0}_{S}-K^{0}_{L}$ asymmetries of $D^{0}$ meson
decays are less sensitive to the $SU(3)$ breaking, and thereby $R(D^{0}\to
K^{0}_{S,L}\pi^{0})=R(D^{0}\to K^{0}_{S,L}\eta)=R(D^{0}\to
K^{0}_{S,L}\eta^{\prime})$ Rosner:2006bw . Our results support this conclusion
and extend it to $D^{0}\to PV$ decays. The results on the $PV$ modes depend on
the assumption of $E_{P}=E_{V}$ in the FAT approach, which works well for the
branching fractions at the current stage.

We have listed the results of the diagrammatic approach Bhattacharya:2009ps ;
Cheng:2010ry , the QCD factorization approach lab4 , the diagrammatic approach
with global linear $SU(3)$ breaking analysis Muller:2015lua , the experimental
data lab10  and the FAT approach in Tables 3 for comparison. Our prediction of
$R(D^{0}\to K_{S,L}^{0}\pi^{0})$ is larger than the others by $y_{D}=(0.62\pm
0.08)\%$ due to the $D^{0}-\overline{D}^{0}$ mixing effects involved. The
result of $R(D^{+}\to K_{S,L}^{0}\pi^{+})$ in this work has the same sign with
the experimental data, but opposite to the other theoretical predictions,
because the FAT approach could contain significant flavor $SU(3)$ symmetry
breaking effects compared with Bhattacharya:2009ps ; Cheng:2010ry ; lab4 , and
the latest experimental data of branching fractions have been considered. It
is a similar case for the predictions of $R(D_{s}^{+}\to K_{S,L}^{0}K^{+})$.
In Muller:2015lua , since there are too many parameters to fit limited data,
the uncertainties of predictions on the $K^{0}_{S}-K^{0}_{L}$ asymmetries are
very large.

Table 3: Results on $K_{S}^{0}-K_{L}^{0}$ asymmetries in $D^{0}\to K_{S,L}^{0}\pi^{0}$, $D^{+}\to K_{S,L}^{0}\pi^{+}$ and $D_{s}^{+}\to K_{S,L}^{0}K^{+}$. Our results are compared to other approaches Bhattacharya:2009ps ; Cheng:2010ry ; lab4 ; Muller:2015lua  and the experimental data lab10 .  |  $R$Bhattacharya:2009ps  |  $R$Cheng:2010ry  |  $R$lab4  |  $R$Muller:2015lua  |  $R_{\text{exp}}$lab10  | $R(\text{FAT})$
---|---|---|---|---|---|---
$D^{0}\to K_{S,L}^{0}\pi^{0}$ | $0.107$ | $0.107$ | $0.106$ | $0.09^{+0.04}_{-0.02}$ | $0.108\pm 0.035$ | $0.113\pm 0.001$
$D^{+}\to K_{S,L}^{0}\pi^{+}$ | $-0.005\pm 0.013$ | $-0.019\pm 0.016$ | $-0.010\pm 0.026$ |  | $0.022\pm 0.024$ | $0.025\pm 0.008$
$D_{s}^{+}\to K_{S,L}^{0}K^{+}$ | $-0.002\pm 0.009$ | $-0.008\pm 0.007$ | $-0.008\pm 0.007$ | $0.11^{+0.04}_{-0.14}$ |  | $0.012\pm 0.006$

##  V Conclusions

The effect of interference between the CF and DCS amplitudes results in the
$K^{0}_{S}-K^{0}_{L}$ asymmetries in $D\to K^{0}_{S,L}f$ decays. We present
the formulas of the $K^{0}_{S}-K^{0}_{L}$ asymmetries, $R(f)$, and calculate
them in the FAT approach in which significant nonperturbative effects and the
$SU(3)$ asymmetry breaking effects are involved. The branching fractions of
the decay modes with $K_{L}^{0}$ are predicted. The results of $R(D^{0}\to
K^{0}_{S,L}\pi^{0})$ and $R(D^{+}\to K^{0}_{S,L}\pi^{+})$ are in agreement
with experimental data. We first predict the $K^{0}_{S}-K^{0}_{L}$ asymmetries
in the decay modes with vector mesons in the final states. Furthermore, we
first consider the effect of $D^{0}-\overline{D}^{0}$ mixing in the study of
$K^{0}_{S}-K^{0}_{L}$ asymmetries in neutral $D$-meson decays. It is found
that $R(D^{0}\to K^{0}_{S,L}f^{0}_{CP})=2\tan^{2}\theta_{C}+y_{D}=0.113\pm
0.001$, where $y_{D}$ is the $D^{0}-\overline{D}^{0}$ mixing parameter, with
the value of $(0.62\pm 0.08)\%$ and cannot be neglected. Our predictions will
be tested by the future experiments with higher precision, like BESIII.
Besides, we find all the $K^{0}_{S}-K^{0}_{L}$ asymmetries in the $D^{0}$
decays are identical to each other. Therefore it is suggested to measure all
of them and combine the results to test the effect of $D^{0}-\overline{D}^{0}$
mixing.

###### Acknowledgements.

This work is partially supported by National Natural Science Foundation of
China under Grants No. 11347027 and No. 11505083 and the Fundamental Research
Funds for the Central Universities under Grant No. lzujbky-2015-241.

##  Appendix A Branching fractions in global fitting

We list the experimental data and our predictions of the channels we used to
determine the universal parameters in the FAT approach in Tables 4 and 5. The
global fitting of the $D\to PP$ and $D\to PV$ modes is separate. There are 30
observables to fix 11 free parameters in the $D\to PP$ modes and 37
observables to fix 13 free parameters in the $D\to PV$ modes. The
$\rho^{0}-\omega$ mixing,

$|\rho^{0}\rangle=|\rho^{0}_{I}\rangle-\varepsilon|\omega_{I}\rangle,\qquad|\omega\rangle=\varepsilon|\rho^{0}_{I}\rangle+|\omega_{I}\rangle,$
(33)

is considered in the $D\to PV$ modes to conform with the undated data of
$\mathcal{B}(D^{0}\to\pi^{0}\omega)$ and $\mathcal{B}(D^{+}\to\pi^{+}\omega)$
b , where $|\rho^{0}_{I}\rangle$ and $|\omega_{I}\rangle$ denote the isospin
eigenstates of $\rho^{0}$ and $\omega$ and $\varepsilon$ is chosen to be
$0.12$ as in FAT2 .

Table 4:  The branching fractions we have used in the global fitting of $D\to PP$ modes, compared with our predictions. All data in this table are obtained from PDG PDG . The $D\to K^{0}_{S,L}f$ modes are not included but listed in Table 1.  Modes | $\mathcal{B}_{\text{exp}}$ | $\mathcal{B}_{\text{FAT}}$ | Modes | $\mathcal{B}_{\text{exp}}$ | $\mathcal{B}_{\text{FAT}}$
---|---|---|---|---|---
$D^{0}\to\pi^{+}K^{-}$ |  (3.93$\pm$0.04)$\%$ |  ($3.82\pm 0.96$)% | $D_{s}^{+}\to\pi^{+}\eta$ |  (1.70$\pm$0.09)$\%$ |  ($1.96\pm 0.44$)%
$D_{s}^{+}\to\pi^{+}\eta^{\prime}$ |  (3.94$\pm$0.25)$\%$ |  ($4.67\pm 0.62$)% | $D^{0}\to\pi^{+}\pi^{-}$ |  (1.421$\pm$0.025)‰ |  ($1.418\pm 0.093$)‰
$D^{0}\to K^{+}K^{-}$ |  (4.01$\pm$0.07)‰ |  ($3.92\pm 0.95$)‰ | $D^{0}\to K^{0}_{S}K^{0}_{S}$ |  (0.18$\pm$0.04)‰ |  (0.20$\pm 0.03$)‰
$D^{0}\to\pi^{0}\pi^{0}$ |  (0.826$\pm$0.035)‰ |  ($0.707\pm 0.029$)‰ | $D^{0}\to\pi^{0}\eta$ |  (0.69$\pm$0.07)‰ |  (0.99$\pm 0.08$)‰
$D^{0}\to\pi^{0}\eta^{\prime}$ |  (0.91$\pm$0.14)‰ |  (0.66$\pm 0.04$)‰ | $D^{0}\to\eta\eta$ |  (1.70$\pm$0.20)‰ |  (1.27$\pm 0.25$)‰
$D^{0}\to\eta\eta^{\prime}$ |  (1.07$\pm$0.26)‰ |  (1.43$\pm 0.21$)‰ | $D^{+}\to\pi^{+}\pi^{0}$ |  (1.24$\pm$0.06)‰ |  (1.04$\pm 0.07$)‰
$D^{+}\to K^{0}_{S}K^{+}$ |  (2.95$\pm$0.15)‰ |  (3.06$\pm 1.18$)‰ | $D^{+}\to\pi^{+}\eta$ |  (3.66$\pm$0.22)‰ |  (2.80$\pm 0.42$)‰
$D^{+}\to\pi^{+}\eta^{\prime}$ |  (4.84$\pm$0.31)‰ |  (3.89$\pm 0.22$)‰ | $D_{s}^{+}\to\pi^{0}K^{+}$ |  (0.63$\pm$0.21)‰ |  (0.69$\pm 0.03$)‰
$D_{s}^{+}\to K^{0}_{S}\pi^{+}$ |  (1.22$\pm$0.06)‰ |  (1.04$\pm 0.13$)‰ | $D_{s}^{+}\to K^{+}\eta$ |  (1.77$\pm$0.35)‰ |  (0.91$\pm 0.20$)‰
$D_{s}^{+}\to K^{+}\eta^{\prime}$ |  (1.8$\pm$0.6)‰ |  (3.1$\pm 0.4$)‰ | $D^{0}\to\pi^{-}K^{+}$ |  (1.399$\pm$0.027)‱ |  (1.550$\pm 0.086$)‱
$D^{+}\to\pi^{0}K^{+}$ |  (1.89$\pm$0.25)‱ |  (1.73$\pm 0.13$)‱ | $D^{+}\to K^{+}\eta$ |  (1.12$\pm$0.18)‱ |  (0.67$\pm 0.17$)‱
$D^{+}\to K^{+}\eta^{\prime}$ |  (1.83$\pm$0.23)‱ |  (1.72$\pm 0.19$)‱ |  |  |
Table 5:  Same as Table 4 but for the $D\to PV$ decays, in which $\mathcal{B}(D^{0}\to\pi^{0}\omega)$ and $\mathcal{B}(D^{+}\to\pi^{+}\omega)$ are taken from b , $\mathcal{B}(D^{0}\to\eta\omega)$ from R. Kass , and the others are obtained from PDG PDG .  Modes | $\mathcal{B}_{\text{exp}}$ | $\mathcal{B}_{\text{FAT}}$ | Modes | $\mathcal{B}_{\text{exp}}$ | $\mathcal{B}_{\text{FAT}}$
---|---|---|---|---|---
$D^{0}\to\pi^{+}K^{*-}$ |  ($5.43\pm 0.44$)% |  (5.72$\pm$0.80)% | $D^{0}\to\pi^{0}\overline{K}^{*0}$ |  ($3.75\pm 0.29$)% |  (3.75$\pm$0.27)%
$D^{0}\to K^{-}\rho^{+}$ |  ($11.1\pm 0.9$)% |  (10.6$\pm$0.6)% | $D^{0}\to\eta\overline{K}^{*0}$ |  ($0.96\pm 0.30$)% |  (0.39$\pm$0.13)%
$D^{+}\to\pi^{+}\overline{K}^{*0}$ |  ($1.57\pm 0.13$)% |  (1.71$\pm$0.33)% | $D_{s}^{+}\to\pi^{+}\rho^{0}$ |  ($0.020\pm 0.012$)% |  (0.002$\pm$0.001)%
$D_{s}^{+}\to\pi^{+}\omega$ |  ($0.24\pm 0.06$)% |  (0.17$\pm$0.05)% | $D_{s}^{+}\to\pi^{+}\phi$ |  ($4.5\pm 0.4$)% |  (3.4$\pm$0.7)%
$D_{s}^{+}\to K^{+}\overline{K}^{*0}$ |  ($3.92\pm 0.14$)% |  (4.06$\pm$0.50)% | $D_{s}^{+}\to\eta\rho^{+}$ |  ($8.9\pm 0.8$)% |  (9.1$\pm$1.6)%
$D_{s}^{+}\to\eta^{\prime}\rho^{+}$ |  ($5.8\pm 1.5$)% |  (1.4$\pm$0.4)% | $D^{0}\to\pi^{+}\rho^{-}$ |  ($5.09\pm 0.34$)‰ |  (4.34$\pm 0.59$)‰
$D^{0}\to\pi^{0}\rho^{0}$ |  ($3.82\pm 0.29$)‰ |  (4.06$\pm$0.29)‰ | $D^{0}\to\pi^{0}\omega$ |  ($0.117\pm 0.035$)‰ |  (0.130$\pm$0.031)‰
$D^{0}\to\pi^{0}\phi$ |  ($1.35\pm 0.10$)‰ |  (1.09$\pm$0.08)‰ | $D^{0}\to\pi^{-}\rho^{+}$ |  ($10.0\pm 0.6$)‰ |  (9.4$\pm$0.6)‰
$D^{0}\to K^{+}K^{*-}$ |  ($1.62\pm 0.15$)‰ |  (1.96$\pm$0.31)‰ | $D^{0}\to K^{-}K^{*+}$ |  ($4.50\pm 0.30$)‰ |  (4.76$\pm$0.31)‰
$D^{0}\to\eta\omega$ |  ($2.21\pm 0.23$)‰ |  (1.92$\pm$0.35)‰ | $D^{0}\to\eta\phi$ |  ($0.14\pm 0.05$)‰ |  (0.20$\pm$0.06)‰
$D^{+}\to\pi^{+}\rho^{0}$ |  ($0.84\pm 0.15$)‰ |  (0.54$\pm$0.06)‰ | $D^{+}\to\pi^{+}\omega$ |  ($0.279\pm 0.059$)‰ |  (0.326$\pm$0.108)‰
$D^{+}\to\pi^{+}\phi$ |  ($5.66^{+0.19}_{-0.21}$)‰ |  (5.60$\pm$0.44)‰ | $D^{+}\to K^{+}\overline{K}^{*0}$ |  ($3.84^{+0.14}_{-0.23}$)‰ |  (3.42$\pm$0.68)‰
$D^{+}\to K^{0}_{S}K^{*+}$ |  ($17\pm 8$)‰ |  (5$\pm$1)‰ | $D_{s}^{+}\to\pi^{+}K^{*0}$ |  ($2.13\pm 0.36$)‰ |  (3.04$\pm$0.53)‰
$D_{s}^{+}\to K^{+}\rho^{0}$ |  ($2.5\pm 0.4$)‰ |  (2.1$\pm$0.3)‰ | $D_{s}^{+}\to K^{+}\phi$ |  ($0.164\pm 0.041$)‰ |  (0.142$\pm$0.052)‰
$D^{0}\to\pi^{-}K^{*+}$ |  ($3.45^{+1.80}_{-1.02}$)‱ |  (4.44$\pm$0.31)‱ | $D^{+}\to\pi^{+}K^{*0}$ |  ($3.9\pm 0.6$)‱ |  (3.7$\pm$0.3)‱
$D^{+}\to K^{+}\rho^{0}$ |  ($2.1\pm 0.5$)‱ |  (2.2$\pm$0.4)‱ | $D_{s}^{+}\to K^{+}K^{*0}$ |  ($0.90\pm 0.51$)‱ |  (0.23$\pm$0.03)‱

## References

  * (1) M. Artuso, B. Meadows, and A. A. Petrov, Annu. Rev. Nucl. Part. Sci. 58, 249 (2008).
  * (2) C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40, 100001 (2016).
  * (3) Y. Amhis et al., arXiv:1612.07233 and online update at http://www.slac.stanford.edu/xorg/hfag.
  * (4) A. L. Kagan and M. D. Sokoloff, Phys. Rev. D 80, 076008 (2009).
  * (5) H. -n. Li, C. -D. Lu, and F. -S. Yu, Phys. Rev. D 86, 036012 (2012).
  * (6) Q. Qin, H. n. Li, C. D. Lu, and F. -S. Yu, Phys. Rev. D 89, 054006 (2014).
  * (7) H. Y. Cheng and C. W. Chiang, Phys. Rev. D 81, 074021 (2010).
  * (8) H. Y. Cheng and C. W. Chiang, Phys. Rev. D 86, 014014 (2012).
  * (9) H. Y. Cheng, C. W. Chiang, and A. L. Kuo, Phys. Rev. D 93, 114010 (2016).
  * (10) S. Müller, U. Nierste, and S. Schacht, Phys. Rev. D 92, 014004 (2015).
  * (11) B. Bhattacharya and J. L. Rosner, Phys. Rev. D 77, 114020 (2008).
  * (12) B. Bhattacharya and J. L. Rosner, Phys. Rev. D 79, 034016 (2009); 81, 099903(E) (2010).
  * (13) B. Bhattacharya and J. L. Rosner, Phys. Rev. D 81, 014026 (2010).
  * (14) J. F. Donoghue, E. Golowich, B. R. Holstein, and J. Trampetic, Phys. Rev. D 33, 179 (1986).
  * (15) F. Buccella, M. Lusignoli, and A. Pugliese, Phys. Lett. B 379, 249 (1996).
  * (16) Z. z. Xing, Phys. Rev. D 55, 196 (1997).
  * (17) A. F. Falk, Y. Grossman, Z. Ligeti, and A. A. Petrov, Phys. Rev. D 65, 054034 (2002).
  * (18) H. Y. Cheng and C. W. Chiang, Phys. Rev. D 81, 114020 (2010).
  * (19) J. L. Rosner, Phys. Rev. D 60, 114026 (1999).
  * (20) I. I. Y. Bigi and H. Yamamoto, Phys. Lett. B 349, 363 (1995).
  * (21) Q. He et al. (CLEO Collaboration), Phys. Rev. Lett. 100, 091801 (2008).
  * (22) W. Zheng, in Pos CHARM, 075 (2016).
  * (23) D. N. Gao, Phys. Lett. B 645, 59 (2007).
  * (24) D. N. Gao, Phys. Rev. D 91, 014019 (2015).
  * (25) M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B 591, 313 (2000).
  * (26) Y. -Y. Keum, H. -n. Li, and A. I. Sanda, Phys. Lett. B 504, 6 (2001); Phys. Rev. D 63, 054008 (2001); C. -D. Lu, K. Ukai, and M. -Z. Yang, Phys. Rev. D 63, 074009 (2001); C. -D. Lu and M. -Z. Yang, Eur. Phys. J. C 23, 275 (2002).
  * (27) C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. Lett. 87, 201806 (2001); Phys. Rev. D 65, 054022 (2002).
  * (28) R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 116, 191601 (2016).
  * (29) Z. Z. Xing, Phys. Lett. B 353, 313 (1995).
  * (30) Y. Grossman, A. L. Kagan, and Y. Nir, Phys. Rev. D 75, 036008 (2007).
  * (31) L. -L. Chau, Phys. Rept. 95, 1 (1983).
  * (32) G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
  * (33) F. S. Yu, X. X. Wang, and C. D. Lu, Phys. Rev. D 84, 074019 (2011).
  * (34) H. -n. Li and S. Mishima, Phys. Rev. D 83, 034023 (2011).
  * (35) G. P. Lepage and S. J. Brodsky, Phys. Lett. B 87, 359 (1979).
  * (36) T. Feldmann, P. Kroll, and B. Stech, Phys. Rev. D 58, 114006 (1998).
  * (37) A. Bharucha, D. M. Straub, and R. Zwicky, J. High Energy Phys. 08 (2016), 098.
  * (38) J. Koponen et al. (HPQCD Collaboration), arXiv:1208.6242.
  * (39) J. L. Rosner, Phys. Rev. D 74, 057502 (2006).
  * (40) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 116, 082001 (2016).
  * (41) R. Kass, in Proceedings of Europhysics Conference on High Energy Physics (Krakow, Poland, 2009), http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=84.


# A Semi-Supervised Deep Learning Approach to Dataset Collection for Query-by-
Humming Task

###### Abstract

Query-by-Humming (QbH) is a task that involves finding the most relevant song
based on a hummed or sung fragment. Despite recent successful commercial
solutions, implementing QbH systems remains challenging due to the lack of
high-quality datasets for training machine learning models. In this paper, we
propose a deep learning data collection technique and introduce Covers and
Hummings Aligned Dataset (CHAD), a novel dataset that contains 18 hours of
short music fragments, paired with time-aligned hummed versions. To expand our
dataset, we employ a semi-supervised model training pipeline that leverages
the QbH task as a specialized case of cover song identification (CSI) task.
Starting with a model trained on the initial dataset, we iteratively collect
groups of fragments of cover versions of the same song and retrain the model
on the extended data. Using this pipeline, we collect over 308 hours of
additional music fragments, paired with time-aligned cover versions. The final
model is successfully applied to the QbH task and achieves competitive results
on benchmark datasets. Our study shows that the proposed dataset and training
pipeline can effectively facilitate the implementation of QbH systems.

##  1 Introduction

Query-by-Humming (QbH) is a well-known task in Music Information Retrieval. It
aims to enable users to find a particular song within a retrieval system by
providing a small audio segment of their voice or humming as a query. Such
systems rely on a large database of songs and display the most similar matches
to the user’s query.

One significant benefit of the QbH system compared to other music search
systems [1] is that users do not have to play a copy of the song or recall its
lyrics. Instead, they can hum or sing the melody of the desired song, and the
system will use advanced audio processing and deep learning techniques to
locate it.

A similar task to QbH is Cover Song Identification (CSI) task [2, 3, 4]. CSI
aims to identify cover songs performed by different artists as versions of
original songs within a music database. Although CSI systems often rely on
neural networks, traditional QbH systems mainly utilize audio processing and
music information retrieval techniques like pitch estimation, note extraction,
and time series matching [5, 6, 7]. The main reason QbH lacks deep learning
models is the absence of large datasets for training. This is primarily due to
the high cost and limited availability of humming/singing data for QbH
compared to CSI, where multiple versions of the same song are sufficient for
training. Additionally, QbH requires the alignment of humming/singing
fragments with the original versions of the song.

To overcome the challenges of limited data, we propose a novel dataset CHAD -
Covers and Hummings Aligned Dataset. This dataset contains groups of time-
aligned music fragments, primarily consisting of vocal segments from popular
songs. Time alignment is the process of synchronizing a fragment from a
humming or cover version of a song with its corresponding fragment from the
original version to have the same temporal structure. The groups are separated
into two categories: one with humming fragments collected via crowdsourcing
and another with cover fragments collected using a semi-supervised training
pipeline. We use this dataset to train our deep learning model for matching
audio fragments with similar melodies using metric learning paradigm. We
demonstrate that these techniques can also be successfully applied in the QbH
task, achieving results comparable to the best performing scores on popular
QbH benchmarks. Furthermore, we evaluate our model’s performance on a large
internal song database, showing its ability to generalize to a wider range of
songs.

The paper is structured as follows. Section 2 briefly reviews existing
approaches to the QbH task. Section 3 describes the proposed deep learning
model and training method for the QbH task. Section 4 outlines the dataset and
semi-supervised data collection pipeline. Section 5 describes the experiments
conducted on public and private data. Finally, Section 6 concludes the paper.

##  2 Related Work

QbH systems typically have two components: audio transcription and search
modules. Many approaches in QbH research have focused on designing effective
representations of hummings that can be easily matched with MIDI targets. Some
standard methods include using Hidden Markov Models [5] to transcribe hummings
into a sequence of symbols, discretizing fundamental frequency into semitones
[6], and transcribing hummings into a note-like structure using pitch,
interval, and duration features [7].

Once the humming has been transcribed into a format that can be compared to
database entries, the search module is responsible for finding the most
relevant songs. Dynamic Time Warping [6, 8] has been a popular algorithm for
comparing the humming query to MIDI-audio entries in the database. This
algorithm finds the minimum path between the discretized humming and the MIDI-
audio database. Another approach, top-down melody matching [9], involves
dynamically aligning the humming query with a song from the database. A third
approach, progressive filtering [10], involves multiple stages of song
recognition with increasingly complex recognition mechanisms. These algorithms
serve to match humming queries with songs in a database effectively. In the
approach [11], authors use melody extraction network to extract robust
features from audios and match them with songs from database using an ensemble
of melody matching algorithms.

In contrast to QbH, the latest research on the Cover Song Identification (CSI)
task has been focused on deep learning-based techniques. A popular approach in
CSI is to use deep neural networks for audio representation and metric
learning for similarity search. In [12], the authors use Constant-Q Transform
(CQT) of audio and train a modified version of ResNet with two losses -
triplet loss for intra-class compactness and classification loss for inter-
class discrimination. In the next study, [13], the authors improve results by
integrating the PCA module into the fully-connected layer of ResNet. Metric
learning is widely used in CSI. It is shown that different model and loss
architectures like Siamese Network [3], triplet loss [12], and contrastive
loss [4] can produce competitive results.

In [2], the authors use VGG on CQT features with variable length to tackle the
problem of tempo changes of the cover songs. In [14], the authors use an audio
signal’s Mel-Frequency Cepstrum Coefficients (MFCC) as the representation.
They build cross-similarity matrices between songs and collect the nearest
neighbors of each song based on these matrices.

Several datasets are available for CSI tasks [15, 16, 17, 18]. These datasets
contain audio features alongside music metadata and provide researchers with a
way to evaluate and validate their models without collecting large amounts of
audio data.

##  3 Model

Figure 1: Audio encoder model. Vocal part is extracted from the input
waveform. Then, either $f0$ or $CQT$ features are calculated on the vocal
part. Finally, the features are processed by a convolutional encoder model
and, then, the output embeddings are normalized.

Our encoder model, $M$, is presented in Figure 1 and inspired by [4]. The
whole fingerprints extraction pipeline can be described in the following
steps:

  1. 1.

The first step of the encoding process is to extract the vocal part of the
audio waveform $y$ using a pre-trained audio source separation model $V(.)$.
The model is applied only to cover fragments since humming fragments do not
contain any accompaniments. We used Spleeter [19] as a model due to its high-
speed performance.

  2. 2.

The vocal part of the audio is then sent to the feature extractor model. In
this study, two different extracted feature types are used: the first is the
fundamental frequency ($f0$) extracted using the CREPE model [20], which is
considered a robust representation of the melody. The second is Constant-Q
Transform (CQT) as its faster alternative. The melody is crucial for search as
it contains essential song information while ignoring irrelevant singing
person details.

  3. 3.

The extracted feature matrix is then separated into overlapping segments,
called analysis windows with length $W$ and step $H$, which are fed separately
to a convolutional encoder $F(.)$ ResNet18 [21]. The final layer of the
encoder is a $\mathcal{L}_{2}$-normalization layer $G(.)$, which normalizes
the output of the encoder along the embedding dimension.

  4. 4.

The output fingerprints $Z=\\{z_{i}\\}_{i=1...T}$, where $T$ is the total
number of fingerprints for a waveform and 128 is its dimension size.

We use the metric learning method similar to [4] as a learning framework. To
form a batch of audio fragments for training, we randomly sample $K$ groups of
time-aligned audio fragments. By group, we refer to a collection of original
song and humming/singing fragments. Then, we select $n$ random audio fragments
from each group and extract a random analysis window of size $W$. Since our
data is aligned, all windows from each group will represent the variations of
the same data. Afterward, we apply our model and extract in total $N=K\cdot n$
fingerprints for $n$ in each group.

Our loss is defined as follows:

$\ell=-\sum_{k=1}^{K}\sum_{z_{k}^{i},z_{k}^{j}\in
Z_{k}}\log\frac{\exp(\frac{sim(z_{k}^{i},z_{k}^{j})}{\tau})}{\sum_{z_{l}\not\in
Z_{k}}\exp(\frac{sim(z_{k}^{i},z_{l})}{\tau})},$ (1)

where $Z_{k}=\\{z_{k}^{0},\dots z_{k}^{n-1}\\}$ is the group of fingerprints,
$z_{k}^{i}$ and $z_{k}^{j}$ are different fingerprints from $Z_{k}$,
$z_{l}\not\in Z_{k}$ stands for all fingerprints not in a given group $k$,
$sim(z_{i},z_{j})=z_{i}^{T}z_{j}$ is the similarity function, and $\tau$ is a
temperature parameter. The final loss is computed across all possible positive
pairs and averaged afterward.

##  4 Dataset

Figure 2: Semi-supervised training and data collection pipeline used to train
the initial model, and iteratively gather new aligned audio fragments and
retrain the model.

This section describes the process of collecting a dataset for the QbH task,
its statistics, and its limitations.

###  4.1 Semi-supervised pipeline

Input : $y^{o}$ \- original song;

$Y^{c}$ \- set of cover songs;

$d_{min}$ \- fragment’s minimal length;

$d_{max}$ \- fragment’s maximal length;

$D_{p}$ \- set of pause lengths;

$L_{db}$ \- set of dB levels;

$\alpha_{corr}$ \- threshold value to exclude same fragments;

Output : $\mathbb{F}$ \- set of groups of aligned fragments from original and
cover songs

1 $\mathbb{F}^{o},\mathbb{F}^{o}_{prev}\leftarrow\\{\\},\\{\\}$ ;

2 $M\leftarrow$ initialize_model$(.)$;

3 $rms\leftarrow$ rms$(y^{o})$;

4 foreach  _$d_{p}\in D_{p}$_ do

5 foreach  _$l_{db}\in L_{db}$_ do

6 $m_{silence}\leftarrow$ find_silence_mask$(rms,l_{db})$;

7 $\mathbb{F}^{o}\leftarrow$ split_by_silence$(y^{o},m_{silence})$;

8 $\mathbb{F}^{o}\leftarrow$
merge_fragments$(\mathbb{F}^{o},d_{p},d_{min},d_{max})$;

9 $\mathbb{F}^{o}_{emb}\leftarrow M(\mathbb{F}^{o})$;

10 $\mathcal{A}_{corr}\leftarrow$
build_correlation_matrix$(\mathbb{F}^{o}_{emb})$;

11 $\mathbb{F}^{o}\leftarrow$
find_unique_fragments$(\mathbb{F}^{o},\mathcal{A}_{corr},\alpha_{corr})$;

12 $\mathbb{F}^{o},\mathbb{F}^{o}_{prev}\leftarrow
max(\mathbb{F}^{o},\mathbb{F}^{o}_{prev})$;

13

14 end foreach

15

16 end foreach

17$\mathbb{F}\leftarrow\\{\\}$;

18 foreach  _$f^{o}\in\mathbb{F}^{o}$_ do

19 foreach  _$y^{c}\in\mathbb{Y}^{c}$_ do

20 $f^{o}_{emb}\leftarrow M(f^{o})$;

21 $y^{c}_{emb}\leftarrow M(y^{c})$;

22 $\mathbb{F}^{c}\leftarrow$
cross_correlation($y^{c},y^{c}_{emb},f^{o}_{emb}$);

23 $\mathbb{F}^{c}\leftarrow$
filter($\mathbb{F}^{c},\beta_{rel},\beta_{irrel}$);

24 $\mathbb{F}\leftarrow\mathbb{F}\cup(f^{o},\mathbb{F}^{c})$

25 end foreach

26

27 end foreach

28

Algorithm 1 Aligned fragments extraction algorithm of data collection
pipeline.

Figure 2 presents our proposed semi-supervised pipeline. The dataset used in
this study is structurally divided into two parts: $\mathcal{H}$ and
$\mathcal{C}$. The first part, $\mathcal{H}$, consists of original music
fragments $f^{o}$ paired with time-aligned humming/singing fragments $f^{h}$,
making groups $F^{h}$. The $f^{o}$ fragments are represented by various vocal
and instrumental parts of music clips. The $f^{h}$ fragments were collected
using a crowdsourcing service Yandex.Toloka. The second part of the dataset,
$\mathcal{C}$, was created by collecting the 100 most popular songs from the
Billboard Charts for each year from 1960 to 2020. For each song, up to 10
cover versions were retrieved from YouTube top results using query "{song
name} {artist name} cover".

Because $\mathcal{H}$ already has groups of time-aligned fragments, we can
train the initial encoder model $M_{0}$ with this data. However, $\mathcal{C}$
only contains groups of full song versions instead of time-aligned fragments,
so extracting fragments from these groups is necessary. We propose Algorithm 1
for this task. This algorithm is designed to extract the maximum amount of
unique fragments from the original versions of the songs and find the
corresponding aligned fragments from cover versions of the songs in
$\mathcal{C}$. The algorithm can be described in three stages:

Initialization stage

  1. 1.

As input, the algorithm takes the vocal part of the original song $y^{o}$ and
a group of cover songs $Y^{c}$. Additionally, the algorithm takes the minimal
and maximal length of the fragment $d_{min}$ and $d_{max}$, respectively, the
set of dB levels $L_{db}$ by which to count the region in song as silent or
non-silent, the set of maximal pause lengths $D_{p}$ between adjacent
fragments in a song separated by silence to be considered as one fragment, and
threshold values $\alpha_{corr}$, $\beta_{rel}$, and $\beta_{irrel}$ to
exclude unwanted fragments from the output set.

  2. 2.

Initialize empty sets of unique fragments $\mathbb{F}^{o}$ and
$\mathbb{F}^{o}_{prev}$, the encoder model $M$, and $rms$ of the waveform
$y^{o}$ (lines 1-3).

Fragmentation stage

  1. 1.

To find the best combination of $D_{p}$ and $L_{db}$ to yield $\mathbb{F}^{o}$
of maximal size, start two loops by iterating over these sets (lines 1-5).

  2. 2.

Compute the binary mask of non-silent regions $m_{silence}$ using $rms$ and
$l_{db}\in L_{db}$ and find a set of initial fragments by splitting the
waveform $y^{o}$ using this mask. Then, merge adjacent fragments, the pause
between which is less than $d_{p}\in D_{p}$. Additionally, the length of such
fragments should satisfy the condition
$d_{min}<|f|<d_{max},f\in\mathbb{F}^{o}$. (lines 1-8).

  3. 3.

Apply model $M_{i}$ to the found fragments and extract the fingerprints. Then,
build the correlation matrix $\mathcal{A}_{corr}$ based on the fragments’
fingerprints $\mathbb{F}^{o}_{emb}$ and exclude the ones with a correlation
higher than the threshold $\alpha_{corr}$. We used the maximum of cross-
correlation function to measure the correlation of fingerprints with different
lengths (lines 1-11).

  4. 4.

Find the parameters of dB levels and pause lengths that yield the maximum
amount of unique fragments (line 1).

Matching stage

  1. 1.

Once the unique fragments from the original version of the song
$\mathbb{F}^{o}$ are extracted, initialize the empty set $\mathbb{F}$ to be
filled with groups of the time-aligned fragments from original and cover songs
and iterate over each found original fragment $f^{o}\in{\mathbb{F}^{o}}$ and
each cover song $y^{c}\in Y_{c}$ (lines 1-17).

  2. 2.

Extract fingerprints from original fragment $f^{o}_{emb}$ and cover song
$y^{c}_{emb}$ using $M$. Search for the same fragments in the cover song using
a cross-correlation function and peak detection algorithm (lines 1-19).

  3. 3.

Filter out noise cover fragments by establishing two thresholds:

    1. (a)

The cover fragments with correlation above $\beta_{rel}$ are considered
relevant, indicating a high level of certainty that the content of the cover
fragment is similar to that of the original fragment.

    2. (b)

The cover fragments with correlation below $\beta_{irrel}$ are considered
irrelevant fragments and are excluded. Fragments with a correlation between
these two thresholds are counted as uncertain and require double-checking via
additional crowdsourcing.

Save the gathered groups of aligned fragments (lines 1-22).

We apply this pipeline to song batches of $\mathcal{C}$, which generates new
groups of aligned data. These groups are then added to $\mathcal{H}$, and the
model, $M$, is retrained on the newly gathered data. In such a way, we first
train $M_{0}$ on initial humming data, then iteratively update our model from
$M_{i}\rightarrow M_{i+1}$ and fill our dataset with new data.

For the unique fragments extraction algorithm, we set $d_{min}=8$,
$d_{max}=20$, $D_{p}=\\{0.5,1,1.5\\}$ seconds, $L_{db}=\\{52,56,60,64,68\\}$
dB, $\alpha_{corr}=0.8$. When searching for fragments in cover versions of
songs, we set the optimal thresholds to $\beta_{rel}=0.5$ and
$\beta_{irrel}=0.3$. All fragments from the same group have equal duration to
retain the time-alignment consistency.

###  4.2 Statistics

We call the collected dataset Covers and Hummings Aligned Dataset (CHAD). Here
are the dataset’s statistics:

  * •

CHAD contains $5494$ original songs, $31630$ cover songs, and $5164$ hummings
fragments.

  * •

The total number of audio fragments is $81781$, which amounts to over $270$
hours of singing/humming audio fragments and $51$ hours of original song
fragments. The group size varies from 2 to 31, with an average size of 6
fragments.

  * •

In $\mathcal{H}$, the duration of the fragments ranges from $4$ to $20$
seconds, with a mean of $11.06\pm 2.67$ seconds, and a total for original
fragments - 2.12 hours, and for humming fragments - 15.83 hours.

  * •

In $\mathcal{C}$, the duration ranges from $8$ to $20$ seconds, with a mean of
$14.66\pm 2.03$ seconds, and a total for original fragments - 49.54 hours, and
cover fragments - 259.03 hours, where 194.53 hours are for fragments with
correlation above $\beta_{rel}$, and 64.50 hours are for fragments with
correlation between $\beta_{rel}$ and $\beta_{irrel}$.

  * •

Additionally, the metadata is collected. It includes YouTube video ID, title,
author, correlation value, and whether the fragment is double-checked. The
dataset’s audio IDs, metadata, start and end timestamps and data download
script are available in our GitHub
repository111https://github.com/amanteur/CHAD.

###  4.3 Limitations

However, our semi-supervised pipeline has some limitations. First, it can only
extract vocal data, and the algorithm needs modification to extract
instrumental segments. Second, the number of covers is limited, as there are
usually fewer cover versions for non-popular songs. Lastly, there will still
be some noisy unrelated fragments in the final set due to the automatic
validation threshold. Future research could explore using generative networks
[22] to overcome these limitations.

##  5 Experiments

| Method | Top-$10$ hit rate $\uparrow$
---|---|---
Jang[23] | Thinkit | Subtask 2 | Jang Real | MTG-QBH [24]
Ours | metric learning(CREPE) | 0.921 | 0.966 | 0.959 | 0.868 | 0.883
metric learning($CQT$) | 0.840 | 0.786 | 0.866 | 0.867 | 0.747
Stasiak [8] |  $f0$-matching | 0.948 | 0.907 | 0.968 | - | -
ACRCloud | proprietary | 0.990 | 0.986 | 0.972 | - | -
Table 1: Evaluation of model $M_{short}$ trained on dataset
$\mathcal{C}+\mathcal{H}$ with two types of features - CREPE and $CQT$.
Evaluation is provided on MIREX datasets - Jang, Thinkit, and Subtask 2, and
datasets - Jang Real, and MTG-QBH, which are more applicable to real-world
scenarios.

###  5.1 Experimental setup

Input features. We used CREPE [20] activations as $f0$ features, yielding
output features with a size of $(360,T)$. We further enhanced the robustness
of the melody feature by trimming it to include only $3$ octaves around its
mean pitch, following the approach used in [25]. Additionally, we downscaled
this representation to the size $(80,\frac{T}{4})$. However, we encountered
issues with the slow speed of the melody extraction model during evaluation,
rendering the overall approach unscalable. To address this, we incorporated
$CQT$ features into our model, extracted with the following parameters: $12$
bins per octave with a total of $7$ octaves, Hann window, hop length $512$,
and a sampling rate of $16$ kHz.

Augmentations. We found an optimal set of augmentations to every batch of
waveform fragments, which included continuous pitch shifting (with a shift
range of -4.0 to 4.0 semitones and probability of 0.5), time stretching (with
a stretch rate range of 0.8 to 1.25 and probability of 0.8), SpliceOut [26]
(with 10 random intervals of 500 frames and probability of 0.8), mixing with
other audio samples in the batch (with an SNR range of 5 to 10 dB and
probability of 0.8), and adding background noises (with an SNR range of 3 to
30 dB and probability of 0.8).

Model. We discovered that as the length of a hummed or sung recording
increases, the tempo/rhythm becomes more mismatched from the original song. So
we trained two models with different analysis window lengths ($W$) and hop
sizes ($S$): $M_{short}$ for shorter recordings (up to $15$ sec) with $W$=$3$
sec and $S$=$0.25$ sec, and $M_{long}$ for longer recordings with $W$=$8$ sec
and $S$=$0.64$ sec. Both models used a vanilla ResNet18 encoder model, with
output embeddings of size $(128,T)$, where $T$ is the number of fingerprints.

Training setup. We trained the encoder model using the ADAM optimizer, with a
learning rate of $lr=0.001$ and a batch size of $32$ for $100$ epochs. We used
the NT-Xent Loss [27] with a temperature of $t=0.05$. We employed the Multi-
similarity miner [28] and an adaptive batch sampler to improve convergence
speed. The batch sampler selects up to $4$ fragments with a random starting
point for each fragments group. We trained the model under two settings: only
on the $\mathcal{C}$ part and on both $\mathcal{C}+\mathcal{H}$ parts of CHAD.
Models were trained on 1 NVIDIA GeForce RTX 2080 Ti 12 Gb.

To evaluate the performance of our model, we conducted a series of
experiments, which involved:

  * •

Experiments on the MIREX QbH datasets [29], specifically the Roger Jang and
ThinkIt datasets, where MIDI recordings were used as references. The MIR-QBSH
corpus of Roger Jang consists of 4431 query hummings and 48 original MIDI
files, while the Thinkit corpus contains 355 queries and 106 original MIDI
files. The song database was constructed according to MIREX QbH Challenge
standards, with 2600 MIDI files. These experiments aimed to find the ground-
truth MIDI by a given query humming.

  * •

Experiments on MIREX QbH datasets according to Subtask 2 testing protocol in
MIREX evaluation system. In this protocol, queries are also considered as
"versions" of ground truth, and the objective is to retrieve all variants
related to a searched ground truth by given query humming.

  * •

Experiments on a dataset of real recordings, which included MIREX Roger Jang
Dataset with all MIDI files replaced with real recordings extracted from
YouTube videos (Jang Real), and MTG-QbH [24] dataset with $118$ queries and
$118$ original songs. The additional database comprises 1886 random songs from
the internal dataset to serve as imposter songs.

  * •

Experiments on a large-scale internal database (DB90K) containing more than
90k real song recordings. For this experiment, we used two types of queries:
$126$ humming fragments for $126$ songs collected by our team as search-by-
humming setup and $2000$ singing fragments from karaoke recordings gathered
from the DAMP-VPB dataset [30] as search-by-singing setup. In the latter case,
we selected $5$ of $16$ original songs and their sung performances and
manually split them into fragments.

For all real recordings, we extract the vocal part beforehand. Also, we ensure
that CHAD does not contain any songs that are also present in the evaluation
datasets. This was achieved by excluding such songs from the training set.

Retrieval. We use two variants of sequence matching methods at the retrieval
phase: maximum Pearson correlation coefficient (Corr) or Dynamic Time Warping
(DTW) [31]. For the large-scale experiment on DB90K, we use a two-step search
procedure with a first step of fast retrieval of preliminary candidates using
the FAISS Approximate Nearest Neighbors (ANN) algorithm with Euclidean
distance followed by a second step of reranking. After further analysis, we
discovered that Euclidean distance and Cosine distance yielded similar
results. To maintain simplicity, we chose to use Euclidean distance. The ANN
search returns the top $5000$ candidates, which are reranked based on the
Pearson correlation score.

Metrics. We follow the MIREX evaluation protocols [29] for the QbH task and
compute the mean of the Top-$n$ hit rate for every humming/singing fragment.
There was only one related song in the database for every query fragment.

###  5.2 Results

We compare our model $M_{short}$ trained on $\mathcal{C}+\mathcal{H}$ with 2
best performing methods according to the latest available result of MIREX QbH
Challenge [32]. The first one [8] is based on $f0$-matching technique. The
second one is a proprietary method for which only scores were reported in the
leaderboard. We use only one of our models ($M_{short}$ on CREPE and CQT
features) in this experiment as most query fragments are shorter than 16
seconds. We use DTW for matching feature sequences on MIDI-based datasets and
Corr for non-MIDI datasets as we found that correlation coefficient gives
performance improvement on real data.

The results are summarized in Table 1. Our model demonstrates competitive
though slightly inferior performance on the given benchmarks. On real (non-
MIDI) data our implementation of [8] produced near-random results which can be
explained by the difficulty of tracking and matching $f0$ in real music
recordings. Also, we see that while using $CQT$ features led to a performance
drop, it is not prohibitively large so $CQT$ features can be used when
computing $f0$ is infeasible.

Table 2 shows the scalability of our approach in the experiment with DB90k. We
do not track the top-1 hit rate as the database contains several versions of
the same song. Table 2(a) reports the results of the search-by-humming setup.
We used $M_{short}$, $M_{long}$, and their combination model $M_{fused}$,
which worked on a simple rule: $M_{short}$ was used for hummings shorter than
15 seconds, while $M_{long}$ was used otherwise. All presented models are
trained with $CQT$ features. We observed that $M_{fused}$ worked better than
$M_{short}$ and $M_{long}$ separately in all scenarios. Comparing models
trained on $\mathcal{C}$ and $\mathcal{C}+\mathcal{H}$, the accuracy gap
suggests that training on real humming data is crucial for search-by-humming
setup. In Table 2(b), we report our results for search-by-singing setup with
$M_{fused}$ on DAMP-VPB. Our model, trained on both $\mathcal{C}$ and
$\mathcal{C}+\mathcal{H}$, retrieves the correct songs with high precision,
with no performance drop observed for the model trained on $\mathcal{C}$
alone, due to the dominance of sung fragments in our training dataset.

Additionally, we evaluate the retrieval speed of our models $M_{short}$ and
$M_{long}$ on DB90k, as shown in Table 3. We find that $M_{long}$ performs
better than $M_{short}$ in both search steps (ANN and Reranking) due to its
ability to process longer humming recordings and thus require less processing
of fragments. Our results demonstrate the scalability and efficiency of our
search system in efficiently achieving high-precision results.

Partition | Model | Top-$n$ hit rate$\uparrow$
---|---|---
$100$ | $10$ | $5$ | $3$
$\mathcal{C}$ | $M_{short}$ | 0.643 | 0.548 | 0.524 | 0.476
$M_{long}$ | 0.412 | 0.277 | 0.270 | 0.262
$M_{fused}$ | 0.759 | 0.621 | 0.603 | 0.517
$\mathcal{C}+\mathcal{H}$ | $M_{short}$ | 0.659 | 0.595 | 0.571 | 0.484
$M_{long}$ | 0.595 | 0.508 | 0.413 | 0.389
$M_{fused}$ | 0.776 | 0.707 | 0.691 | 0.586
(a) Results on humming queries.

Partition | Model | Top-$n$ hit rate$\uparrow$
---|---|---
$100$ | $10$ | $5$ | $3$
$\mathcal{C}$ | $M_{fused}$ | 0.931 | 0.904 | 0.885 | 0.865
$\mathcal{C}+\mathcal{H}$ | 0.923 | 0.899 | 0.885 | 0.856
(b) Results on singing queries.

Table 2: Evaluation on DB90K with humming and singing fragments using models $M_{short}$, $M_{long}$, and their fusion model $M_{fused}$ trained on $\mathcal{C}$ and $\mathcal{C}+\mathcal{H}$ with $CQT$ features. Model | Search step, s
---|---
ANN | Reranking
$M_{short}$ | 1.41 $\pm$ 0.57 | 5.37 $\pm$ 0.87
$M_{long}$ | 0.52 $\pm$ 0.11 | 2.39 $\pm$ 0.43
Table 3: Query search speed on DB90K using models $M_{short}$ and $M_{long}$
trained on $\mathcal{C}+\mathcal{H}$ with $CQT$ features using 32 CPU.

##  6 Conclusions

In this paper, we propose a novel dataset CHAD alongside a semi-supervised
data collection and training pipeline for a Query-by-Humming system. We show
that cover songs could be used to train query-by-humming models with
competitive performance. Although the model trained on open data performs well
on sung queries, the pure search-by-humming setup requires adding a portion of
real humming data into the training set for acceptable performance. The main
disadvantage of the proposed approach is that it cannot be used for searching
instrumental tracks. One possible solution to this problem would lie in the
field of dominant melody extraction and generative networks and is left for
future research.

## References

  * [1] A. Wang, “An industrial strength audio search algorithm,” in _ISMIR_ , 2003.
  * [2] Z. Yu, X. Xu, X. Chen, and D. Yang, “Learning a representation for cover song identification using convolutional neural network,” _CoRR_ , vol. abs/1911.00334, 2019. [Online]. Available: http://arxiv.org/abs/1911.00334
  * [3] G. Doras and G. Peeters, “A prototypical triplet loss for cover detection,” _CoRR_ , vol. abs/1910.09862, 2019. [Online]. Available: http://arxiv.org/abs/1910.09862
  * [4] S. Chang, D. Lee, J. Park, H. Lim, K. Lee, K. Ko, and Y. Han, “Neural audio fingerprint for high-specific audio retrieval based on contrastive learning,” _CoRR_ , vol. abs/2010.11910, 2020. [Online]. Available: https://arxiv.org/abs/2010.11910
  * [5] E. Unal, E. Chew, P. G. Georgiou, and S. S. Narayanan, “Challenging uncertainty in query by humming systems: A fingerprinting approach,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 16, no. 2, pp. 359–371, 2008.
  * [6] R. A. Putri and D. P. Lestari, “Music information retrieval using query-by-humming based on the dynamic time warping,” in _2015 International Conference on Electrical Engineering and Informatics (ICEEI)_ , 2015, pp. 65–70.
  * [7] L. Lu, H. You, and H.-J. Zhang, “A new approach to query by humming in music retrieval,” in _IEEE International Conference on Multimedia and Expo, 2001. ICME 2001._ , 2001, pp. 595–598.
  * [8] B. Stasiak, “Follow that tune – adaptive approach to dtw-based query-by-humming system,” _Archives of Acoustics_ , vol. 39, pp. 467 –, 01 2014.
  * [9] X. Wu, M. Li, J. Liu, J. Yang, and Y. Yan, “A top-down approach to melody match in pitch contour for query by humming,” 2006.
  * [10] J.-S. R. Jang and H.-R. Lee, “A general framework of progressive filtering and its application to query by singing/humming,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 16, no. 2, pp. 350–358, 2008.
  * [11] M. Ulfi and R. Mandala, “Improving query by humming system using frequency-temporal attention network and partial query matching,” in _2022 9th International Conference on Advanced Informatics: Concepts, Theory and Applications (ICAICTA)_ , 2022, pp. 1–6.
  * [12] X. Du, Z. Yu, B. Zhu, X. Chen, and Z. Ma, “Bytecover: Cover song identification via multi-loss training,” _CoRR_ , vol. abs/2010.14022, 2020. [Online]. Available: https://arxiv.org/abs/2010.14022
  * [13] X. Du, K. Chen, Z. Wang, B. Zhu, and Z. Ma, “Bytecover2: Towards dimensionality reduction of latent embedding for efficient cover song identification,” in _ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2022, pp. 616–620.
  * [14] C. J. Tralie and P. Bendich, “Cover song identification with timbral shape sequences,” _CoRR_ , vol. abs/1507.05143, 2015. [Online]. Available: http://arxiv.org/abs/1507.05143
  * [15] D. P. Ellis and G. E. Poliner, “Identifying ‘cover songs’ with chroma features and dynamic programming beat tracking,” in _2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP ’07_ , vol. 4, 2007, pp. IV–1429–IV–1432.
  * [16] T. Bertin-Mahieux and D. Ellis, “Large-scale cover song recognition using hashed chroma landmarks,” in _Proceedings of IEEE WASPAA_. New Platz, NY: IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), 2011.
  * [17] Y. Bayle, L. Maršík, M. Rusek, M. Robine, P. Hanna, K. Slaninová, J. Martinovic, and J. Pokorný, “Kara1k: A karaoke dataset for cover song identification and singing voice analysis,” in _2017 IEEE International Symposium on Multimedia (ISM)_ , 2017, pp. 177–184.
  * [18] F. Yesiler, C. J. Tralie, A. A. Correya, D. F. Silva, P. Tovstogan, E. Gómez, and X. Serra, “Da-tacos: A dataset for cover song identification and understanding,” in _ISMIR_ , 2019.
  * [19] R. Hennequin, A. Khlif, F. Voituret, and M. Moussallam, “Spleeter: a fast and efficient music source separation tool with pre-trained models,” _Journal of Open Source Software_ , vol. 5, no. 50, p. 2154, 2020, deezer Research. [Online]. Available: https://doi.org/10.21105/joss.02154
  * [20] J. W. Kim, J. Salamon, P. Li, and J. P. Bello, “Crepe: A convolutional representation for pitch estimation,” 2018. [Online]. Available: https://arxiv.org/abs/1802.06182
  * [21] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” _CoRR_ , vol. abs/1512.03385, 2015. [Online]. Available: http://arxiv.org/abs/1512.03385
  * [22] C. Frank, “The machine learning behind hum to search,” https://ai.googleblog.com/2020/11/the-machine-learning-behind-hum-to.html, accessed: 2022-10-23.
  * [23] J.-S. R. Jang, “Qbsh: A corpus for designing qbsh (query by singing/humming) systems.” [Online]. Available: http://www.cs.nthu.edu.tw/~jang
  * [24] J. Salamon, J. Serrà, and E. Gómez, “Tonal representations for music retrieval: From version identification to query-by-humming,” _International Journal of Multimedia Information Retrieval, special issue on Hybrid Music Information Retrieval_ , vol. 2, pp. 45–58, 03 2013.
  * [25] G. Doras and G. Peeters, “Cover detection using dominant melody embeddings,” _CoRR_ , vol. abs/1907.01824, 2019. [Online]. Available: http://arxiv.org/abs/1907.01824
  * [26] A. Jain, P. R. Samala, D. Mittal, P. Jyothi, and M. Singh, “Spliceout: A simple and efficient audio augmentation method,” _CoRR_ , vol. abs/2110.00046, 2021. [Online]. Available: https://arxiv.org/abs/2110.00046
  * [27] K. Sohn, “Improved deep metric learning with multi-class n-pair loss objective,” in _Advances in Neural Information Processing Systems_ , D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, Eds., vol. 29. Curran Associates, Inc., 2016. [Online]. Available: https://proceedings.neurips.cc/paper/2016/file/6b180037abbebea991d8b1232f8a8ca9-Paper.pdf
  * [28] X. Wang, X. Han, W. Huang, D. Dong, and M. R. Scott, “Multi-similarity loss with general pair weighting for deep metric learning,” 2019.
  * [29] “MIREX QBSH challenge results,” https://www.music-ir.org/mirex/wiki/2021:Query_by_Singing/Humming, accessed: 2022-10-23.
  * [30] I. Smule, “DAMP-VPB: Digital Archive of Mobile Performances - Smule Vocal Performances Balanced,” Nov. 2017. [Online]. Available: https://doi.org/10.5281/zenodo.2616690
  * [31] M. Müller, “Dynamic time warping,” _Information Retrieval for Music and Motion_ , vol. 2, pp. 69–84, 01 2007.
  * [32] “MIREX QBSH 2016 results,” https://www.music-ir.org/mirex/wiki/2016:MIREX2016_Results, accessed: 2022-10-23.