# Understanding the non-trivial isoscalar pseudoscalar structures in the
$K_{S}K_{S}\pi^{0}$ spectra in the $J/\psi$ radiative decay

Yin Cheng1,2,3111E-mail address<EMAIL_ADDRESS>Lin Qiu1,2222E-mail
address<EMAIL_ADDRESS>and Qiang Zhao1,2333E-mail address:
<EMAIL_ADDRESS>1 Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China  2 University of Chinese
Academy of Sciences, Beijing 100049, China  3 Institute of Theoretical
Physics,
Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

Initiated by the recent observation of a flattened lineshape of
$IJ^{PC}=00^{-+}$ around $1.4\sim 1.5$ GeV in the $K_{S}K_{S}\pi^{0}$
invariant mass spectrum by BESIII, we make a systematic partial wave analysis
of $J/\psi\to\gamma\eta_{X}\to\gamma K\bar{K}\pi$ based on an isobaric
approach. We demonstrate that in the scenario of the first radial excitations
of the isoscalar pseudoscalar from the $K\bar{K}\pi$ threshold to about 1.6
GeV the non-trivial $K_{S}K_{S}\pi^{0}$ invariant mass spectrum can be
explained by the coupled-channel effects with the presence of the triangle
singularity mechanism. It shows that a combined fit of the three-body and two-
body spectra can be achieved which suggests that the one-state solution around
$1.4\sim 1.5$ GeV proposed before still holds well. In particular, we show
that the coupled-channel effects between the two most important quasi-two-body
decay channels, $K^{*}\bar{K}+c.c.$ and $a_{0}(980)\pi$, can be well described
by taking into account the one-loop corrections in the isobaric approach. This
is because the isoscalar pseudoscalar states are coupled to the
$K^{*}\bar{K}+c.c.$ and $a_{0}(980)\pi$ channels via the $P$ and $S$ waves,
respectively. As a consequence, the coupled-channel effects can be largely
absorbed into the redefinition of the tree-level effective couplings with the
transition amplitudes computed to the order of one-loop corrections. Then, the
coupled-channel effects can be estimated by the contributions from the one-
loop rescattering amplitudes in comparison with the tree-level ones, where we
find that the rescattering contributions from the $P$-wave into the $S$-wave,
or vice verse, are apparently suppressed in the kinematic region near
threshold.

##  I Introduction

The controversial results concerning the nature of the isoscalar pseudoscalars
$\eta(1405/1475)$ have existed for a long time. The key question is whether
they are indeed two separate states or just one state in this mass region, and
the final answer will have a strong impact on our understanding of the low-
energy QCD properties. Because of this, tremendous efforts were made in both
experiment and theory aiming at disentangling those mysterious issues related
to the isoscalar pseudoscalar states around $1.2\sim 1.5$ GeV. Historically,
the first evidence for $\eta(1405)/\eta(1475)$ was from $p\bar{p}$
annihilations at rest into $(K\bar{K}\pi)\pi^{+}\pi^{-}$ Baillon et al.
(1967). In this experiment only one resonance enhancement was seen in the
$0^{-+}$ channel, which was called E-meson at that time. Later, with higher
statistics from MARK III Bai et al. (1990) and DM-2 Augustin et al. (1992), it
showed that there was a strong asymmetric signal in the $K\bar{K}\pi$
invariant mass spectrum, which seemed to be an overlap of two Breit-Wigner
(BW) resonances. The E852 Collaboration Adams et al. (2001) and the Obelix
Collaboration at LEAR Nichitiu et al. (2002) also confirmed the two-resonances
results, and since then a two-state solution, i.e. $\eta(1405)$ and
$\eta(1475)$, was introduced in order to account for the non-trivial lineshape
in the $K\bar{K}\pi$ invariant mass spectrum. However, it should be noted that
the masses and widths of these two states are very different in these analyses
Bai et al. (1990); Augustin et al. (1992); Adams et al. (2001); Nichitiu et
al. (2002).

For the first radial excitation of $J^{PC}=0^{-+}$, there are enough states to
fill the SU(3) nonet, i.e. $\pi(1300)$, $K(1460)$, $\eta(1295)$ and
$\eta(1405)/\eta(1475)$ Workman et al. (2022). The claim that $\eta(1405)$ and
$\eta(1475)$ are two different states, indicates a supernumerary of this SU(3)
nonet and the existence of possible exotic hadrons beyond the conventional
quark model. In the early studies, the calculations in the flux tube model
suggest that the mass of the ground-state pseudoscalar glueball should be
around $1.4$ GeV Faddeev et al. (2004). Hence, one of these two close states
could be the candidate for the pseudoscalar glueball. Such a possibility has
stirred up a lot of efforts on understanding their structures Donoghue et al.
(1981); Close and Monaghan (1981); Close (1988); Amsler and Tornqvist (2004);
Barnes et al. (1982); Close et al. (1997); Masoni et al. (2006). However, the
glueball assignment with the mass around $1.4$ GeV is not favored by the later
lattice QCD (LQCD) simulations. Both quenched Chen et al. (2006); Bali et al.
(1993); Morningstar and Peardon (1999); Chowdhury et al. (2015) and unquenched
calculations Richards et al. (2010); Sun et al. (2018) show that the ground-
state pseudoscalar glueball should have a mass around $2.4\sim 2.6$ GeV.
Furthermore, considering the mixing between the $q\bar{q}$ states and glueball
with the axial vector anomaly dynamics, it was shown that the physical mass of
the pseudoscalar glueball would not be lighter than $1.8$ GeV Qin et al.
(2018); Mathieu and Vento (2010); Gabadadze (1998). These analyses suggest
that the “extra pseudoscalar state” does not seem to be a plausible candidate
for the ground-state $0^{-+}$ glueball. Meanwhile, one has to understand the
abnormal structure observed in $K\bar{K}\pi$ and different peak positions
extracted in different decay channels.

A breakthrough of this puzzling question on the $\eta(1405)$ and $\eta(1475)$
was triggered by the observation of the abnormally large isospin-breaking
effects in $J/\psi\to\gamma+3\pi$ at BESIII Ablikim et al. (2012). In this
decay a narrow peak with a width of about 8 MeV, indicating the $K^{+}K^{-}$
and $K^{0}\bar{K}^{0}$ threshold effects, was observed in the invariant mass
spectrum of $\pi^{+}\pi^{-}$ and $\pi^{0}\pi^{0}$ in the decay of a single
state $\eta(1405)/\eta(1475)\to 3\pi$. Recognizing that a triangle singularity
(TS) has played a crucial role of enhancing the isospin-violating decays for
$\eta(1405)/\eta(1475)\to 3\pi$, it was then realized that the TS mechanism
should also play an important role in the decays of $\eta(1405)/\eta(1475)$ to
$K\bar{K}\pi$ and $\eta\pi\pi$ of which the interference effects can lead to
changes to the lineshapes Wu et al. (2012). It was then demonstrated
systematically that a single state around 1.4 GeV is sufficient to account for
all the available experimental data in various channels Wu et al. (2013); Du
and Zhao (2019); Cheng and Zhao (2022); Aceti et al. (2012); Achasov et al.
(2015). Such a state can be naturally accommodated by the first radial
excitation nonet and hence no additional state is needed.

It is necessary to briefly review the experimental measurements of the
isoscalar pseudoscalar states in the first radial excitation energy region in
order to pin down key issues concerning our understanding of the isoscalar
pseudoscalar spectra in the mass region of $1.2\sim 1.5$ GeV. The first fact
to note is that, so far, only one Breit-Wigner structure was observed in the
invariant mass spectra of $\eta\pi\pi$ Augustin et al. (1990); Bolton et al.
(1992); Bai et al. (1999); Anisovich et al. (2001); Acciarri et al. (2001);
Amsler et al. (2004); Ablikim et al. (2011), $3\pi$ Ablikim et al. (2012), and
$\gamma V(V=\rho,\phi)$ Coffman et al. (1990); Amsler et al. (2004); Bai et
al. (2004); Ablikim et al. (2018), although one also notices that the peak
positions of the single enhancement are slightly different in different decay
channels. In the $\eta\pi\pi$ and $3\pi$ spectra, the BW masses are located
around $1.4$ GeV, while in the $\gamma\rho$ spectrum Coffman et al. (1990);
Bai et al. (2004), the peak is located around $1.42\sim 1.44$ GeV. In Ref.
Ablikim et al. (2018), the BW mass of $\eta(1405/1475)$ in the $\gamma\phi$
spectrum is located at around $1.475$ GeV, and in the latest analysis of the
same decay channel but with higher statistics Ablikim et al. (2024), the mass
is found to be $1422\pm 2.1^{+5.9}_{-7.8}$ MeV. This could be an indication
that there must be controversial issues among these existing data. Comparing
the analyses between Refs. Ablikim et al. (2018) and Ablikim et al. (2024), it
implies that the isoscalar pseudoscalar state favors a lower mass around
$1.42$ GeV instead of $1.475$ GeV. Secondly, the different mass positions
around $1.4\sim 1.5$ GeV observed in different decay channels do not
necessarily lead to a solution of two states. As demonstrated in Refs. Wu et
al. (2012, 2013); Du and Zhao (2019); Cheng and Zhao (2022) the TS mechanism
can provide interferences to shift mass positions and distort the lineshapes.
Furthermore, since different decay channels may involve different sources of
background, their interferences may also shift the peak positions in the
invariant mass spectra. Actually, these single peaks observed in different
channels, if were from different states, would raise a critical question, “why
these two states with the same quantum numbers and similar masses cannot
couple to these channels simultaneously?” While these crucial issues have been
addressed systematically from different aspects by treating $\eta(1405)$ and
$\eta(1475)$ as the same state in the pseudoscalar nonet of the first radial
excitations Wu et al. (2013); Qin et al. (2018); Du and Zhao (2019); Cheng and
Zhao (2022), we still look for new evidences for such a scenario.

Recently, BESIII Collaboration published their high-statistics analysis of
$J/\psi\to\gamma K_{S}K_{S}\pi^{0}$ Ablikim et al. (2023). A non-trivial
lineshape of the $0^{-+}$ partial wave is found in the $K_{S}K_{S}\pi^{0}$
spectrum around $1.25-1.6$ GeV. Unlike the previous low-statistic data from
MARK III Bai et al. (1990) and DM-2 Augustin et al. (1992) that the
$K\bar{K}\pi$ lineshape showed an asymmetric resonant structure, a relatively
flat lineshape in the range of $1.4\sim 1.5$ GeV is observed. This astonishing
result again raises questions on the nature of $\eta(1405)$ and $\eta(1475)$,
which has also motivated our combined analysis of $J/\psi\to\gamma
K\bar{K}\pi$ and $\gamma\eta\pi\pi$ in this work. On the one hand, it is a
crucial case for examining whether the anomalous structure arises from any
unusual mechanism beyond the first radial excitation multiplets. On the other
hand, if there is no additional mechanisms, e.g. glueball state, to play a
role, then how such an abnormal lineshape is produced. We also note that the
accumulated experimental data also call for a unified treatment of the
coupled-channel effects with three-body unitarity. A first try of this
direction can be found in Refs. Nakamura et al. (2023, 2024). However, it
should be noted that since the mass of $J/\psi$ is much larger than the mass
thresholds of $\eta\pi\pi$, $K\bar{K}\pi$, and the two-body thresholds
$a_{0}(980)\pi$ and $K\bar{K}^{*}+c.c.$, a coupled-channel approach with
three-body unitarity requires a reliable treatment of the two-body scattering
amplitudes into a relatively large momentum transfer region.

Concerning the two-body coupled-channel in $J/\psi\to\gamma K\bar{K}\pi$ and
$\gamma\eta\pi\pi$, one notices that the most important two-body channels are
$a_{0}(980)\pi$ and $K\bar{K}^{*}+c.c.$ They involve the two-body $K\bar{K}$
interactions in the $S$ wave and $K\pi$ in both $S$ and $P$ wave. Also, one
notices that the dynamically generated pole in the $S$-wave $K\pi$ scattering
is a broad structure deeply into the complex energy plane, while the $P$-wave
pole structure, i.e. $K^{*}$ is relatively a narrow peak. It suggests that
these two partial waves can be well separated in the two-body channel. For the
coupled-channel between $a_{0}(980)\pi$ and $K\bar{K}^{*}+c.c.$ channels,
since the $K\bar{K}\pi$ are all in the relative $S$ wave in the
$a_{0}(980)\pi$ channel, while they all in a relative $P$ wave in the
$K\bar{K}^{*}$ channel, the effects from the three-body unitarity appear to be
limited. We will come back to this point later with phenomenological analysis.
A detailed study with the full three-body unitarity will be reported in an
independent work.

Before proceeding to the detailed analysis, some main features of the BESIII
measurement of $J/\psi\to\gamma K_{S}K_{S}\pi^{0}$ should be noted. In Ref.
Ablikim et al. (2023) the partial wave lineshapes of the
$(KK)_{\text{S-wave}}\pi$ and $(K\pi)_{\text{P-wave}}\bar{K}$ modes for
$J^{PC}=0^{-+}$ are presented where two resonant structures can be identified
with an apparent mass shift of about $100$ MeV. This seems to be a strong
evidence for two isoscalar pseudoscalars which can respectively decay into
$(KK)_{\text{S-wave}}\pi$ and $(K\pi)_{\text{P-wave}}\bar{K}$. It is claimed
that two pseudoscalars, i.e. $\eta(1405)$ and $\eta(1475)$, are present, and
they can both decay into $K\bar{K}\pi$ by the $(K\bar{K})_{\text{S-wave}}\pi$
and $(K\pi)_{\text{P-wave}}\bar{K}$ modes. In the analysis of Ref. Ablikim et
al. (2023), $(K\bar{K})_{\text{S-wave}}\pi$ represents the spectrum from the
combined transitions via $a_{0}(980)\pi$ and $(K\bar{K})_{\text{S-phsp}}\pi$.
Note that “phsp” stands for the phase space which comes from the direct decay
process without an intermediate resonance. Notation
$(K\pi)_{\text{P-wave}}\bar{K}$ represents the spectrum from the combined
transitions via $K\bar{K}^{*}+c.c.$ and $(K\pi)_{\text{P-phsp}}\bar{K}$. Note
that the $(KK)_{\text{S-wave}}\pi$ and $(K\pi)_{\text{P-wave}}\bar{K}$ modes
involves different partial waves from the intermediate isobars. Nevertheless,
the mass of the initial pseudoscalar state is not far away from the thresholds
of the intermediate isobars, i.e. $a_{0}(980)\pi$ and $K\bar{K}^{*}+c.c.$ A
well-known phenomenon is that the open $S$ and $P$-wave can evolve differently
in terms of the energy in the near-threshold regime. When they contribute to
the same final state, the combined spectrum may deviate from a single-particle
BW distribution. On top of these effects the TS, as a special final-state
interaction (FSI) mechanism, also plays a crucial role of producing unique
features that can be probed by experimental measurements.

In fact, the difficulty for any two-state solution is that it cannot explain
why only one BW structure is seen in the $\eta\pi\pi$ channel. If there were
indeed two individual states, i.e. $\eta(1405)$ and $\eta(1475)$, and they
respectively couple to the $a_{0}(980)\pi$ and $K\bar{K}^{*}+c.c.$ channels,
the presence of the TS mechanism will dictate that these two states will both
contribute to the final $\eta\pi\pi$. Their interference will produce
structures deviated from the BW structure similar to that seen in the
$K\bar{K}\pi$ channel. Nevertheless, the radiative decays into $\gamma V$ via
the strange state of $s\bar{s}$ will imply that
$BR(\eta(1475)\to\gamma\phi)>BR(\eta(1475)\to\gamma\rho^{0})$ and that of
$\gamma\omega$. However, the experimental measurement shows that the
$\gamma\rho^{0}$ channel has the largest branching ratio. This cannot be
explained if there exist two independent states around 1.4 GeV and one of
these contains a sizeable glueball component.

In order to further disentangle the $\eta(1405/1475)$ puzzle, in this paper we
try to describe the non-trivial $K\bar{K}\pi$ spectra, including the two-body
$K\pi$ and $K\bar{K}$ spectra measured by BESIII Ablikim et al. (2023), with
the one-state scenario of $\eta(1405/1475)$. Namely, we assume that there is
only one isoscalar pseudoscalar state around $1.4\sim 1.5$ GeV. We will show
that the multiple intermediate channels with different thresholds, e.g.
$K^{*}\bar{K}$, $\kappa\bar{K}$ and $a_{0}\pi$, can produce the non-trivial
resonant lineshape. In the one-state scheme, we use $\eta(1405)$ to denote all
signals related to either $\eta(1405)$ or $\eta(1475)$, and treat $\eta(1295)$
and $\eta(1405)$ as the first radial excitation states of $\eta$ and
$\eta^{\prime}$. In Sec. II, we first introduce the theoretical model which
describes the $J/\psi\to\gamma K\bar{K}\pi$ via the isoscalar pseudoscalars in
the range of $1.25\sim 1.6$ GeV. In Sec. III, we will present our numerical
results with discussions. A brief summary will be given in Sec. VI.

##  II Formalism

###  II.1 An isobaric analysis scheme

In the one-state scenario, $\eta(1295)$ and $\eta(1405)$ are naturally
assigned to the first radial excitations of $\eta-\eta^{\prime}$, and their
mixing can be described as follows Wu et al. (2012, 2013); Du and Zhao (2019);
Cheng and Zhao (2022):

$\displaystyle\eta(1295)$ $\displaystyle=$
$\displaystyle\cos\alpha_{P}n\bar{n}-\sin\alpha_{P}s\bar{s},$ (1)
$\displaystyle\eta(1405)$ $\displaystyle=$
$\displaystyle\sin\alpha_{P}n\bar{n}+\cos\alpha_{P}s\bar{s},$ (2)

where $n\bar{n}\equiv(u\bar{u}+d\bar{d})/\sqrt{2}$, and
$\alpha_{P}\equiv\theta_{p}+\arctan\sqrt{2}$ with $\theta_{p}$ the SU(3)
flavor singlet and octet mixing angle. Such a mixing scheme is similar to that
of the $\eta$ and $\eta^{\prime}$. This is a natural expectation since the
radial excitation only changes the spatial part of the $\eta$ and
$\eta^{\prime}$ wavefunctions in the constituent quark scenario. Although the
transitions via the U(1)A anomaly coupling may change when the $0^{-+}$
$q\bar{q}$ has different spatial configurations, the overall effects should
not change the mixing pattern for the first radial excitation multiplets.

For simplicity, we label $\eta_{X}=(\eta(1295)$, $\eta(1405))$,
$\eta_{L}=\eta(1295)$, and $\eta_{H}=\eta(1405)$, respectively, in the rest of
this work. We define $\tilde{g}_{\eta_{L}}$ and $\tilde{g}_{\eta_{H}}$ as the
couplings of $J/\psi$ radiative decays to $\eta(1295)$ and $\eta(1405)$,
respectively. Hence, as discussed in our previous work Cheng and Zhao (2022),
$\tilde{g}_{\eta_{L}}$ and $\tilde{g}_{\eta_{H}}$ have the relation of:

$\displaystyle\frac{\tilde{g}_{\eta_{H}}}{\tilde{g}_{\eta_{L}}}=\frac{\sqrt{2}\sin\alpha_{P}+R\cos\alpha_{P}}{\sqrt{2}\cos\alpha_{P}-R\sin\alpha_{P}},$
(3)

where $R$ is an SU(3) flavor symmetry breaking factor which is estimated to be
$f_{\pi}/f_{K}\simeq 0.7\sim 0.9$, and $R=0.8$ is adopted in this work. The
mixing angle $\alpha_{P}$ is within the range of $38^{\circ}\sim 44^{\circ}$,
which covers the range of the mixing angle for $\eta$ and $\eta^{\prime}$.
Note that in the study of the line shape of $K\bar{K}\pi$/$\eta\pi\pi$ spectra
it is the ratio of these two couplings that matters. With $R$ and $\alpha_{P}$
the ratio of $\tilde{g}_{L}$ to $\tilde{g}_{H}$, as well as the ratio between
$g_{\eta_{L}K^{*}\bar{K}}$ and $g_{\eta_{H}K^{*}\bar{K}}$ can be determined
Cheng and Zhao (2022).

There are three two-body decay channels with different open thresholds that
can contribute to the $K\bar{K}\pi$ final states: $K^{*}\bar{K}+c.c.$,
$\kappa\bar{K}+c.c.$ and $a_{0}(980)\pi$. The schematic diagrams for the
decays of $\eta_{X}\to K\bar{K}\pi$ and $\eta\pi\pi$ in our model are shown in
Fig. 1 and 2, respectively. Note that the $\eta_{X}$ couplings to the
intermediate $VP$ and $PS$ are via either $P$ or $S$-wave transitions. Their
coupled-channel effects can be estimated by the rescattering amplitudes of
$\eta_{X}\to VP\to PS$ and $\eta_{X}\to PS\to VP$. Namely, by comparing the
one-loop amplitudes (including the TS mechanism) with the tree-level
amplitudes, one can learn approximately the order of magnitude of the couple-
channel effects. If the one-loop corrections are sizeable and important, it
means that a full coupled-channel calculation with three-body unitarity is
required. Otherwise, a reasonably well description of the partial-wave data
can be expected with the one-loop corrections considered. It implies a
relatively small contributions from the three-body unitarization. We would
like to emphasize that the fitting of the line shapes of the two-body partial-
wave spectra is a crucial check of the self-consistency of our treatment 444We
shall also report the comparison between this phenomenological analysis and
the full coupled-channel calculations with three-body unitarity in a
forthcoming work. Our preliminary results show that these two analyses are in
good agreement with each other..

Figure 1: Tree and Loop level diagrams of $\eta_{X}$ decays to $K\bar{K}\pi$.

In Ref. Du et al. (2022), the leading order vertex corrections from the TS
mechanism to the bare coupling between $\eta_{X}$ and $K^{*}\bar{K}$ are
calculated and the corrections turn out to be small. We also calculate the
triangle-loop diagrams of $a_{0}\pi$ and $\kappa\bar{K}+c.c.$ channels
scattering to $K^{*}\bar{K}+c.c.$, and find that they are of ${\cal
O}(10^{-3})$ compared to the tree-level contributions of these two channels.
Due to the small vertex corrections to the effective couplings of $\eta_{X}\to
K^{*}\bar{K}+c.c.$, we can reasonably define the effective couplings
$g_{\eta_{X}K^{*}\bar{K}}$ as the input parameter for the loop amplitudes.
Their values should not deviate much from the bare couplings. Thus, the SU(3)
flavor symmetry relation will approximately hold. Based on these
considerations, we only consider the $K^{*}\bar{K}+c.c.$ loop contributions at
the one-loop level in the $K\bar{K}\pi$ channel (see Fig. 1). The intermediate
$a_{0}\pi/\kappa\bar{K}+c.c.$ contributions are introduced through the tree-
level transitions, where $a_{0}(980)$ and $\kappa$ are dynamically generated
states due to the $K\bar{K}$ and $K\pi$ scatterings, respectively. For
simplicity, we adopt the parametrized BW forms for the propagators of
$a_{0}(980)$ and $\kappa$ in the calculations. As mentioned earlier, our
purpose here is to examine whether the isobaric analysis can describe the data
or not. By adopting the parametrized forms for the intermediate $a_{0}(980)$
and $\kappa$ propagators the results should also expose dynamical aspects
which are crucial for the $K\bar{K}\pi$ channel.

###  II.2 Effective Lagrangians and transition amplitudes

We use effective Lagrangians to describe all the strong vertices appearing in
the decay transitions. In principle, the hadronic couplings can be arranged by
the SU(3) symmetry and then their relative strengths and phases are fixed,
especially for those couplings between genuine $q\bar{q}$ states, such as the
couplings between $\eta_{X}$ and $K^{*}\bar{K}$ pair:
$g_{\eta_{L}K^{*}\bar{K}}$ and $g_{\eta_{H}K^{*}\bar{K}}$. There are two types
of hadronic couplings involved here, i.e., $VPP$ and $SPP$, for which the
corresponding effective Lagrangians are as follows:

$\displaystyle\mathcal{L}_{VPP}$ $\displaystyle=$ $\displaystyle
ig_{VPP}\text{Tr}[(P\partial_{\mu}P-\partial_{\mu}PP)V^{\mu}],$ (4)
$\displaystyle\mathcal{L}_{SPP}$ $\displaystyle=$ $\displaystyle
g_{SPP}\text{Tr}[SPP],$ (5)

where $S$, $P$ and $V$ stand for the scalar, pseudoscalar and vector fields,
respectively, of the SU(3) flavor multiplet, and they have the following
forms:

$S=\begin{pmatrix}\frac{\sigma+a_{0}(980)}{\sqrt{2}}&a_{0}^{+}&\kappa^{+}\\\
a_{0}^{-}&\frac{\sigma-a_{0}(980)}{\sqrt{2}}&\kappa^{0}\\\
\kappa^{-}&\bar{\kappa}^{0}&f_{0}(980)\end{pmatrix},$ (6)

$P=\left(\begin{array}[]{ccc}\frac{\sin\alpha_{P}\eta^{\prime}+\cos\alpha_{P}\eta+\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\sin\alpha_{P}\eta^{\prime}+\cos\alpha_{P}\eta-\pi^{0}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K^{0}}&\cos\alpha_{P}\eta^{\prime}-\sin\alpha_{P}\eta\\\
\end{array}\right),$ (7)

and

$V=\left(\begin{array}[]{ccc}\frac{\omega+\rho^{0}}{\sqrt{2}}&\rho^{+}&{K^{*}}^{+}\\\
\rho^{-}&\frac{\omega-\rho^{0}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\\\ \end{array}\right).$ (8)

For the $\eta_{X}$ coupling to $SP$, the Lagrangian has the same form as Eq.
(5). For convenience, we write it as

$\displaystyle\mathcal{L}_{XSP}$ $\displaystyle=$ $\displaystyle
g_{XSP}\text{Tr}[XSP]\ .$ (9)

Assuming that the first radial excitation nonet has the same form as Eq. (7),
the strong couplings of $\eta_{X}$ can in general be expressed as an overall
coupling constant multiplied by a factor contributed by the mixing, i.e.

$\displaystyle\mathcal{L}_{\eta(1405)K^{*0}\bar{K}^{0}}$ $\displaystyle=$
$\displaystyle
ig_{\eta(1405)K^{*0}\bar{K}^{0}}(\bar{K}^{0}\partial_{\mu}\eta(1405)-\eta(1405)\partial_{\mu}\bar{K}^{0})(K^{*0})^{\mu}$
(10) $\displaystyle\equiv$ $\displaystyle
ig_{XVP}(\frac{\sin\alpha_{P}}{\sqrt{2}}R-\cos\alpha_{P})(\bar{K}^{0}\partial_{\mu}\eta(1405)-\eta(1405)\partial_{\mu}\bar{K}^{0})(K^{*0})^{\mu}\
,$

and

$\displaystyle\mathcal{L}_{\eta(1295)K^{*0}\bar{K}^{0}}$ $\displaystyle=$
$\displaystyle
ig_{\eta(1295)K^{*0}\bar{K}^{0}}(\bar{K}^{0}\partial_{\mu}\eta(1295)-\eta(1295)\partial_{\mu}\bar{K}^{0})(K^{*0})^{\mu}$
(11) $\displaystyle\equiv$ $\displaystyle
ig_{XVP}(\frac{\cos\alpha_{P}}{\sqrt{2}}R+\sin\alpha_{P})(\bar{K}^{0}\partial_{\mu}\eta(1295)-\eta(1295)\partial_{\mu}\bar{K}^{0})(K^{*0})^{\mu}\
,$

where $g_{XVP}$ is the overall coupling between a radial excitation
pseudoscalar $(q\bar{q})_{0^{-+}}$ and $VP$. By expanding Eq. (4) we obtain
the specific expression of the effective Lagrangian for the $K^{*}K\pi$
vertex:

$\displaystyle\mathcal{L}_{K^{*0}K^{0}\pi^{0}}=i\frac{g_{VPP}}{\sqrt{2}}(\pi^{0}\partial_{\mu}K^{0}-K^{0}\partial_{\mu}\pi^{0})(K^{*0})^{\mu},$
(12)

where the coupling $g_{VPP}=4.52$ can be determined by the experimental data
for $K^{*}\to K\pi$ Workman et al. (2022).

The processes of $J/\psi\to\gamma\eta_{X}\to\gamma
K\bar{K}\pi/\gamma\eta\pi\pi$ between $1.25\sim 1.6$ GeV energy region are
mediated by $\eta(1295)$ and $\eta(1405)$ dominantly. Although the production
of $\eta(1295)$ in $J/\psi$ radiative decay is highly suppressed, its
interference may not be small and its inclusion is necessary for understanding
the non-trivial $K\bar{K}\pi$ spectrum. The invariant mass spectrum of the
three-body final states $abc$ will have the following form:

$\displaystyle\frac{d\Gamma_{J/\psi\to\gamma
abc}}{d\sqrt{s}}=\frac{2s\Gamma_{J/\psi\to\gamma\eta_{X}}(s)}{2m_{J/\psi}}\int
d\Phi_{abc}\bigg{|}\frac{\tilde{g}_{\eta_{L}}\mathcal{M}_{\eta_{L}\to
abc}(s)}{s-m^{2}_{\eta_{L}}+i\sqrt{s}\Gamma_{\eta_{L}}(s)}+\frac{\tilde{g}_{\eta_{H}}\mathcal{M}_{\eta_{H}\to
abc}(s)}{s-m^{2}_{\eta_{H}}+i\sqrt{s}\Gamma_{\eta_{H}}(s)}+\mathcal{M}_{NR}\bigg{|}^{2},$
(13)

where

$\displaystyle\Gamma_{J/\psi\to\gamma\eta_{X}}=\frac{(m_{J/\psi}^{2}-s)^{3}}{48\pi^{2}m_{J/\psi}^{2}}.$
(14)

In the above expressions the masses of $J/\psi$, $\eta(1295)$, and
$\eta(1405)$ are denoted by $m_{J/\psi}$, $m_{\eta_{L}}$ and $m_{\eta_{H}}$,
respectively; $s$ is the energy square of the $abc$ three-body system, which
corresponds to $K\bar{K}\pi$ or $\eta\pi\pi$ in this study; $\Phi_{abc}$ is
the phase space for $\eta_{X}\to abc$; $\mathcal{M}_{\eta_{X}\to abc}(s)$ is
the amplitude of $\eta_{X}\to abc$ with an energy dependence of $s$. We
explicitly include a non-resonant contribution for $J/\psi$ directly decaying
into $\gamma K\bar{K}\pi$ via the amplitude $\mathcal{M}_{NR}$, and a constant
coupling $g_{J/\psi\gamma K\bar{K}\pi}$ is introduced in order to account for
the smooth background contribution.

For an accurate description of the propagators of $\eta(1295)$ and
$\eta(1405)$, we use the energy dependent widths $\Gamma_{\eta_{X}}(s)$.
Because the dominant decay channels of $\eta_{X}$ are the strong decays to
$K\bar{K}\pi$ and $\eta\pi\pi$, the energy dependent widths of $\eta_{X}$ can
be defined as:

$\displaystyle\Gamma_{\eta_{X}}(s)=\Gamma_{\eta_{X}\to
K\bar{K}\pi}(s)+\Gamma_{\eta_{X}\to\eta\pi\pi}(s).$ (15)

For the $\eta\pi\pi$ channel, we argue that the most dominant process is via
the $a_{0}\pi$ channel. Hence, up to one-loop corrections, the diagrams of
$\eta_{X}\to\eta\pi\pi$ are shown in Fig. 2.

Figure 2: Tree and loop level diagrams of $\eta_{X}\to a_{0}\pi\to\eta\pi\pi$.

As follows, we will compute the transition amplitude,
$\mathcal{M}_{\eta_{X}\to abc}(s)$, for $\eta_{X}$ decays into $K\bar{K}\pi$
and $\eta\pi\pi$ based on the mechanisms illustrated in Figs. 1 and 2.

We first focus on the $J/\psi\to\gamma K^{0}_{S}K^{0}_{S}\pi^{0}$ channel
which is recently measured by BESIII Ablikim et al. (2023). In the
experimental analysis the total $K\bar{K}\pi$ spectrum, partial wave spectra
of $\eta_{X}\to(K^{0}_{S}K^{0}_{S})_{\text{S-wave}}\pi^{0}$ and
$\eta_{X}\to(K^{0}_{S}\pi^{0})_{\text{S-wave}}K^{0}_{S}$ are presented. When
describing the spectra of $K^{0}_{S}\bar{K}^{0}_{S}\pi^{0}$,
$(K^{0}_{S}\bar{K}^{0}_{S})_{\text{S-wave}}\pi^{0}$ and
$(K^{0}_{S}\pi^{0})_{\text{P-wave}}{K}^{0}_{S}$, the amplitudes in Eq. (13)
are $\mathcal{M}_{\eta_{X}\to K\bar{K}\pi}$,
$\mathcal{M}_{\eta_{X}\to\pi(K\bar{K})_{S}}$ and $\mathcal{M}_{\eta_{X}\to
K(K\pi)_{P}}$, respectively, and their detailed expressions will be given
later.

Table 1: The notations of the relevant couplings. $\tilde{G}_{\eta_{X}K^{*0}\bar{K}^{0}}$ | $\tilde{G}_{\eta_{X}\kappa^{0}\bar{K}^{0}}$ | $\tilde{G}_{\eta_{X}a_{0}\pi}$ | $\tilde{G}_{K^{*0}K^{0}\pi^{0}}$ | $\tilde{G}_{\kappa^{0}K^{0}\pi^{0}}$ | $\tilde{G}_{a_{0}K^{0}\bar{K}^{0}}$
---|---|---|---|---|---
$g_{\eta_{X}K^{*0}\bar{K}^{0}}e^{i\theta_{\eta_{X}K^{*}\bar{K}^{0}}}$ | $g_{\eta_{X}\kappa^{0}\bar{K}^{0}}e^{i\theta_{\eta_{X}\kappa^{0}\bar{K}^{0}}}$ | $g_{\eta_{X}a_{0}\pi^{0}}e^{i\theta_{\eta_{X}a_{0}\pi^{0}}}$ | $g_{K^{*0}K^{0}\pi^{0}}e^{i\theta_{K^{*0}K^{0}\pi^{0}}}$ | $g_{\kappa^{0}K^{0}\pi^{0}}e^{i\theta_{\kappa^{0}K^{0}\pi^{0}}}$ | $g_{a_{0}K^{0}\bar{K}^{0}}e^{i\theta_{a_{0}K^{0}\bar{K}^{0}}}$

Now we come to the formulae of amplitudes and the energy-dependent widths of
$\eta_{X}\to abc$ in detail. To denote the four-vector momenta of the
particles, we adopt the following notations for the kinematic variables:
$p_{X}(\eta_{X})$, $p_{a}(K)$, $p_{b}(\pi)$, $p_{c}(\bar{K})$ denote the four-
vector momenta in the $K\bar{K}\pi$ channel, and $p_{a}(\pi^{0}/\pi^{+})$,
$p_{b}(\eta)$, $p_{c}(\pi^{0}/\pi^{-})$ for those in the $\eta\pi\pi$ channel.

In order to describe the off-shell effects of the couplings of $\eta_{X}$ to a
meson pair, we introduce form factors for the $\eta_{X}$ coupling vertices,
i.e. $\eta_{X}K^{*}\bar{K}$, $\eta_{X}\kappa\bar{K}$ and $\eta_{X}a_{0}\pi$:

$\displaystyle{\cal F}(s,\lambda)$ $\displaystyle=$
$\displaystyle\exp\biggl{(}\frac{-(s-m^{2}_{\eta_{X}})^{2}}{\lambda^{4}}\biggr{)},$
(16)

where $\lambda$ is a cut-off parameter to be determined by the experimental
data. Consider that the couplings $\eta_{X}K^{*}\bar{K}$,
$\eta_{X}\kappa\bar{K}$ and $\eta_{X}a_{0}\pi$ involve different partial waves
and the properties of these final state particles are very different. We
release $\lambda$ to be different in different channels. Namely,
$\lambda_{K^{*}}$, $\lambda_{\kappa}$ and $\lambda_{a_{0}}$, are three more
parameters in addition to Eq. (38) in $\eta_{X}\to K^{*}\bar{K}$,
$\kappa\bar{K}$ and $a_{0}\pi$, respectively. It should be noted that the
introduction of these form factor parameters is based on phenomenological
considerations instead of rigorous proof. Our strategy is to take into account
the non-trivial properties of $a_{0}(980)$ and $\kappa$ which cannot be
accommodated by the conventional $q\bar{q}$ structure in the quark model.

For the $K\bar{K}\pi$ decay channel, the amplitude for Fig. 1(a) for the
$K^{0}\bar{K}^{0}\pi$ channel can be expressed as

$\displaystyle i\mathcal{M}_{K^{*}\bar{K}}(s)=$ $\displaystyle-i$
$\displaystyle\biggl{[}\tilde{G}_{\eta_{X}K^{*0}\bar{K}}\tilde{G}_{K^{*0}K^{0}\pi}\frac{(2p_{X}-p_{ab})_{\mu}(g^{\mu\nu}-\frac{p_{ab}^{\mu}p_{ab}^{\nu}}{p_{ab}^{2}})(p_{ab}-2p_{b})_{\nu}}{p_{ab}^{2}-m_{K^{*}}^{2}+im_{K^{*}}\Gamma_{K^{*}}}$
$\displaystyle+$
$\displaystyle\tilde{G}_{\eta_{X}\bar{K}^{*0}K}\tilde{G}_{\bar{K}^{*0}\bar{K}^{0}\pi}\frac{(2p_{X}-p_{bc})_{\mu}(g^{\mu\nu}-\frac{p_{bc}^{\mu}p_{bc}^{\nu}}{p_{bc}^{2}})(p_{bc}-2p_{b})_{\nu}}{p_{bc}^{2}-m_{K^{*}}^{2}+im_{K^{*}}\Gamma_{K^{*}}}\biggr{]}\exp\biggl{(}\frac{-(s-m^{2}_{\eta_{X}})^{2}}{\lambda_{K^{*}}^{4}}\biggr{)},$

with $p_{ab}\equiv(p_{a}+p_{b})$ and $p_{bc}\equiv(p_{b}+p_{c})$; We also
define the following quantities:

$\displaystyle p_{ac}=p_{a}+p_{c},\qquad s_{ab}=p_{ab}^{2},\qquad
s_{bc}=p_{bc}^{2},\qquad s_{ac}=p_{ac}^{2},$

and will continue to use this notation in the subsequent calculations. We note
in advance that $\tilde{G}$’s in Eq. (II.2) and in amplitudes given later
denote the coupling constants with a relative phase angle, i.e.
$\tilde{G}\equiv ge^{i\theta}$. We will discuss this in the next Subsection.

The amplitudes of Fig. 1(b) and (c) for $\eta_{X}\to K^{0}\bar{K}^{0}\pi$ can
be expressed, respectively, as

$\displaystyle
i\mathcal{M}_{\kappa\bar{K}}(s)=-\biggl{[}\tilde{G}_{\eta_{X}\kappa\bar{K}}\tilde{G}_{\kappa
K\pi}G_{\kappa}(s_{ab})+\tilde{G}_{\eta_{X}\bar{\kappa}K}\tilde{G}_{\bar{\kappa}\bar{K}\pi}G_{\kappa}(s_{bc})\biggr{]}\exp\biggl{(}\frac{-(s-m^{2}_{\eta_{X}})^{2}}{\lambda_{\kappa}^{4}}\biggr{)}\
,$ (18)

and

$\displaystyle
i\mathcal{M}_{a_{0}\pi}(s)=-\tilde{G}_{\eta_{X}a_{0}\pi}\tilde{G}_{a_{0}K^{0}\bar{K}^{0}}G_{a_{0}}(s_{ac})\exp\biggl{(}\frac{-(s-m^{2}_{\eta_{X}})^{2}}{\lambda_{a_{0}}^{4}}\biggr{)}\
,$ (19)

where $G_{\kappa}(s_{bc})$ and $G_{a_{0}}(s_{ac})$ are the propagators of
$\kappa$ and $a_{0}$, respectively.

For the broad scalar $\kappa$, we use the commonly adopted BW parametrization
form in order to reduce the complexities in the loop calculations:

$\displaystyle
G_{\kappa}(s)=\frac{i}{s-m^{2}_{\kappa}+i\sqrt{s}\Gamma_{\kappa}},$ (20)

where the PDG values for the resonant parameters are adopted, i.e.
$m_{\kappa}=0.85$ GeV, $\Gamma_{\kappa}=0.47$ GeV Workman et al. (2022). For
the propagator $G_{a_{0}}$ we adopt the same formulae as in the Ref. Du and
Zhao (2019).

Similarly, the tree-level amplitude for the $\eta\pi^{0}\pi^{0}$ decay channel
shown in Fig. 2(a) can be expressed as the following form:

$\displaystyle
i\mathcal{M}^{\prime}_{a_{0}\pi}=-\frac{1}{\sqrt{2}}\tilde{G}_{\eta_{X}a_{0}\pi}g_{a_{0}\eta\pi}\bigg{[}G_{a_{0}}(s_{ab})+G_{a_{0}}(s_{bc})\bigg{]}\exp\biggl{(}\frac{-(s-m^{2}_{\eta_{X}})^{2}}{\lambda_{a_{0}}^{4}}\biggr{)}.$
(21)

For the loop transitions illustrated in Figs. 1 and 2, the kinematic variables
of the loop diagrams are denoted in Fig. 3. The mass and momentum of the
$i$-th internal particle are labelled as $m_{i}$ and $p_{i}$. To cut off the
ultra-violet (UV) divergence in the loop integrals, we include a commonly
adopted form factor to regularize the integrand,

$\displaystyle\mathcal{F}(\mathbf{p}^{2}_{i})=\prod_{i}\exp\bigg{(}-\frac{\mathbf{p}^{2}_{i}}{\Lambda^{2}}\bigg{)},$
(22)

where $\Lambda=0.8$ GeV is the cutoff energy and its typical value is around
the $\rho$ mass; $\mathbf{p}_{i}$ is the three-vector momentum of the $i$-th
particle in the loops in the c.m. frame of the initial particle.

Hence, without considering the couplings, the general form of the triangle
loop amplitude $\mathcal{I}$ can be expressed as the following:

$\displaystyle\mathcal{I}_{VP(S)}(s_{R})=i\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{(2p_{X}-p_{1})_{\mu}(-g^{\mu\nu}+\frac{p^{\mu}_{1}p^{\nu}_{1}}{p^{2}_{1}})(p_{1}-2p^{\prime})_{\nu}}{(p^{2}_{1}-m^{2}_{1})(p_{2}^{2}-m^{2}_{2})(p_{3}^{2}-m^{2}_{3})}G_{S}(s_{R})\mathcal{F}(\mathbf{p}^{2}_{i}),$
(23)

where the subscript $VP(S)$ denotes the intermediate $K^{*}\bar{K}$
rescattering into a scalar (S) which will then decays into two pseudoscalars.
Note that $p^{\prime}$ is the momentum of the external particle which is not
from the scalar meson decay.

Figure 3: Conventions for the kinematics of the loop diagrams for resonance
decaying to $bc$. $s_{R}$ is the invariant mass square of the intermediate
resonance which decays to the final states, therefore, $s_{R}$ could possibly
be $s_{ab}$, $s_{ac}$ or $s_{bc}$.

The triangle amplitude of $\eta_{X}\to K^{*}\bar{K}(\bar{\kappa})\to
K^{0}\bar{K}^{0}\pi^{0}$ is written as:

$\displaystyle
i\mathcal{M}_{K^{*}\bar{K}(\bar{\kappa})}(s)=\tilde{G}_{\eta_{X}K^{*0}\bar{K}}\tilde{G}_{K^{*0}\bar{K}^{0}\pi^{0}}\tilde{G}^{2}_{\kappa^{0}\bar{K}^{0}\pi^{0}}\biggl{(}[\mathcal{I}^{N}(s_{ab})+\mathcal{I}^{N}(s_{cb})]+2[\mathcal{I}^{C}(s_{ab})+\mathcal{I}^{C}(s_{cb})]\biggr{)},$
(24)

where superscripts $C$ and $N$ in the loop function $\mathcal{I}$ indicate the
charged and neutral loop, respectively; The subscript $VP(S)$ of $\mathcal{I}$
has been omitted for brevity.

For the process of $\eta_{X}\to K^{*}\bar{K}(a_{0})\to
K^{0}\bar{K}^{0}\pi^{0}$ in Fig. 1(e), the amplitude is written as:

$\displaystyle
i\mathcal{M}_{K^{*}\bar{K}(a_{0})}=2\tilde{G}_{\eta_{X}K^{*0}\bar{K}}\tilde{G}_{K^{*0}\bar{K}^{0}\pi^{0}}\tilde{G}^{2}_{a_{0}K^{0}\bar{K}^{0}}[\mathcal{I}^{C}(s_{ac})+\mathcal{I}^{N}(s_{ac})].$
(25)

For the processes of $\eta_{X}\to K^{*}\bar{K}(a_{0})\to\eta\pi^{0}\pi^{0}$ in
Fig. 2(b), the amplitude is written as

$\displaystyle
i\mathcal{M^{\prime}}_{K^{*}\bar{K}(a_{0})}=\frac{2}{\sqrt{2}}\tilde{G}_{\eta_{X}K^{*0}\bar{K}}\tilde{G}_{K^{*0}\bar{K}^{0}\pi^{0}}\tilde{G}_{a_{0}K^{0}\bar{K}^{0}}g_{a_{0}\eta\pi}[\mathcal{I}^{N}(s_{ab})+\mathcal{I}^{N}(s_{bc})+\mathcal{I}^{C}(s_{ab})+\mathcal{I}^{C}(s_{bc})].$
(26)

Finally, we collect the amplitudes and express the total amplitudes for each
partial wave as follows:

$\displaystyle\mathcal{M}_{\eta_{X}\to K\bar{K}\pi}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{K^{*}\bar{K}}(s)+\mathcal{M}_{\kappa\bar{K}}(s)+\mathcal{M}_{a_{0}\pi}(s)+\mathcal{M}_{K^{*}\bar{K}(\kappa)}(s)+\mathcal{M}_{K^{*}\bar{K}(a_{0})}(s),$
(27) $\displaystyle\mathcal{M}_{\eta_{X}\to K(K\pi)_{P}}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{K^{*}\bar{K}}(s),$ (28)
$\displaystyle\mathcal{M}_{\eta_{X}\to\pi(K\bar{K})_{S}}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\kappa\bar{K}}(s)+\mathcal{M}_{a_{0}\pi}(s)+\mathcal{M}_{K^{*}\bar{K}(\kappa)}(s)+\mathcal{M}_{K^{*}\bar{K}(a_{0})}(s),$
(29) $\displaystyle\mathcal{M}_{\eta_{X}\to\eta\pi\pi}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\prime}_{a_{0}\pi}(s)+\mathcal{M}^{\prime}_{K^{*}\bar{K}(a_{0})}(s).$
(30)

The partial widths of the $K\bar{K}\pi$ and $\eta\pi\pi$ channels are

$\displaystyle\Gamma_{\eta_{X}\to K\bar{K}\pi}(s)=\frac{6}{2\sqrt{s}}\int
d\Phi_{K^{0}\bar{K}^{0}\pi}|\mathcal{M}_{\eta_{X}\to K\bar{K}\pi}(s)|^{2}$
(31)

and

$\displaystyle\Gamma_{\eta_{X}\to\eta\pi\pi}(s)=\frac{3}{2\sqrt{s}}\int
d\Phi_{\eta\pi^{0}\pi^{0}}|\mathcal{M}_{\eta_{X}\to\eta\pi\pi}(s)|^{2},$ (32)

respectively, where $\Phi_{abc}$ is the phase space of $\eta_{X}\to abc$.

###  II.3 Parameters and fitting scheme

With the amplitudes defined in the previous Subsection parameters to be fitted
by experimental data include coupling constants and phase angles between
different transition amplitudes, i.e. $\tilde{G}\equiv ge^{i\theta}$. These
couplings and phase angles are listed in Table. 1. The phase angles arise from
the fact that the interactions are introduced at the hadronic level and the
interacting hadrons are not elementary particles. In particular, these
coupling channels, $K^{*}\bar{K}$, $a_{0}\pi$ and $\kappa\bar{K}$, involve
hadrons with very different properties. By setting the overall couplings
$g_{XVP}$, $g_{XSP}$, $g_{VPP}$, $g_{SPP}$ as real positive numbers, the
relative phase angles will partially account for hadronic effects including
the coupled-channel effects with three-body unitarity, in the numerical
fitting.

To be more specific, we clarify the following considerations in treating the
parameters:

  * •

Since $\eta(1295)$ and $\eta(1405)$ are treated as the first radial excitation
states of the $q\bar{q}$ nonet their couplings to the same final states can be
connected by the SU(3) flavor symmetry. For instance, the $\eta_{H}/\eta_{L}$
couplings to $K^{*}\bar{K}$ can be connected by Eqs. (10) and (11) with the
common coupling $g_{XVP}$, and
$\theta_{\eta_{L}K^{*}\bar{K}}=\theta_{\eta_{H}K^{*}\bar{K}}$ can be set.

  * •

Scalars $a_{0}$ Baru et al. (2004); Janssen et al. (1995); Lohse et al.
(1990); Oller and Oset (1997); Weinstein and Isgur (1990) and $\kappa$ Black
et al. (1998); Descotes-Genon and Moussallam (2006); Doring and Meissner
(2012) are broadly accepted as dynamically generated states by the
$K\bar{K}-\eta\pi$ and $K\pi$ scatterings, respectively. Putting them in the
SU(3) multiplet of Eq. (6) does not suggest that the SU(3) relation still
holds between the $\eta_{X}a_{0}\pi$ and $\eta_{X}\kappa\bar{K}$ couplings.
However, we can assume that the couplings of $\eta_{L}$ and $\eta_{H}$ to the
same final states can still be connected by Eqs. (10) and (11). Therefore, we
treat $g_{Xa_{0}P}$ and $g_{X\kappa P}$ as different parameters, and their
corresponding phase angles, $\theta_{\eta_{L}a_{0}\pi}\
(=\theta_{\eta_{H}a_{0}\pi})$ and $\theta_{\eta_{L}\kappa\bar{K}}\
(=\theta_{\eta_{H}\kappa\bar{K}})$ are also different from each other.

Note that it is the relative phase angles among the transition amplitudes that
can produce measurable effects via interferences. Therefore, we redefine the
phase angles as follows:

$\displaystyle\Theta_{K^{*}\bar{K}}$ $\displaystyle=$
$\displaystyle\theta_{\eta_{X}K^{*}\bar{K}}+\theta_{K^{*0}\bar{K}^{0}\pi^{0}},$
(33) $\displaystyle\Theta_{\eta_{X}\kappa\bar{K}}$ $\displaystyle=$
$\displaystyle\theta_{\eta_{X}\kappa^{0}\bar{K}^{0}}+\theta_{\kappa^{0}K^{0}\pi},$
(34) $\displaystyle\Theta_{\kappa L}$ $\displaystyle=$ $\displaystyle
2\theta_{\kappa^{0}K^{0}\pi},$ (35) $\displaystyle\Theta_{\eta_{X}a_{0}\pi}$
$\displaystyle=$
$\displaystyle\theta_{\eta_{X}a_{0}\pi^{0}}+\theta_{a_{0}K^{0}\bar{K}^{0}},$
(36) $\displaystyle\Theta_{a_{0}L}$ $\displaystyle=$ $\displaystyle
2\theta_{a_{0}K\bar{K}}\ ,$ (37)

where $\theta_{K^{*}\bar{K}}=0^{\circ}$ is set as the reference phase angle.

The couplings $g_{K^{*}\bar{K}\pi}$, $g_{a_{0}\eta\pi}$, $g_{a_{0}K\bar{K}}$
and $g_{\kappa K\pi}$ are extracted from experimental data. We assume that the
$K\pi$ final state saturates the widths of $K^{*}$ and $\kappa$. Meanwhile, we
choose $g_{a_{0}K^{0}\bar{K}^{0}}=-2.24$ GeV Du and Zhao (2019) and
$g_{a_{0}\eta\pi}=2$ GeV Teige et al. (1999). Here, we adopted a relatively
smaller value for $g_{a_{0}\eta\pi}$ in order to reduce the decay width for
$\eta_{H}\to\eta\pi\pi$ as much as possible. With the adopted couplings, these
two channels saturate the $a_{0}$ width up to $50$ MeV, which is the lower
limit listed in PDG Workman et al. (2022).

Hence, we have a set of $12$ independent parameters to be fitted by data:

$\displaystyle m_{\eta_{H}},\quad\alpha_{P},\quad g_{XVP},\quad
g_{Xa_{0}P},\quad g_{X\kappa P},$
$\displaystyle\Theta_{\eta_{X}a_{0}\pi},\quad\Theta_{a_{0}L},\quad\Theta_{\eta_{X}\kappa\bar{K}},\quad\Theta_{\kappa
L},\quad\lambda_{K^{*}},\quad\lambda_{a_{0}},\quad\lambda_{\kappa}$ (38)

##  III Numerical results

The BESIII measurement of $J/\psi\to\gamma K_{S}K_{S}\pi^{0}$ has provided the
most precise partial-wave data for the $K_{S}K_{S}\pi^{0}$ spectrum Ablikim et
al. (2023). In Ref. Ablikim et al. (2023) the $1^{++}$ partial wave is also
included in the $\gamma K_{S}K_{S}\pi^{0}$ spectrum. Since the $1^{++}$ and
$0^{-+}$ partial waves can be well separated by the angular distribution of
the recoiled photon we can focus on the $0^{-+}$ wave in this analysis. Thus,
our effort here is to examine whether our model can describe the $0^{-+}$
partial wave or not. Meanwhile, we will also examine its impact on the
$\eta\pi\pi$ decay channel.

For the two-body spectra, i.e. we extract the $0^{-+}$ components by
subtracting the $1^{++}$ components from the total events that shown in Fig. 4
in Ref. Ablikim et al. (2023). With the differential decay widths and the
corresponding amplitudes listed in Eqs. (13) and (27)-(29) we perform a
combined fitting of the two-body $K\bar{K}$, $K\pi$ and three-body
$K\bar{K}\pi$ spectra measured by BESIII, including the partial-wave
lineshapes of the $(KK)_{\text{S-wave}}\pi$ and the
$(K\pi)_{\text{P-wave}}\bar{K}$ modes Ablikim et al. (2023). The best fitted
parameters and the partial widths of $\eta_{L}$ and $\eta_{H}$ are listed in
Table. 2, and the corresponding best fitted 3-body and 2-body spectra are
shown in Figs. 4 and 5, respectively.

Our model includes both $\eta(1295)$, $\eta(1405)$ and the non-resonant
process $J/\psi\to\gamma K\bar{K}\pi$. As follows, we first take a look at the
fitted spectra and then discuss the fitted parameters. In Fig. 4(a), we plot
our best fitted lineshapes of the spectra of the total $K\bar{K}\pi$ events
and the two different decay mode: $(KK)_{\text{S-wave}}\pi$ and
$(K\pi)_{\text{P-wave}}K$. The data of the $K\bar{K}\pi$ spectra for
$K\bar{K}\pi$, $(KK)_{\text{S-wave}}\pi$ and $(K\pi)_{\text{P-wave}}K$ are
obtained from the BESIII analysis Ablikim et al. (2023), which are
respectively marked as dark blue solid circles, orange solid triangles and
green solid squares. The corresponding fitted results are shown in the same
color and as solid, dot-dashed, and dashed lines, respectively. As shown in
Fig. 4(a), the flat peak in the $K\bar{K}\pi$ spectrum and the shifted peaks
in the $(KK)_{\text{S-wave}}\pi$ and $(K\pi)_{\text{P-wave}}K$ modes can be
well described. The reason that the peak positions are shifted in the
$(KK)_{\text{S-wave}}\pi$ and $(K\pi)_{\text{P-wave}}K$ modes is because these
two modes belong to different partial waves in their quasi-two-body decays.
For the the $(KK)_{\text{S-wave}}\pi$ mode it can receive contributions
directly from the $S$-wave coupling of $\eta_{X}\to a_{0}(980)\pi$. In the
isobaric approach the quasi-two-body $S$ wave will increase faster than the
quasi-two-body $P$ wave when the threshold opens. This explains that the
$(KK)_{\text{S-wave}}\pi$ (dominated by the $a_{0}(980)\pi$ transition) has a
forwardly shifted peak position than the $(K\pi)_{\text{P-wave}}K$ mode
(dominated by the $K^{*}\bar{K}$ transition), and such a behavior can
naturally explain the observed $K\bar{K}\pi$ lineshape.

To demonstrate the role played by each component of the transition amplitude,
we also plot the exclusive contributions from each component in Fig. 4(b),
i.e. the $K\bar{K}\pi$, $(KK)_{\text{S-wave}}\pi$ and
$(K\pi)_{\text{P-wave}}K$ spectra for $\eta(1295)$, $\eta(1405)$, and the non-
resonant contributions, respectively. As shown in Fig. 4(b), because of the
suppressed production rate and small partial width of $\eta(1295)\to
K\bar{K}\pi$, the contributions from $\eta(1295)$ is rather small when
compared to the contributions of $\eta(1405)$. This feature is consistent with
the existing analyses in experiment in various processes.

Figure 4: Three-body invariant mass spectra of $K\bar{K}\pi$ with
$J^{PC}=0^{-+}$: (a) The best fitted $K\bar{K}\pi$ spectra around $1.25\sim
1.6$ GeV. Symbols of dark blue solid circles, orange solid triangles, and
green solid squares are data Ablikim et al. (2023) for the total $0^{-+}$
events, $(K\bar{K})_{\text{S-wave}}\pi$ events, and
$(K\pi)_{\text{P-wave}}\bar{K}$ events, respectively. The corresponding
theoretical fitting results are denoted by the dark blue solid line, orange
dot-dashed line, and green dashed line, respectively. (b) The exclusive
$K\bar{K}\pi$ spectra from different components as noted by the legends.

The transition mechanism can also be understood by the two-body invariant mass
spectra of $K\pi$ and $K\bar{K}$. As shown in Fig. 5 (a) the
$(K\pi)_{\text{P-wave}}$ (dashed line) has clear contributions from the
$K^{*}$, while the dotted line indicates the contributions from the
transitions of $(KK)_{\text{S-wave}}$, of which the projection to the $K\pi$
spectrum corresponds to the $K\pi$ in an $S$ wave. Both the intermediate
$\kappa\bar{K}+c.c.$ and $a_{0}(980)\pi$ can contribute here. Due to the large
width of $\kappa$ only a broad structure similar to the phase space can be
seen.

In Fig. 5 (b) the $K\bar{K}$ invariant mass spectra are plotted, where the
solid line is the fitted result in comparison with the extracted data from
BESIII Ablikim et al. (2023). The dashed and dotted lines denote the
$(K\pi)_{\text{P-wave}}$ and $(KK)_{\text{S-wave}}$ spectra, respectively.
Note that in the $(KK)_{\text{S-wave}}$ spectrum no apparent signal from
$a_{0}(980)$ can be seen here. This feature sheds some light on the $K\bar{K}$
threshold phenomenon. Firstly, the absence of the $a_{0}(980)$ signals in the
$(KK)_{\text{S-wave}}$ spectrum reflects the relatively small branching ratio
for $\eta_{H}\to a_{0}(980)\pi$ compared to $\eta_{H}\to K^{*}\bar{K}$, and
the limited phase space. However, if looks closely, one can still see an
immediate rise of the dotted line at the $K\bar{K}$ threshold.

Secondly, it is interesting to realize that the threshold enhancement produced
by the $K^{*}\bar{K}$ transitions has explicitly indicated the TS kinematics,
and revealed the non-trivial role played by the TS mechanism. For the leading
tree-level transition of $\eta_{H}\to K^{*}\bar{K}+c.c.\to K\bar{K}\pi$, the
elastic $K\bar{K}$ rescattering, although fulfils the TS kinematics, would not
produce measurable effects as proven by the Schmidt theorem Schmid (1967).
However, due to the presence of the $a_{0}(980)$ pole, which can be
interpreted as the dynamically generated state of the $K\bar{K}$ scatterings,
the Schmidt theorem is eventually violated and the TS mechanism can produce
measurable effects. As shown in Fig. 5 (b) the TS amplitude can interfere with
the tree-level amplitudes of the $a_{0}(980)\pi$ and $K^{*}\bar{K}$
transitions, and produce the predominant near-threshold enhancement. This
mechanism is crucial for describing the extracted data 555In Ref. Ablikim et
al. (2023) the two-body invariant mass spectra $K\bar{K}$ and $K\pi$ are
extracted by an isobaric parametrization which is an assumption made in the
data analysis. Because of this treatment some model-dependent features with
the two-body spectra may be inevitable. Although it may raise concerns about
the output of fitting the two-body spectra, it is actually a proof that the
one-state isobaric picture around 1.4 GeV can be accommodated by the data. It
also implies that the three-body unitarity approach would not drastically
change such a phenomenon. We will report the full analysis with three-body
unitarity in a forthcoming work. from BESIII Ablikim et al. (2023).

In the scenario of the first radial excitations the best fitted mass of
$\eta_{H}$ (i.e. $\eta(1405)$) is about $1.44$ GeV. The fitting error of
$m_{\eta_{H}}$ is very small, implying that $m_{\eta_{H}}$ is well constrained
by the data. This mass value is consistent with the peak positions observed in
the $\gamma V$ spectra. Combining the mechanisms introduced in this analysis,
it is likely that the resonance lineshapes can be distorted and the peak
positions can be shifted due to the coupled-channel interferences and TS
mechanism.

Table 2: The best fitted parameters and the corresponding partial widths of $\eta_{L}$ and $\eta_{H}$. Parameters | Values | SU(3) relation
---|---|---
$\alpha_{P}$ | $41.7^{\circ}\pm 0.34^{\circ}$ |
$m_{\eta_{H}}$ |  $1.436\pm 0.001$(GeV) |
$g_{XVP}$ | $5.76\pm 0.26$ | -
$g_{\eta_{H}K^{*0}\bar{K}^{0}}$ | $-2.13\pm 0.09$ | $g_{XVP}(\frac{\sin\alpha_{P}}{\sqrt{2}}R-\cos\alpha_{P})$
$g_{\eta_{L}K^{*0}\bar{K}^{0}}$ | $6.26\pm 0.29$ | $g_{XVP}(\frac{\cos\alpha_{P}}{\sqrt{2}}R+\sin\alpha_{P})$
$g_{Xa_{0}P}$ |  $2.65\pm 0.04$ GeV | -
$g_{\eta_{H}a_{0}\pi}$ |  $2.49\pm 0.04$ GeV | $\sqrt{2}g_{Xa_{0}P}\sin\alpha_{P}$
$g_{\eta_{L}a_{0}\pi}$ |  $2.8\pm 0.04$ GeV | $\sqrt{2}g_{Xa_{0}P}\cos\alpha_{P}$
$g_{X\kappa P}$ |  $0.57\pm 0.07$ GeV | -
$g_{\eta_{H}\kappa\bar{K}}$ |  $0.63\pm 0.07$ GeV | $g_{X\kappa P}(\frac{\sin\alpha_{P}}{\sqrt{2}}R+\cos\alpha_{P})$
$g_{\eta_{L}\kappa\bar{K}}$ |  $-0.14\pm 0.02$ GeV | $g_{X\kappa P}(\frac{\cos\alpha_{P}}{\sqrt{2}}R-\sin\alpha_{P})$
$\lambda_{a_{0}}$ |  $0.51\pm 0.01$ (GeV) |
$\lambda_{\kappa}$ |  $2.74\pm 0.55$ (GeV) |
$\lambda_{K^{*}}$ |  $0.67\pm 0.01$ (GeV) |
$\theta_{\eta_{X}K^{*}\bar{K}}+\theta_{K^{*}\bar{K}^{0}\pi^{0}}$ |  $0^{\circ}$ (Baseline ) |
$\theta_{\eta_{X}a_{0}\pi}$ | $140^{\circ}\pm 5^{\circ}$ |
$\theta_{\eta_{X}\kappa\bar{K}}$ | $102^{\circ}\pm 5^{\circ}$ |
$\theta_{a_{0}K\bar{K}}$ | $27.7^{\circ}\pm 2.3^{\circ}$ |
$\theta_{\kappa K\pi}$ | $40.1^{\circ}\pm 2.6^{\circ}$ |
$\Gamma(\eta_{H}\to K\bar{K}\pi)$ |  $88.8$ MeV |
$\Gamma(\eta_{H}\to\eta\pi\pi)$ |  $64.0$ MeV |
$\Gamma(\eta_{L}\to K\bar{K}\pi)$ |  $20.0$ MeV |
$\Gamma(\eta_{L}\to\eta\pi\pi)$ |  $76.7$ MeV |

Figure 5: Theoretical calculations of the two-body spectra in comparison with
the experimental data Ablikim et al. (2023). (a) $K\pi$ invariant mass
spectrum; (b) $K\bar{K}$ invariant mass spectrum. The solid lines are the
total spectra, and the dashed and dotted lines are the contributions from the
$(K\pi)_{\text{P-wave}}\bar{K}$ and $(K\bar{K})_{\text{S-wave}}\pi$
transitions, respectively.

Figure 6:  Three-body invariant mass spectra of $\eta\pi\pi$ with
$J^{PC}=0^{-+}$ in the $J/\psi$ radiative decays: (a) The $\eta\pi\pi$
spectrum (solid line) predicted with the best fitted parameters. The data are
from Bolton et al. (1992). (b) The exclusive contributions of $\eta_{L}$
(dashed line) and $\eta_{H}$ (solid line) in the $\eta\pi\pi$ spectrum.

Note that the phase difference between Fig. 1(c) and (e) is the same as that
between Fig. 2(a) and (b). Hence, using the parameters fitted by the
$K\bar{K}\pi$ data, the $J/\psi\to\gamma\eta\pi\pi$ spectra can be predicted.
We assume $a_{0}\pi\to\eta\pi\pi$ is dominant because the $a_{0}\pi$ threshold
is more closer to the $\eta_{X}$ mass region than the $\sigma\eta$ and
$f_{0}(980)\eta$ ones, and the TS enhancement only appear in the rescattering
diagram of $\eta_{X}\to K^{*}\bar{K}\to a_{0}\pi$. We do not include the non-
resonance $\eta_{X}\to\eta\pi\pi$ contributions at this moment. In Fig. 6(a)
the calculated $\eta\pi\pi$ spectra are illustrated and compared with the
experimental data from Ref. Bolton et al. (1992). It shows that the
$\eta(1405)$ peak is shifted forward and close to $1.4$ GeV, which is
consistent with the existing measurements Augustin et al. (1990); Bolton et
al. (1992); Bai et al. (1999); Anisovich et al. (2001); Acciarri et al.
(2001); Amsler et al. (2004); Ablikim et al. (2011, 2012). We also plot the
exclusive contributions from $\eta(1295)$ and $\eta(1405)$ in Fig. 6(b), where
a clear $\eta(1295)$ structure can be identified due to the large partial
widths of $\eta(1295)\to\eta\pi\pi$. This feature appears to be consistent
with the measurement of Ref. Bolton et al. (1992), where an enhancement around
the $\eta(1295)$ mass region with $J^{PC}=0^{-+}$ seems to be present in the
$\eta\pi\pi$ spectrum. Note that the experimental data Bolton et al. (1992)
have quite large uncertainties. It would be necessary to have a high-precision
measurement of the $\eta\pi\pi$ spectrum with partial wave analysis.

As mentioned earlier that in Fig. 6 we have not yet included the
$\sigma\pi/f_{0}\eta\to\eta\pi\pi$ and non-resonant contributions in
$J/\psi\to\gamma\eta\pi\pi$. These processes might bring subtle modifications
to the $\eta\pi\pi$ spectra, although the main conclusions are intact. In this
isobaric approach to the one-loop corrections, we obtain the branching faction
$Br(J/\psi\to\gamma\eta(1405)\to\gamma
K\bar{K}\pi)/Br(J/\psi\to\gamma\eta(1405)\to\gamma\eta\pi\pi)\simeq 1.74$ by
integrating the $\sqrt{s}$ distributions from $1.35$ to $1.55$ GeV. This value
turns out to be smaller than the PDG average, which is about $4.5\sim 7.9$
Workman et al. (2022). This may not be surprising since the average has
included data without the partial-wave analysis. We should mention that if we
try to fit this ratio to the PDG averaged value, the description of the
$(K\bar{K})_{\text{S-wave}}\pi$ spectrum will get worse.

In our isobaric approach $\eta_{H}$ and $\eta_{L}$ are treated as the first
radial excitations of the conventional quark model pseudoscalar states. Thus,
their couplings to the same final states can be related with each other via
the SU(3) flavor symmetry as shown in Table 2. However, it should also be
noted that the fitting results show that the $a_{0}(980)\pi$ and
$\kappa\bar{K}+c.c.$ channel have different couplings to the initial state
$\eta_{X}$. This suggests that $a_{0}(980)$ and $\kappa$ cannot be pure
$q\bar{q}$ state. This should be an evidence that they are dynamically
generated states in the $S$-wave $K\bar{K}$ and $K\pi$ scatterings,
respectively.

##  IV Summary

In this work we make a systematic partial wave analysis of
$J/\psi\to\gamma\eta_{X}\to\gamma K\bar{K}\pi$ based on an isobaric approach.
This effort is initiated by the recent observation of a flattened lineshape
around $1.4\sim 1.5$ GeV in the $K_{S}K_{S}\pi^{0}$ invariant mass spectrum by
BESIII Ablikim et al. (2023), which may be regarded as a strong evidence for
two close-by $0^{-+}$ isoscalar pseudoscalars. We demonstrate that with one
state (i.e. the first radial excitation state of $\eta^{\prime}$) around
$1.4\sim 1.5$ GeV the non-trivial $K_{S}K_{S}\pi^{0}$ invariant mass spectrum
can be explained by the coupled-channel effects with the presence of the TS
mechanism. We show that a combined fit of the three-body and two-body spectra
can be achieved which suggests that the one-state solution proposed by Refs.
Wu et al. (2012, 2013); Du and Zhao (2019); Cheng and Zhao (2022) still holds
well. Although it should also be recognized that a more rigorous approach
taking into account the coupled-channel effects with three-body unitarity is
needed for further clarifying the underlying dynamics, it shows that the
results from the isobaric approach can still provide a consistent prescription
of the observables in $J/\psi\to\gamma\eta_{X}\to\gamma K\bar{K}\pi$.

Our fitted results give the mass of $\eta_{H}$ at about $1.43$ GeV, which is
consistent with the peak positions in the current data of the $\gamma V$
spectra, where only one BW structure is observed. In particular, the mass
positions of the BW structures in $\gamma\rho$ and $\gamma\phi$ are close to
each other.

Our model also predicts the $0^{-+}$ $\eta\pi\pi$ spectrum, which is in good
agreements with the experimental data, though the data precision needs further
improvement. In our model, the branching ratio fraction,
$Br(J/\psi\to\gamma\eta(1405)\to\gamma
K\bar{K}\pi)/Br(J/\psi\to\gamma\eta(1405)\to\gamma\eta\pi\pi)\simeq 1.74$, is
extracted and it indicates a sizeable partial width of
$\eta_{H}\to\eta\pi\pi$. This value turns out to be smaller than the PDG
averaged one listed in the $J/\psi$ radiative decays Workman et al. (2022).
However, it should be noted that most of the measurements adopted in the PDG
average were not obtained with the PWA. Future combined PWA analyses of
$J/\psi\to\gamma V$ and $\eta\pi\pi$ with high-statistics data are strongly
demanded. We also stress that since these combined properties of the $0^{-+}$
partial wave in our analyses are not in apparent contradiction with the first
radial excitation scenario Wu et al. (2012, 2013), it makes the space for
accommodating a low-mass pseudoscalar glueball very limited.

###### Acknowledgements.

Useful discussions with M.-C. Du and J.-J. Wu are acknowledged. This work is
supported, in part, by the National Natural Science Foundation of China (Grant
No. 11521505), the DFG and NSFC funds to the Sino-German CRC 110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076), National Key Basic Research Program of China under
Contract No. 2020YFA0406300, and Strategic Priority Research Program of
Chinese Academy of Sciences (Grant No. XDB34030302). Part of the numerical
calculations was supported by HPC Cluster of ITP-CAS.

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# Instructions for *ACL Proceedings

Ibtihel Amara1, Vinija Jain2,3† Aman Chadha2,3†
1McGill University 2Stanford University 3Amazon AI

# DCS: Dynamic Corrective Self-Distillation for Better Fine-Tuning of
Pretrained Models

Ibtihel Amara1, Vinija Jain2,3† Aman Chadha2,3†
1McGill University 2Stanford University 3Amazon AI

###### Abstract

We tackle the challenging issue of aggressive fine-tuning encountered during
the process of transfer learning of pre-trained language models (PLMs) with
limited labeled downstream data. This problem primarily results in a decline
in performance on the subsequent task. Inspired by the adaptive boosting
method in traditional machine learning, we present an effective dynamic
corrective self-distillation (DCS) approach to improve the fine-tuning of the
PLMs. Our technique involves performing a self-distillation mechanism where,
at each iteration, the student model actively adapts and corrects itself by
dynamically adjusting the weights assigned to individual data points. This
iterative self-correcting process significantly enhances the overall fine-
tuning capability of PLMs, leading to improved performance and robustness. We
conducted comprehensive evaluations using the GLUE benchmark demonstrating the
efficacy of our method in enhancing the fine-tuning process for various PLMs
across diverse downstream tasks.

DCS: Dynamic Corrective Self-Distillation for Better Fine-Tuning of Pretrained
Models

Ibtihel Amara1, Vinija Jain2,3† Aman Chadha2,3† 1McGill University 2Stanford
University 3Amazon AI

22footnotetext: Work does not relate to position at Amazon.

##  1 Introduction

There has been a remarkable advancement in the field of Natural Language
Processing (NLP) in the past few years, thanks to the introduction of pre-
trained language models (PLMs). Recent PLMs like BERT Devlin et al. (2019),
RoBERTa Zhuang et al. (2021), XLNet Yang et al. (2019) have revolutionized and
shaped the landscape of the field of NLP by demonstrating significant progress
across different downstream applications such as machine translation, reading
comprehension, and question and answering. Standard practice involves fine-
tuning PLMs directly on labeled data from these tasks. Yet, when faced with
limited downstream data, known as aggressive fine-tuning Jiang et al. (2020),
the risk of model overfitting and reduced generalization capacity emerges.
Addressing this challenge has spurred various approaches, encompassing hyper-
parameter tuning heuristics Howard and Ruder (2018); Peters et al. (2019),
additional layer integration Houlsby et al. (2019); Stickland and Murray
(2019), improved training strategies Chen et al. (2020), and noise-induced
fine-tuning methods Wu et al. (2022). Notably, techniques such as adapters,
hypernetworks, LoRA, QLoRA, and GLoRA have gained traction for efficient
parameter updates when data is scarce Houlsby et al. (2019); Hu et al. (2021);
Dettmers et al. (2023); Chavan et al. (2023).

Figure 1: Dynamic Corrective Self-Distillation Framework. DCS iteratively
adjusts the weights on the data samples at every epoch. It puts more emphasis
on teacher-student discordant (i.e., non-agreeing) samples. We refer to
agreement between teacher and student as the level of consensus in their
predictions. The distillation component ensures the proper guidance of the
student network throughout its optimization process.

Surprisingly, the potential of distillation as a fine-tuning tool has been
somewhat overlooked. Despite its simplicity, distillation offers substantial
utility in guiding and optimizing fine-tuning processes. In this work, we
introduce Dynamic Corrective Self-distillation (DCS), a straightforward
approach inspired by adaptive boosting in machine learning. DCS iteratively
adjusts sample weights, prioritizing instances where the teacher and student
model disagrees. This emphasis on challenging examples helps improve
individual model performance and accuracy in subsequent iterations.

  * •

We propose a flexible and adaptable fine-tuning framework called DCS based on
distillation-boosting fusion, which helps large PLMs in avoiding overfitting
problems.

  * •

We introduce a self-corrective training framework where the student model is
guided by the knowledge of a teacher network.

  * •

Our proposed technique yields a gain in performance of more than 2% on the
GLUE benchmark when compared to vanilla fine-tuning.

Model | MNLI | QQP | RTE | QNLI | MRPC | CoLA | SST | STS | Avg.
---|---|---|---|---|---|---|---|---|---
BERT (base) Devlin et al. (2019) | 84.4 | 90.9 | 67.7 | 91.5 | 87.1 | 58.1 | 93.0 | 89.4 | 82.76
BERT (base) + DCS | 84.8 | 91.2 | 71.8 | 91.8 | 87.7 | 59.7 | 93.2 | 89.5 | 83.71
RoBERTa (base) Wu et al. (2022) | 87.5 | 91.7 | 77.1 | 92.7 | 90.1 | 62.9 | 94.5 | 90.8 | 85.91
RoBERTa (base) + DCS | 87.6 | 91.8 | 81.2 | 93.0 | 90.7 | 63.4 | 94.2 | 90.9 | 86.60
XLNET Yang et al. (2019) | 86.6 | 91.2 | 72.9 | 91.6 | 88.1 | 59.6 | 94.4 | 89.6 | 84.25
XLNET + DCS | 86.7 | 91.5 | 74.4 | 91.7 | 89.2 | 60.5 | 95.1 | 88.5 | 84.70
ELECTRA Clark et al. (2020) | 88.4 | 91.7 | 75.2 | 92.9 | 88.2 | 64.2 | 94.9 | 90.1 | 85.70
ELECTRA + DCS | 88.7 | 92.0 | 84.5 | 93.5 | 90.4 | 70.4 | 95.2 | 91.1 | 88.22

Table 1: Comparison between DCS and Vanilla fine-tuning applied to widely used
Pretrained Language Models. The best results are in bold. The results show
that DCS leads to substantial improvements across all tasks and among the
various PLMs. All DCS values represent the mean values over 3 random seeds.

Methods | CoLA | RTE | MRPC | STS-B | Avg | $\Delta$
---|---|---|---|---|---|---
Vanilla FT | 63.13 | 70.18 | 90.77 | 89.61 | 78.42 | 0.00
Weight Decay | 63.63 | 71.99 | 90.93 | 89.82 | 79.09 | +0.67
Top-K FT | 62.63 | 70.90 | 91.09 | 89.97 | 78.65 | +0.23
Mixout | 63.60 | 72.12 | 91.29 | 89.99 | 79.26 | +0.84
RecAdam | 64.33 | 71.63 | 90.85 | 89.86 | 79.17 | +0.75
R3F | 64.13 | 72.28 | 91.18 | 89.61 | 79.30 | +0.88
CHILD FT(F) | 63.71 | 72.02 | 91.22 | 90.18 | 79.29 | +0.87
CHILD FT(D) | 64.92 | 73.14 | 91.42 | 90.18 | 79.92 | +1.50
DCS (Ours) | 64.38 | 74.36 | 91.58 | 89.95 | 80.06 | +1.65

Table 2: Comparison between DCS with other existing fine-tuning methods Daumé
III (2007); Houlsby et al. (2019); Lee et al. (2020); Chen et al. (2020);
Aghajanyan et al. (2021); Xu et al. (2021). Most values are taken from Xu et
al. (2021). $\Delta$ refers to the performance gain w.r.t VFT. Our findings
show DCS performs comparably or even better than other fine-tuning methods.

##  2 Dynamic Corrective Self-Distillation

To address the issue of limited data for finetuning, we draw inspiration from
the adaptive boosting technique in machine learning and propose a dynamic
process that iteratively adjusts the sample weights at each epoch of the
training process. To achieve accurate weight adjustments, we rely on the
knowledge distillation technique Hinton et al. (2015). This technique involves
employing a teacher network to guide a student model with the same
architecture and capacity. In Figure 1, we provide a visual representation of
the DCS framework. Initially, DCS assigns equal weights to each data point
within the downstream dataset. However, as the learning process progresses,
DCS modifies these weights. Specifically, greater emphasis is placed on
samples where the student and teacher networks diverge in their predictions.
In other words, samples that the student network mispredicts compared to the
teacher’s prediction are assigned higher weights. By incorporating this
mechanism, DCS effectively leverages the discrepancy between the student and
teacher networks to enhance its learning capability. KD training is performed
according to the following objective function.

$L_{Total}=\alpha L_{CE}+(1-\alpha)L_{KD}$ (1)
$\vspace{-1.5mm}L_{KD}=-\sum_{i=1}^{N}p(x_{i})log(q(x_{i}))\vspace{1mm}$ (2)

where $p_{i}$ and $q_{i}$ are the class probabilities generated by the teacher
and student For the DCS technique, each sample weight is adjusted according to
the teacher and student’s agreement. For this, the $L_{KD}$ is replaced with
the following:

$\displaystyle L_{KD}=-\sum_{i=1}^{N}w_{i}p(x_{i})log(q(x_{i}))$ (3)

such that $w_{i}=\lambda$ if $\hat{y}_{i}^{T}\neq\hat{y}_{i}^{S}$, otherwise
$w_{i}=1$. $\lambda$ is a hyper-parameter higher than 1 $(\lambda>>1)$.
$\hat{y}_{i}^{T}$ and $\hat{y}_{i}^{S}$ are the predictions of the teacher and
the student, respectively, for a sample point $x_{i}$. We provide the steps
for performing DCS in Algorithm A.1.
How can self-KD help finetune the student? Recent studies by Furlanello et al.
Furlanello et al. (2018) and Stanton et al. Stanton et al. (2021) have
highlighted the potential of self-distillation, where the students created
through this process can actually surpass the performance of their teachers.
This intriguing outcome stems from the fact that distillation inherently
involves simplifying the student’s model based on the teacher’s knowledge. The
crucial point here is that it doesn’t need an exact match between the student
and the teacher. In fact, striving for a perfect match would hinder the
student’s ability to learn and outperform the teacher. These research works
Stanton et al. (2021); Furlanello et al. (2018) study the significance of not
merely mimicking the teacher but rather distilling and imparting vital
knowledge in a manner that amplifies the student’s performance.
Weighing discordant samples. Introducing sample weighting has been
demonstrated to enhance the KD process Lu et al. (2021). More specifically,
DCS assigns increased weights to the discordant samples, amplifying their
impact in the subsequent stages of the student model training. This technique
steers the student model toward challenging samples by leveraging the
teacher’s soft labels while concurrently adhering to the guidance provided by
the hard labels (i.e., the ground truth labels).

##  3 Experiments and Results

###  3.1 Experimental Setup

We conduct our experiments on various datasets from the GLUE benchmark Wang et
al. (2018), a set of datasets containing different tasks such as natural
language inference, sentiment analysis, and sentence similarity. Following
previous research works, we fine-tune our pretrained models on the training
set and directly report results on the dev set. We use the pretrained models
and codes provided by HuggingFace Wolf et al. (2020). Appendix A.3 elaborates
on our hyperparameter sweep setup.

###  3.2 Results

Comparison to Vanilla Fine-Tuning. We present compelling results of DCS on
eight GLUE tasks using four commonly used PLMs. Following the evaluation
methodology of related works Xu et al. (2021); Wu et al. (2022), we compare
the performance of DCS against vanilla fine-tuning. Table 1 clearly
demonstrates that DCS substantially outperforms vanilla fine-tuning across all
tasks, affirming its superiority and potential for enhancing PLM fine-tuning.
On average, DCS achieves an approximate 1% improvement across all GLUE
benchmark tasks using the BERT base model, and a substantial 2.5% average
score increase on ELECTRA. Notably, DCS shows significant performance gains on
smaller downstream tasks as well. The RTE dataset, the smallest in the GLUE
benchmark with only 2.5K samples, exhibits a remarkable boost with DCS. Across
various PLMs, we observe notable increases, such as nearly 8% with ELECTRA and
approximately 4% with RoBERTa and BERT for RTE.
Comparison to Existing FT Methods. In this section, we assess the performance
and the effectiveness of our proposed fine-tuning technique DCS. For this, we
follow the work in Xu et al. (2021) and proceed to compare prior methods on
the BERT-large model and report the mean values across different seeds.
Results of this comparison are shown in Table 2. We observe that all fine-
tuning techniques improve upon vanilla fine-tuning. Our technique, which is
considered a simple and straightforward technique compared to the existing
methods, is as good as or even substantially outperforms these fine-tuning
schemes. Mainly we observe a similar trend as in Table 1, where DCS depicts a
significant increase on RTE (the smallest dataset within the GLUE benchmark).

Figure 2: Effect of different weighting strategies of DCS on the performance.
Results show that leveraging the discordant teacher-student samples benefits
our DCS fine-tuning framework.

###  3.3 Ablation and Analyses

Model and Technique | RTE | CoLA | MRPC
---|---|---|---
BERT DCS | 71.11 | 59.61 | 87.58
BERT DCS (w/o weighting) | 70.39 | 58.38 | 87.25
ELECTRA DCS | 84.47 | 70.57 | 90.49
ELECTRA DCS (w/o weighting) | 83.75 | 70.06 | 90.19

Table 3: Ablation study of DCS. Without weighting involves distillation
without the dynamic self-correction component. Removing both weighting and
distillation components leads to vanilla fine-tuning. Results emphasize the
importance of combining both components for enhancing the performance of our
fine-tuning framework.

Components of DCS. The DCS technique comprises two key components that
synergistically enhance its effectiveness: (1) The sample re-weighting
mechanism, serving as a self-corrective element within DCS, and (2) The
offline distillation component, which plays a dual role. Firstly, it
identifies samples that require increased attention, and secondly, it
continues to guide the student network during the learning process. To assess
the significance of these components, we conducted an ablation study, the
results of which are presented in Table 3. When removing the self-corrective
mechanism (referred to as DCS without weighting), we transition to sole
distillation. However, this leads to a noticeable performance decrease
compared to DCS. On average, we observe a reduction of more than 1% in RTE
performance for both BERT base (from 71.11% downto 70.39%) and ELECTRA (from
84.47% downto 83.75%) models. Furthermore, eliminating the distillation
component returns us to the vanilla fine-tuning technique. As previously
observed, DCS consistently outperforms vanilla fine-tuning, highlighting the
superiority of our approach.
Leveraging Disagreement. In this section, we investigate the benefits of
assigning higher weights to discordant student-teacher samples in DCS and how
different weighting strategies impact its performance. Figure 2 presents a
comparison of various weighting strategies: DCS-reverse, which prioritizes
concordant teacher-student samples, and DCS-random, which assigns weights
randomly. Analyzing the performance of BERT-base on different datasets, we
observe that leveraging the discrepancies between teacher and student networks
improves the performance and robustness of DCS. Mainly, DCS-reverse performs
the least, suggesting that focusing on concordant samples hampers
generalization and performance. Conversely, emphasizing non-agreeing samples
between the teacher and student enhances the student’s understanding and
potential for improved generalization, resulting in performance gains.

###  3.4 Hyperparameter Sensitivity

Influence of the hyperparameter $\alpha$. In KD, $\alpha$ handles the amount
of contribution for each of the losses (i.e., cross-entropy and KD). We run
different experiments with different values of $\alpha$. As shown in Figure 3,
DCS shows higher performance for smaller to intermediate values of $\alpha$,
which suggests that the student fine-tuned model benefits from the
distillation component of the optimization process. Relevant details can be
found in A.2.

Figure 3: Sensitivity of DCS to $\alpha$. The results highlight the importance
of teacher guidance through distillation. Higher $\alpha$ values indicate a
stronger influence on the KD loss, while lower values emphasize the
distillation loss. Interestingly, moderate alpha values lead to significant
performance improvements, while further increases in alpha show decreasing
values.

##  4 Conclusion

In this paper, we introduce DCS, a framework for enhancing the fine-tuning of
pre-trained language models (PLMs) on downstream tasks. By incorporating a
dynamic self-correction mechanism triggered via self-distillation, DCS
achieves remarkable performance gains over vanilla fine-tuning and previous
approaches. Through extensive experiments on various tasks from the GLUE
benchmark, we demonstrate the superior performance of DCS across four
different pre-trained language models. Notably, DCS not only outperforms
existing methods but also significantly enhances the generalization ability of
the fine-tuned models. One of the main advantages of using our proposed method
DCS is flexibility. Unlike other existing methods such as parameter-efficient
fine-tuning (PEFT) methods and noise addition, DCS does not limit the
flexibility of the model and its ability to adapt to each downstream task.

##  5 Limitations

One main limitation of the DCS method is its computational complexity. DCS
involves training a teacher model (of similar size as the student network)
using the direct fine-tuning method on the downstream tasks. Then, we use this
teacher to better guide a student model using the adaptive corrective approach
via knowledge distillation. However, we argue that teacher models (i.e.,
vanilla fine-tuned) are abundant, meaning that we can easily find a directly
fine-tuned teacher network in public repositories like Hugging face hubs.
Therefore, one may not need to consider this as a limitation.

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## Frequently Asked Questions (FAQs)

  * ✽

How do you determine the optimal teacher network to guide the student model
effectively during the process of distillation?

➠

The DCS framework does not require a specific type of teacher model or high-
performing teacher model. Any pre-fine-tuned teacher model can be used within
the DCS framework. In our experiments, we opted for self-distillation, where
the teacher and student networks have similar architectures and sizes. The
performance of the teacher model does not need to be exceptional or achieve
high accuracy. It simply needs to be trained for a few iterations without full
convergence. In our experiments, we fine-tuned the teachers for a limited
number of epochs (2 epochs). This flexibility in the choice of teacher models
allows for broader applicability and ease of implementation within the DCS
framework.

  * ✽

How can we expect the student network to achieve better results on downstream
tasks if we choose a teacher network that is not performing well?

➠

Knowledge distillation can still be beneficial for the student network’s
optimization process. The main purpose of knowledge distillation is to
transfer valuable insights learned by the teacher down to the student
Furlanello et al. (2018); Stanton et al. (2021). Although the teacher network
did not achieve high performance on its own, it still provides valuable
patterns and special relationships in the data. By considering this distilled
knowledge, the student has the ability to improve its performance and achieve
better results on the downstream task. One main thing to note, is that in our
DCS framework, we perform a weighted distillation in which we give higher
weights on discordant samples. The latter can be one of the following
scenarios: (a) The teacher network correctly classified the sample whereas the
student misclassified the sample, or (b) vice versa. In this case, the student
network gets to attend to both scenarios and learn from both the good and the
bad throughout its learning process, which contributes to its effectiveness
and robustness. Furthermore, it is worth mentioning that our student networks
receive guidance not solely from the soft labels provided by the teacher but
also from the hard labels known as the ground truth labels. This dual guidance
alongside the re-weighting strategy serves as an effective regularization
strategy.

  * ✽

What type of knowledge are you using for distilling information from teacher
to student?

➠

In DCS, we solely use response-based knowledge. We mainly exploit the logits
(i.e., the raw pre-softmax outputs) to align the knowledge between the teacher
and the student networks. Our main goal is to propose a straightforward and
less complicated framework for fine-tuning. Therefore, we opted for this type
of knowledge rather than feature based distillation, which can bring more
computational complications to the overall framework. In addition, by aligning
the logits, the student model can gain more insights into the finer details of
the task guided by the teacher model.

  * ✽

DCS is inspired from the adaptive boosting technique in machine learning,
which in the end combines multiple weak learners. Do you do something similar?

➠

For this current version of our work, the same network is being adjusted
(i.e., dynamic self-correction) according to the new weighted samples. For
inference, we use a single network (i.e., the student network). It is possible
to combine multiple versions of our student (i.e. different student model
checkpoints) during inference to have a more robust predictions and outcomes.

##  Appendix A Appendix

This section provides supplementary material in the form of additional
results, implementation details, etc. to bolster the reader’s understanding of
the concepts presented in this work.

###  A.1 DCS Algorithm

Algorithm 1 DCS

1:Load teacher weights $\theta_{T}$

2:Initialize student’s parameters $\theta_{S}$

3:for each epoch do

4: if epoch == 0 then

5: Train the student according to Eq.1

6: using the KD loss in Eq. 2

7: else

8: Re-weight the samples according to

9: the agreement between the teacher

10: and the current state of the student

11: Train the student according to Eq.1

12: using the KD loss in Eq. 3

13: end if

14:end for

The pseudo-code for our DCS framework is outlined in Algorithm 1. A key
requirement is to have a pre-trained teacher model with similar architecture
and size as the student network. While the algorithm shares similarities with
offline knowledge distillation for downstream tasks, the key differentiating
factor is the incorporation of the re-weighting mechanism. The latter plays a
crucial role in identifying discordant teacher-student samples and assigning
them higher weight values, allowing the student network to benefit from the
valuable insights provided by these samples. It is important to note that, the
first epoch of training in DCS is intentionally designed to exclude the re-
weighting mechanism. This is based on the assumption that the student network
has not yet encountered and learned the complexity and intricacies of the
downstream data. By providing this initial epoch as a warm-up phase, the DCS
framework enables the student network to gradually adapt to the complexities
of the downstream task, leading to improved performance and generalization
capabilities.

###  A.2 Influence of the $\lambda$ Hyperparameter

Model $\downarrow$ / $\lambda$ $\rightarrow$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
BERT-DCS | 70.39 | 71.11 | 70.50 | 70.81 | 70.63

Table 4: Influence of the hyper-parameter $\lambda$ on the performance of DCS
on the RTE task.

DCS relies on the re-weighting strategy for self-correction. We perform
experiments with different lambda values and assess their effect on the
performance of DCS. Table 4 shows that the optimal value is $\lambda=2$.
Beyond that, we observe a steady outcome on the DCS.

###  A.3 Hyperparameter Search Values

In this paper, we fine-tune different large pre-trained language models. We
performed a simple grid search to find the optimal hyperparameter settings as
shown in Table 5.

Hyperparameters | Search Values
---|---
Batch size | {8, 16}
Learning rate (lr) | {1e-5, 2e-5, 1e-6, 2e-6}
Distillation temperature (T) | 1
$\alpha$ KD | {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}
sample weighting ($\lambda$) | {2, 3, 4, 5, 6}

Table 5: Hyperparameter search space for all tasks in GLUE.