# Comprehensive study of light axial vector mesons with the presence of
triangle singularity

Meng-Chuan Du1,2111E-mail address<EMAIL_ADDRESS>and Qiang Zhao1,2222E-mail
address<EMAIL_ADDRESS>1 Institute of High Energy Physics and Theoretical
Physics Center for Science Facilities,
Chinese Academy of Sciences, Beijing 100049, P.R. China  2 University of
Chinese Academy of Sciences, Beijing 100049, P.R. China

###### Abstract

We present a systematic study of the productions and decays of light axial
vector mesons with $J^{PC}=1^{+\pm}$ in charmonium decays. In the quark model
scenario, the two axial vector nonets are connected with each other by the
Gell-Mann-Okubo mass relation through the mixing between two $K_{1}$ states
(i.e. $K_{1}(1270)$ and $K_{1}(1400)$) with configurations of ${}^{3}P_{1}$
and ${}^{1}P_{1}$. The mixing angles between $f_{1}$ and $f_{1}^{\prime}$
(i.e. $f_{1}(1285)$ and $f_{1}(1420)$), and between $h_{1}$ and
$h_{1}^{\prime}$ (i.e. $h_{1}(1170)$ and $h_{1}(1415)$) can be reliably
constrained. We then introduce the intermediate $K^{*}\bar{K}+c.c.$ meson loop
transitions in the description of the productions and decays of these axial
vector mesons. The presence of the nearby $S$-wave $K^{*}\bar{K}+c.c.$ to
which these axial vector mesons have strong coupling strengths, turns out to
be crucial for understanding many puzzling questions related to their
productions and decays. This is because that the $S$-wave $K^{*}\bar{K}+c.c.$
rescatterings by the kaon exchange satisfy the triangle singularity (TS)
condition in some of these cases and the TS mechanism can introduce special
interference effects in the exclusive decays of these light axial vector
mesons.

##  I Introduction

The constituent quark model has proved to be successful on the description of
many qualitative features of low-lying hadrons. However, it also shows that it
is challenging for us to provide a quantitative prescription for the hadron
spectroscopy due to the complexity of non-perturbative QCD. A long-standing
puzzling case is the properties of the lowest-lying positive parity vectors,
namely, the axial vector mesons. With many unanswered mysterious questions on
their productions and decays it seems that a consistent picture for the axial
vector nonets as the first orbital excitation states in the light meson
spectroscopy is still far from broadly accepted.

In the Particle Data Group (PDG) Patrignani:2016xqp ; Zyla:2020zbs  the
following nonstrange light axial vector states are identified in experiment,
namely, $a_{1}(1260)$, $f_{1}(1285)$, and $f_{1}(1420)$ as multiplets of
$J^{PC}=1^{++}$, and $b_{1}(1235)$, $h_{1}(1170)$ and $h_{1}(1415)$ 333The
state $h_{1}(1415)$ was labelled as $h_{1}(1380)$ in the PDG before the 2020
Edition Zyla:2020zbs . The change is due to the new data for $J/\psi\to
h_{1}^{\prime}\eta^{\prime}\to K\bar{K}\pi\eta^{\prime}$ from BESIII, where
the resonance parameters have been updated Ablikim:2018ctf .  as multiplets of
$J^{PC}=1^{+-}$. Two strange multiplets, $K_{1}(1270)$ and $K_{1}(1400)$, are
identified as their strange partners, respectively, and then two quark-model
axial vector nonets can be constructed. Since $K_{1}(1270)$ and $K_{1}(1400)$
do not have a fixed $C$-parity their mixing has impact on the nonstrange axial
vector meson masses through the Gell-Mann-Okubo relation, and it has been the
focus of many studies in the literature. While the mixing angle between
$K_{1}(1270)$ and $K_{1}(1400)$ has been well established during the past
decade (see a brief review by Refs. Cheng:2011pb ; Cheng:2013cwa  and
references therein), more puzzling but interesting issues were raised about
the $f_{1}(1285)$ and $f_{1}(1420)$ and they have been proposed to be exotic
states based on their couplings to the $K^{*}\bar{K}+c.c.$ open channel.

In fact, the role played by the $S$-wave $K^{*}\bar{K}+c.c.$ open threshold
has been recognised in almost all the non-$q\bar{q}$ interpretations for the
nonstrange axial vector mesons in the literature. For instance, it was
proposed in Ref. Longacre:1990uc  that $f_{1}(1420)$ could be a
$K^{*}\bar{K}+c.c.$ molecule. In Ref. Roca:2005nm  it was found that the
$S$-wave interactions between the vector and pseudoscalar mesons would lead to
pole structures in the second Riemann sheet. Within this scenario axial vector
states $f_{1}(1285)$, $a_{1}(1420)$, $h_{1}(1170)$, $h_{1}(1385)$ and
$b_{1}(1235)$ were interpreted as dynamically generated states. In particular,
$f_{1}(1285)$ was proposed to be a $K^{*}\bar{K}+c.c.$ molecule, while the
authors found that $f_{1}(1420)$ could not be accommodated by their framework.
In Ref. Liang:2019vhf  an analysis of $J/\psi\to b_{1}(1235)\pi$,
$h_{1}(1170)\eta^{(\prime)}$, and $h_{1}(1380)\eta^{(\prime)}$ was presented
by treating the $C=-1$ axial vector mesons as dynamically generated states by
the pseudoscalar and vector $S$-wave couplings.

Although the $S$-wave $K^{*}\bar{K}+c.c.$ open threshold appears to be crucial
for all the present non-$q\bar{q}$ interpretations, model-dependent
assumptions in different phenomenologies make it difficult to distinguish
those scenarios beyond the simple quark model $q\bar{q}$ categorizations. In
this work we show that the $S$-wave $K^{*}\bar{K}+c.c.$ interactions actually
cut in the problem via its introduction of the triangle singularity (TS)
mechanism into the axial vector meson decays. While this is a key for a self-
consistent description of the axial vector mesons, we will show that by a
self-consistent treatment of the TS mechanism there exists smoking-gun
observables for disentangling the puzzles in the axial vector spectra and a
coherent picture can be obtained.

The TS mechanism in the axial vector decays was recognized with the help of
two recent experimental progresses: (I) The observation of abnormally large
isospin breaking effects in $J/\psi\to\gamma\eta(1405/1475)\to\gamma+3\pi$ by
the BESIII Collaboration BESIII:2012aa , where the $f_{1}(1420)$ should also
contribute. (II) The observation of $a_{1}(1420)$ in the invariant mass
spectrum of $3\pi$ in $\pi^{-}p\to p+3\pi$ by the COMPASS Collaboration
Adolph:2015pws , where the $a_{1}(1420)$ in the vicinity of $a_{1}(1260)$
apparently cannot be accommodated by the quark model nonet. For the former
case it was first proposed in Ref. Wu:2011yx  that the TS mechanism was the
key for understanding the abnormally large isospin violations. In a later
detailed analysis, it was shown that a small contribution from the
$f_{1}(1420)$ should appear in the decay of $J/\psi\to\gamma+3\pi$ Wu:2012pg .
Although the importance of the TS mechanism for $\eta(1405/1475)\to 3\pi$ was
confirmed by Refs. Aceti:2012dj ; Achasov:2015uua , it was emphasized by Ref.
Achasov:2015uua  that the width effects arising from the intermediate $K^{*}$
may dilute the contributions from the TS transitions. A comprehensive analysis
of the $\eta(1405/1475)$ decays into $K\bar{K}\pi$, $\eta\pi\pi$ and $3\pi$
was presented in Ref. Du:2019idk , and it was clarified that apart from the
width effects the TS enhanced $a_{0}(980)\pi$ production should also be
explicitly included. This provides another important isospin breaking source
from the TS mechanism in $\eta(1405/1475)\to 3\pi$ and makes it possible to
coherently investigate those three decay channels in a self-consistent
framework. This has been one of the major motivations of this work since such
a treatment can be applied to $J/\psi\to\gamma
f_{1}(1285)/f_{1}(1420)\to\gamma K\bar{K}\pi$, $\eta\pi\pi$ and $3\pi$, and a
better understanding of $f_{1}(1285)$ and $f_{1}(1420)$ may be achieved. It
should be mentioned that because of the strong enhancement caused by the TS
mechanism, it was proposed by Ref. Debastiani:2016xgg  that the signal for
$f_{1}(1420)$ observed in $pp$ scatterings by the WA102 Collaboration could be
due to the $f_{1}(1285)$ via the TS mechanism. This is an interesting point
since according to the result of Ref. Roca:2005nm  the $f_{1}(1420)$ cannot be
accommodated by their model for dynamically generated state.

Concerning the observation of $a_{1}(1420)$ by the COMPASS Collaboration
Adolph:2015pws  listed above, it is explained to be a tetraquark state in some
studies base on QCD sum rule Chen:2015fwa ; Sundu:2017xct , and ADS/CFT method
Gutsche:2017oro . However, it can be regarded as a natural consequence of the
presence of the TS mechanism in this kinematic region caused by the $S$-wave
isovector coupling between $K^{*}\bar{K}+c.c.$ in the $3\pi$ spectrum 444The
TS mechanism was first pointed out by Q.Z. at Hadron 2013 in Nara when the
COMPASS data were first reported.. An analysis based on the TS mechanism was
carried out in Ref. Ketzer:2015tqa . By considering that the TS mechanism can
be satisfied by the slightly offshell $a_{1}(1260)$, it gave a natural
explanation of the enhancement at about 1.42 GeV in the $3\pi$ channel with
$I=1$. This mechanism was confirmed by Ref. Aceti:2016yeb  where a coupled
channel treatment was emphasized. Actually, the $a_{1}(1420)$ so far has only
been observed in $\pi^{-}p\to p+3\pi$. This can be not only a strong evidence
for the TS mechanism, but also a stringent constraint on a coherent analysis.
Nevertheless, since the $K^{*}\bar{K}+c.c.$ can also couple to the negative
$C$-parity axial vector states, i.e. $h_{1}$, $h_{1}^{\prime}$ and $b_{1}$,
examinations of its impact on the productions and decays of these states can
further well-establish the axial vector meson spectra.

To proceed, In Section II we first give a brief introduction of the axial
mixing scenarios, which are closely correlated with the TS mechanism due to
the strong couplings of the axial vector states to vector and pseudoscalar
mesons. We then make a detailed analysis of the productions and decays of the
two sets of axial vector mesons with the presence of the TS mechanism in
Section III. The numerical results and discussions are presented in Section
IV, and a brief summary is given in Section V.

##  II State mixings and determination of the mixing angles

In the axial vector sector, the mixing angles between $f_{1}(1285)$ and
$f_{1}(1420)$, and between $h_{1}(1170)$ and $h_{1}(1415)$, are correlated
with the mixing between $K_{1}(1270)$ and $K_{1}(1400)$ in the quark model.
Since the mixing angles decide the relative coupling strengths of the two
states in a doublet ($f_{1}(1285)/f_{1}(1420)$ or $h_{1}(1170)/h_{1}(1415)$)
to $K^{*}\bar{K}+c.c.$, we first set up the convention for the mixing schemes,
and discuss the leading-order couplings in the flavor SU(3) symmetry limit.
These couplings will be the input for evaluating the relative production rates
for the axial vector meson productions in $J/\psi$ decays and the impact of
the TS mechanism in these hadronic decays. For simplicity, from now on, we
denote the $f_{1}(1285)$, $f_{1}(1420)$, $h_{1}(1170)$ and $h_{1}(1415)$ by
$f$, $f^{\prime}$, $h$ and $h^{\prime}$, respectively.

We start with the state mixing on the SU(3) basis,

$\displaystyle\left(\begin{array}[]{c}f_{1}^{\prime}\\\ f_{1}\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos{\theta_{f}}&-\sin{\theta_{f}}\\\
\sin{\theta_{f}}&\cos{\theta_{f}}\end{array}\right)\left(\begin{array}[]{c}\tilde{f}_{8}\\\
\tilde{f}_{1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos{\theta_{f}}&-\sin{\theta_{f}}\\\
\sin{\theta_{f}}&\cos{\theta_{f}}\end{array}\right)\left(\begin{array}[]{cc}\sqrt{\frac{1}{3}}&-\sqrt{\frac{2}{3}}\\\
\sqrt{\frac{2}{3}}&\sqrt{\frac{1}{3}}\end{array}\right)\left(\begin{array}[]{c}f_{n}\\\
f_{s}\\\ \end{array}\right)$ (13) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos{\alpha_{f}}&-\sin{\alpha_{f}}\\\
\sin{\alpha_{f}}&\cos{\alpha_{f}}\end{array}\right)\left(\begin{array}[]{c}f_{n}\\\
f_{s}\\\ \end{array}\right),$ (18)

where $\tilde{f}_{1}\equiv(u\bar{u}+d\bar{d}+s\bar{s})/{\sqrt{3}}$,
$\tilde{f}_{8}\equiv(u\bar{u}+d\bar{d}-2s\bar{s})/{\sqrt{6}}$,
$f_{n}\equiv(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and $f_{s}\equiv s\bar{s}$.
Likewise, the mixing angles between $h$ and $h^{\prime}$ states are defined by

$\displaystyle\left(\begin{array}[]{c}h_{1}^{\prime}\\\ h_{1}\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos{\theta_{h}}&-\sin{\theta_{h}}\\\
\sin{\theta_{h}}&\cos{\theta_{h}}\end{array}\right)\left(\begin{array}[]{c}\tilde{h}_{8}\\\
\tilde{h}_{1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos{\theta_{h}}&-\sin{\theta_{h}}\\\
\sin{\theta_{h}}&\cos{\theta_{h}}\end{array}\right)\left(\begin{array}[]{cc}\sqrt{\frac{1}{3}}&-\sqrt{\frac{2}{3}}\\\
\sqrt{\frac{2}{3}}&\sqrt{\frac{1}{3}}\end{array}\right)\left(\begin{array}[]{c}h_{n}\\\
h_{s}\\\ \end{array}\right)$ (31) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos{\alpha_{h}}&-\sin{\alpha_{h}}\\\
\sin{\alpha_{h}}&\cos{\alpha_{h}}\end{array}\right)\left(\begin{array}[]{c}h_{n}\\\
h_{s}\\\ \end{array}\right),$ (36)

where $\tilde{h}_{1}\equiv(u\bar{u}+d\bar{d}+s\bar{s})/{\sqrt{3}}$,
$\tilde{h}_{8}\equiv(u\bar{u}+d\bar{d}-2s\bar{s})/{\sqrt{6}}$,
$h_{n}\equiv(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and $h_{s}\equiv s\bar{s}$. The
relation between $\theta_{f}$ ($\theta_{h}$) and $\alpha_{f}$ ($\alpha_{h}$)
is

$\displaystyle\alpha_{f(h)}=\theta_{f(h)}+\arctan{\sqrt{2}}\ .$ (37)

The mixing angles $\theta_{f(h)}$ encode the mechanisms that contribute to the
mass matrices via the mixings between the SU(3) flavor singlet and octet, i.e.
$\tilde{f}_{1}$ and $\tilde{f}_{8}$ ($\tilde{h}_{1}$ and $\tilde{h}_{8}$).

With the help of the Gell-Mann-Okubo relations (see Appendix for a pedagogic
deduction), the octet masses $m_{\tilde{f}_{8}}^{2}$ and
$m_{\tilde{h}_{8}}^{2}$ can be expressed by the states with isospin 1/2 and 1
in the quark model, respectively, i.e.

$\displaystyle m_{\tilde{f}_{8}}^{2}$ $\displaystyle=$
$\displaystyle\frac{4m_{K_{1A}}^{2}-m_{a_{1}}^{2}}{3},$ $\displaystyle
m_{\tilde{h}_{8}}^{2}$ $\displaystyle=$
$\displaystyle\frac{4m_{K_{1B}}^{2}-m_{b_{1}}^{2}}{3},$ (38)

where $K_{1A}$ and $K_{1B}$ are assigned as the ${}^{3}P_{1}$ and
${}^{1}P_{1}$ states, respectively. The mixing angles, $\theta_{f}$ and
$\theta_{h}$, can thus be calculated by

$\displaystyle\tan{\theta_{f}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{m_{\tilde{f}_{8}}^{2}(m_{f}^{2}+m_{f^{\prime}}^{2}-m_{\tilde{f}_{8}}^{2})-m_{f}^{2}m_{f^{\prime}}^{2}}}{m_{\tilde{f}_{8}}^{2}-m_{f}^{2}}=\frac{m_{f^{\prime}}^{2}-m_{\tilde{f}_{8}}^{2}}{\sqrt{m_{\tilde{f}_{8}}^{2}(m_{f}^{2}+m_{f^{\prime}}^{2}-m_{\tilde{f}_{8}}^{2})-m_{f}^{2}m_{f^{\prime}}^{2}}},$
(39) $\displaystyle\tan{\theta_{h}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{m_{\tilde{h}_{8}}^{2}(m_{h}^{2}+m_{h^{\prime}}^{2}-m_{\tilde{h}_{8}}^{2})-m_{h}^{2}m_{h^{\prime}}^{2}}}{m_{\tilde{h}_{8}}^{2}-m_{h}^{2}}=\frac{m_{h^{\prime}}^{2}-m_{\tilde{h}_{8}}^{2}}{\sqrt{m_{\tilde{h}_{8}}^{2}(m_{h}^{2}+m_{h^{\prime}}^{2}-m_{\tilde{h}_{8}}^{2})-m_{h}^{2}m_{h^{\prime}}^{2}}}.$
(40)

One can see from the above relation that given the masses of the physical
states and the octet masses $m_{\tilde{f}_{8}}^{2}$ and
$m_{\tilde{h}_{8}}^{2}$, the mixing angles can be determined. Note that the
octet masses $m_{\tilde{f}_{8}}^{2}$ and $m_{\tilde{h}_{8}}^{2}$ are
determined by the masses of $K_{1A}$ and $K_{1B}$ which, however, are not the
physical masses. The determination of $\theta_{f(h)}$ is thus correlated with
the mixing between $K_{1A}$ and $K_{1B}$.

The corresponding physical states, $K_{1}(1270)$ and $K_{1}(1400)$, as the
mixing states between $K_{1A}$ and $K_{1B}$ (i.e. ${}^{3}P_{1}$ and
${}^{1}P_{1}$) are expressed as

$\displaystyle\left(\begin{array}[]{c}K_{1}(1270)\\\ K_{1}(1400)\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos{\theta_{K_{1}}}&\sin{\theta_{K_{1}}}\\\
-\sin{\theta_{K_{1}}}&\cos{\theta_{K_{1}}}\end{array}\right)\left(\begin{array}[]{c}K_{1B}\\\
K_{1A}\\\ \end{array}\right).$ (47)

As a result, the mixing angles $\theta_{f(h)}$ are now correlated with the
dynamics for the $K_{1A}$ and $K_{1B}$ mixing via their mixing angle
$\theta_{K_{1}}$.

There have been various approaches for the determination of $\theta_{K_{1}}$
in the literature. The early analysis of $\tau$ decay in Ref. Suzuki:1993yc
gave a two-fold solution, i.e. $|\theta_{K_{1}}|=33^{\circ}$ or $57^{\circ}$.
With the help of the constituent quark model, Ref. Blundell:1995au  predicted
$|\theta_{K_{1}}|\sim 45^{\circ}$. In Ref. Burakovsky:1997dd  a range of
$35^{\circ}\leq|\theta_{K_{1}}|\leq 55^{\circ}$ was obtained in the
nonrelativistic quark model. In Ref. Cheng:2003bn
$\theta_{K_{1}}\sim-58^{\circ}$ was shown to be favored in the charmed meson
decays. Analysis of the axial vector decays into a pseudoscalar and a vector
meson suggested $\theta_{K_{1}}=+(62\pm 3)^{\circ}$ Roca:2003uk , which seems
to be confirmed by the conclusion in Ref. Li:2006we . Later, based on the data
for the $B$ and $f_{1}$ radiative decays, $\theta_{K_{1}}$ was determined to
be $-34\pm 13^{\circ}$ by Ref. Yang:2010ah , together with
$\theta_{f}=(19.4^{+4.5}_{-4.6})^{\circ}$. The analysis of Ref. Cheng:2011pb
also favored $|\theta_{K_{1}}|\sim 35^{\circ}$. The sign of $\theta_{K_{1}}$
was discussed in Ref. Dag:2012zz , which led to $\theta_{K_{1}}=-(39\pm
4)^{\circ}$. A combined analysis including all light axial vector mesons was
made in Ref. Divotgey:2013jba , where a good agreement with the experimental
data was achieved with $\theta_{K_{1}}=-(33.6\pm 4.3)^{\circ}$. In Ref.
Cheng:2013cwa  a brief review of the status of $\theta_{K_{1}}$ suggested that
$\theta_{K_{1}}\sim 33^{\circ}$ (less than $\pi/4$) is more favoured than
$57^{\circ}$ (larger than $\pi/4$). Note that the sign ambiguity was removed
by fixing the relative sign of the coupling constants for $K_{1A}$ and
$K_{1B}$ Cheng:2013cwa . The recent analysis of $B^{+}\to J/\psi K_{1}^{+}$
using a pQCD method also obtained $\theta_{K_{1}}\sim 33^{\circ}$
Zhang:2017cbi .

Taking the PDG Patrignani:2016xqp  values for these charge neutral axial
vector states (see Table 1) and adopting several possible solutions for
$\theta_{K_{1}}$ as inputs, we extract the mixing angles $\theta_{f(h)}$ and
$\alpha_{f(h)}$ ($\alpha_{f(h)}=\theta_{f(h)}+\arctan{\sqrt{2}}$) in Table 2.
One can see that the large range of values for $\theta_{K_{1}}$ leads to also
large uncertainties in the predictions of $\theta_{f(h)}$ and $\alpha_{f(h)}$.

Table 1: Physical masses adopted for the axial vector mesons Patrignani:2016xqp . Mesons | Mass (GeV)
---|---
$a_{1}$ | 1.26
$f_{1}$ | 1.285
$f_{1}^{\prime}$ | 1.426
$b_{1}$ | 1.235
$h_{1}$ | 1.17
$h_{1}^{\prime}$ | 1.423
Table 2: Extracted mixing angles for $\theta_{f(h)}$ and $\alpha_{f(h)}$ with typical values for $|\theta_{K_{1}}|$ as the input. $|\theta_{K_{1}}|$ | $33^{\circ}$ | $34^{\circ}$ | $39^{\circ}$ | $45^{\circ}$ | $57^{\circ}$
---|---|---|---|---|---
$\alpha_{f}-90^{\circ}$ | $-6.84^{\circ}$ | $-5.52^{\circ}$ | $0.82^{\circ}$ | $8.17^{\circ}$ | $22.7^{\circ}$
$\alpha_{h}-90^{\circ}$ | $2.45^{\circ}$ | $1.77^{\circ}$ | $-1.80^{\circ}$ | $-6.55^{\circ}$ | $-18.1^{\circ}$
$\theta_{f}$ | $28.4^{\circ}$ | $29.7^{\circ}$ | $36.1^{\circ}$ | $43.4^{\circ}$ | $58.0^{\circ}$
$\theta_{h}$ | $37.7^{\circ}$ | $37.0^{\circ}$ | $33.5^{\circ}$ | $28.7^{\circ}$ | $17.2^{\circ}$

In order to determine $\theta_{f(h)}$ ($\alpha_{f(h)}$) we look for
alternative constraints on the mixing angles. It seems that the $h_{1}$ and
$h_{1}^{\prime}$ decays into a vector and a pseudoscalar meson can set up a
reasonable constraint.

The Lagrangian describing the vertices between the $1^{+-}$ axial vector meson
(B), vector meson (V) and pseudoscalar meson (P) is given by

$\displaystyle L_{BVP}$ $\displaystyle=$ $\displaystyle
g_{BVP}Tr[B^{\mu}\\{V_{\mu},P\\}],$ (48)

where $g_{BVP}$ is the coupling constant and with the SU(3) flavor symmetry
the coupling fields are

$\displaystyle B^{\mu}$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\cos{\alpha_{h}}h_{1}^{\prime}+\sin{\alpha_{h}}h_{1}}{\sqrt{2}}+\frac{b_{1}^{0}}{\sqrt{2}}&b_{1}^{+}&K_{1B}^{+}\\\
b_{1}^{-}&\frac{\cos{\alpha_{h}}h_{1}^{\prime}+\sin{\alpha_{h}}h_{1}}{\sqrt{2}}-\frac{b_{1}^{0}}{\sqrt{2}}&K_{1B}^{0}\\\
K_{1B}^{-}&\bar{K}_{1B}^{0}&-\sin{\alpha_{h}}h_{1}^{\prime}+\cos{\alpha_{h}}h_{1}\end{array}\right),$
(52) $\displaystyle V^{\mu}$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&\frac{-\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right),$ (56) $\displaystyle P$
$\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\cos{\alpha_{P}}\eta+\sin{\alpha_{P}}\eta^{\prime}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\cos{\alpha_{P}}\eta+\sin{\alpha_{P}}\eta^{\prime}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K}^{0}&\sin{\alpha_{P}}\eta+\cos{\alpha_{P}}\eta^{\prime}\end{array}\right),$
(60)

where we parametrize the mixing between $\eta$ and $\eta^{\prime}$ as

$\displaystyle\left(\begin{array}[]{c}\eta\\\ \eta^{\prime}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos{\alpha_{P}}&-\sin{\alpha_{P}}\\\
\sin{\alpha_{P}}&\cos{\alpha_{P}}\end{array}\right)\left(\begin{array}[]{c}\eta_{n}\\\
\eta_{s}\\\ \end{array}\right).$ (67)

The couplings for an vector meson to two pseudoscalars have the following
form:

$\displaystyle L_{VPP}$ $\displaystyle=$ $\displaystyle
ig_{VPP}Tr[V^{\mu}(\partial_{\mu}PP-P\partial_{\mu}P)],$ (68)

where $g_{VPP}$ is the coupling constant and can be calculated by $K^{*}\to
K\pi$ or $\phi\to K\bar{K}$. With the data for $\phi\to K\bar{K}$
Patrignani:2016xqp , one determines $g_{VPP}=4.52$. To determine the coupling
$g_{BVP}$, we assume that the total width of $h_{1}(1170)$ is saturated by
$h_{1}\to\rho\pi\to\pi^{+}\pi^{-}\pi^{0}$ for which the amplitude reads

$\displaystyle M_{h_{1}\to\rho\pi\to\pi^{+}\pi^{-}\pi^{0}}$ $\displaystyle=$
$\displaystyle
ig_{h_{1}\rho\pi}i^{2}g_{\rho\pi\pi}\epsilon_{\mu}\frac{i}{s_{cd}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}}i[(1+\frac{s_{d}-s_{a}}{s_{cd}})p_{a}^{\mu}+(-1+\frac{s_{d}-s_{a}}{s_{ad}})p_{d}^{\mu}]$
(69) $\displaystyle+$ $\displaystyle
ig_{h_{1}\rho\pi}i^{2}g_{\rho\pi\pi}\epsilon_{\mu}\frac{i}{s_{ab}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}}i[(-1+\frac{s_{a}-s_{b}}{s_{ab}})p_{a}^{\mu}+(1+\frac{s_{a}-s_{b}}{s_{ab}})p_{b}^{\mu}]$
$\displaystyle+$ $\displaystyle
ig_{h_{1}\rho\pi}i^{2}g_{\rho\pi\pi}\epsilon_{\mu}\frac{i}{s_{bd}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}}i[(-1+\frac{s_{b}-s_{d}}{s_{bd}})p_{b}^{\mu}+(1+\frac{s_{b}-s_{d}}{s_{bd}})p_{d}^{\mu}],$

where the momenta of $\pi^{+}$, $\pi^{0}$, $\rho$ and $\pi^{-}$ mesons are
denoted as $p_{a}$, $p_{b}$, $p_{c}$ and $p_{d}$, respectively, and
$s_{ad}\equiv(p_{a}+p_{d})^{2}$, $s_{ab}\equiv(p_{a}+p_{b})^{2}$,
$s_{bd}\equiv(p_{b}+p_{d})^{2}=s_{1}+s_{b}-s_{ad}+s_{a}+s_{d}-s_{ab}$. Given
$\Gamma_{h_{1}\to\rho\pi\to\pi^{+}\pi^{-}\pi^{0}}=0.37$ GeV from the PDG as
input, one extracts $g_{h_{1}\rho\pi}=4.28$ GeV.

Note that the following relation holds,

$g_{h_{1}\rho\pi}\equiv\sqrt{2}g_{BVP}\sin{\alpha_{h}}.$ (70)

It suggests that another relation is needed in order to determine $g_{BVP}$
and $\alpha_{h}$ in the $1^{+-}$ sector. In principle, with the data for
$h_{1}^{\prime}\to\rho\pi\to\pi^{+}\pi^{-}\pi^{0}$, one will be able to
determine these two quantities. However, the branching ratio of
$h_{1}^{\prime}\to\rho\pi\to\pi^{+}\pi^{-}\pi^{0}$ has not been well
established. To get around of this problem, it shows that the total width
measurement may provide a reliable estimate given that it is dominated ( or
nearly saturated) by the $\rho\pi$ and $K^{*}\bar{K}$ channels. Since these
two channels can be related to each other by the SU(3) flavor symmetry, one
has

$\displaystyle g_{h_{1}^{\prime}\rho\pi}$ $\displaystyle=$
$\displaystyle\sqrt{2}g_{BVP}\cos{\alpha_{h}}=g_{h_{1}\rho\pi}\cot{\alpha_{h}}\
,$ (71) $\displaystyle g_{h_{1}^{\prime}K^{*}\bar{K}}$ $\displaystyle=$
$\displaystyle
g_{BVP}\left(\frac{\sqrt{2}}{2}\cos{\alpha_{h}}-\sin{\alpha_{h}}\right)$ (72)
$\displaystyle=$
$\displaystyle\frac{1}{2}g_{h_{1}\rho\pi}(\cot{\alpha_{h}}-\sqrt{2})\ .$ (73)

Then, the total width of $h_{1}^{\prime}$ will be calculated as the sum of the
partial widths of the $\rho\pi$ and $K^{*}\bar{K}$ channels.

The BESIII Collaboration recently measured the total width of
$h_{1}^{\prime}$, i.e. $\Gamma_{h^{\prime}}=90\pm 9.8\pm 17.5$ MeV
Ablikim:2018ctf . We find that with $\theta_{K_{1}}=34^{\circ}$ (i.e.
$\alpha_{h}=91.77^{\circ}$) and $g_{h_{1}\rho\pi}=4.28$ GeV, the parameter
relations give $g_{BVP}=3.03$ GeV and $g_{h_{1}^{\prime}K^{*}\bar{K}}=-3.09$
GeV. Thus, the partial width of $h_{1}^{\prime}\to K^{*}\bar{K}\to
K\bar{K}\pi$ is estimated to be

$\Gamma_{h_{1}^{\prime}\to K^{*}\bar{K}\to K\bar{K}\pi}\simeq
6\Gamma_{h_{1}^{\prime}\to K^{*}\bar{K}\to K^{+}K^{-}\pi^{0}}=55.9\
\text{MeV}.$ (74)

Together with the contributions from the other possible decay channels, the
total width of $h_{1}^{\prime}$ can be reasonably described. We also mention
that with higher values for $|\theta_{K_{1}}|$ in Table 2, e.g.
$|\theta_{K_{1}}|=57^{\circ}$, the partial width for
$h_{1}^{\prime}\to\rho\pi\to\pi^{+}\pi^{-}\pi^{0}$ will be large and in
contradiction with the experimental measurements. So, we adopt
$\theta_{K_{1}}=34^{\circ}$ as the input for the later calculations.

##  III Production and decay of axial vector mesons with $J^{PC}=1^{++}$

###  III.1 Key issues

In our previous study Wu:2012pg  it was shown that the axial vector meson
$f_{1}^{\prime}$ should have contributions to the isospin-violating decay of
$J/\psi\to\gamma+3\pi$ which was measured by BESIII BESIII:2012aa . The
observable of the angular distribution of the pion recoiling the $f_{0}(980)$
should contain a non-negligible $S$-wave contribution apart from the dominant
$P$-wave from the intermediate $\eta$ resonances. Different from the treatment
of Ref. Wu:2012pg  where an arbitrary coupling strength was determined by
fitting the angular distribution, we will quantify the production and decay of
both $f_{1}$ and $f_{1}^{\prime}$ in the $J/\psi$ radiative decays into
$\gamma\eta\pi\pi$, $\gamma K\bar{K}\pi$ and $\gamma+3\pi$ with the presence
of the TS mechanism.

In Ref. APFB-2017  a preliminary result was reported and it showed that the
$f_{1}(1420)$ as a pole structure should account for the $S$-wave enhancements
observed in the invariant mass spectra of $\eta\pi\pi$ and $K\bar{K}\pi$ in
$J/\psi\to\eta\pi\pi$ and $K\bar{K}\pi$, respectively. Later, it was claimed
in Ref. Debastiani:2016xgg  that the $f_{1}(1420)$ enhancement was not a
genuine state and it could be produced by the TS mechanism due to the
$f_{1}(1285)$ pole. This is an interesting scenario since it, on the one hand,
shows that the TS mechanism plays an non-negligible role in various processes
indeed. And on the other hand, it also raises questions on the structure of
$f_{1}(1420)$. In Refs. Roca:2005nm ; Roca:2006tr  a systematic study of the
$S$-wave vector and peudoscalar meson interactions led to dynamically
generated pole structures which can be associated with the PDG listed states,
i.e. $b_{1}(1235),\ h_{1}(1170),\ h_{1}(1380),\ a_{1}(1260)$, and
$f_{1}(1285)$. It was also mentioned there that the $f_{1}(1420)$ could not be
accommodated in their scheme. In Ref. Debastiani:2016xgg  the $f_{1}(1420)$
was proposed to be an enhanced structure by the TS mechanism which was
originated from the $f_{1}(1285)$ as a dynamically generated state by the
$K^{*}\bar{K}+c.c.$ $S$-wave interaction. The authors of Ref.
Debastiani:2016xgg  obtained the lineshape of the invariant mass by fitting
the data from WA102 Barberis:1998by  with an arbitrary strength undetermined.
Although this seems to be consistent with the result of Ref. Roca:2005nm , it
would leave crucial questions in the understanding the overall axial vector
spectrum. In particular, by treating $f_{1}(1285)$ as a dynamically generated
state due to the $S$-wave $K^{*}\bar{K}+c.c.$ interaction would bring
conflicts to some of those experimental observables for $f_{1}^{\prime}$ which
fit quite well the flavor singlet and octet mixing pattern Barberis:1998by ;
Close:1997nm ; Patrignani:2016xqp .

To proceed, we first point out that the dominant decay mode of $f_{1}(1285)$
is $\eta\pi\pi$ with $B.R.(f_{1}\to\eta\pi\pi)=(52.4^{+1.9}_{-2.2})\%$, among
which $B.R.(f_{1}\to a_{0}(980)\pi)=(36\pm 7)\%$ Patrignani:2016xqp . Note
that the branching ratio $B.R.(f_{1}\to a_{0}(980)\pi)=(36\pm 7)\%$ does not
include the contributions from the $a_{0}(980)\to K\bar{K}$ Patrignani:2016xqp
. It means that the real value should be larger than this and it explicitly
indicates the dominant mode of $f_{1}\to a_{0}(980)\pi$. In contrast, the
branching ratio for $f_{1}\to K\bar{K}\pi$ is $B.R.(f_{1}\to
K\bar{K}\pi)=(9.0\pm 0.4)\%$ and the intermediate decay mode of $f_{1}\to
K^{*}\bar{K}+c.c.$ has not been observed. Although the phase space would limit
the partial width of $f_{1}\to K^{*}\bar{K}+c.c.$, it suggests that the
$f_{1}$ coupling to $K^{*}\bar{K}$ does not necessarily to be strong. This
actually jeopardizes the conclusion of Ref. Debastiani:2016xgg  where it has
been assumed that the $K^{*}\bar{K}+c.c.$ decay channel had saturated the
$K\bar{K}\pi$ mode for $f_{1}(1285)$. Another caveat in Ref.
Debastiani:2016xgg  is that the invariant mass spectrum measured by WA102
cannot constrain the relative coupling strength for the production of $f_{1}$
and $f_{1}^{\prime}$ and this will be correlated with the couplings for their
decays into exclusive channels (Note that their mixing angle is also a
variable to be determined). These remind us of a coherent study of both
$f_{1}$ and $f_{1}^{\prime}$ in the exclusive process of the $J/\psi$
radiative decays where both the production and decay mechanisms can be
properly evaluated.

For the coherent study of the axial vector mesons with $C=1^{++}$ the
following observations can be itemized:

  * •

The available experimental results suggest that the $f_{1}^{\prime}$
dominantly decays into the $K\bar{K}\pi$ final state via $K^{*}\bar{K}+c.c.$
and its decays into $\eta\pi\pi$ is much suppressed.

  * •

The $f_{1}$ dominantly decays into $\eta\pi\pi$ via the intermediate
$a_{0}\pi$ process and its decay into $K\bar{K}\pi$ is also via the
$a_{0}\pi$.

  * •

The mixing between $f_{1}$ and $f_{1}^{\prime}$ will affect both production
and decay processes.

  * •

The $S$-wave coupling to $K^{*}\bar{K}+c.c.$ introduces the effects from the
TS mechanism which needs to be coherently investigated for both $f_{1}$ and
$f_{1}^{\prime}$ in exclusive processes.

  * •

The $S$-wave coupling to $K^{*}\bar{K}+c.c.$ implies the TS enhancement in the
isospin-1 channel which refers to the $a_{1}(1420)$.

###  III.2 Productions of the $C=+1$ axial vector mesons in charmonium decays

The isoscalar $f_{1}$ and $f_{1}^{\prime}$ can be produced in the $J/\psi$
radiative decays. In their three-body decays, e.g. $f_{1}\to\eta\pi\pi$ and
$K\bar{K}\pi$, both states can contribute. It thus requires a coherent study
of these two states in each channel. We parametrize the production amplitudes
using the following effective Lagrangian for $J/\psi\to\gamma A$ (here, $A$
denotes $f_{1}$ or $f_{1}^{\prime}$):

$\displaystyle L_{J/\psi\gamma
A}=g_{A}\epsilon_{\mu\nu\rho\sigma}\partial^{\mu}\psi^{\nu}\gamma^{\rho}A^{\sigma}\
,$ (75)

which gives the amplitude

$\displaystyle M_{J/\psi\gamma
A}=g_{A}\epsilon_{\mu\nu\rho\sigma}p_{\psi}^{\mu}\epsilon_{\psi}^{\nu}\epsilon_{\gamma}^{\rho}\epsilon_{A}^{\sigma}\
,$ (76)

where $\epsilon_{\psi}^{\nu}$, $\epsilon_{\gamma}^{\rho}$, and
$\epsilon_{A}^{\sigma}$ are the polarization vectors for $J/\psi$, the photon
and the axial vector meson, respectively. The production strength for $f_{1}$
and $f_{1}^{\prime}$ in the $J/\psi$ radiative decays can be related to each
other in the SU(3) flavor symmetry limit:

$\displaystyle\frac{g_{f}^{\prime}}{g_{f}}=\frac{\langle
f_{1}^{\prime}|\hat{H}_{\gamma}|J/\psi\rangle}{\langle
f_{1}|\hat{H}_{\gamma}|J/\psi\rangle}=\frac{\sqrt{2}\cos{\alpha_{f}}-\sin{\alpha_{f}}}{\sqrt{2}\sin{\alpha_{f}}+\cos{\alpha_{f}}},$
(77)

where we have assumed that
$\langle(s\bar{s})_{1^{++}}|\hat{H}_{\gamma}|J/\psi\rangle=\langle(u\bar{u})_{1^{++}}|\hat{H}_{\gamma}|J/\psi\rangle=\langle(d\bar{d})_{1^{++}}|\hat{H}_{\gamma}|J/\psi\rangle$.
Namely, we neglect the SU(3) symmetry breaking with the couplings for the
light pair creations. Then, the amplitude of
$J/\psi\to\gamma(f_{1}+f_{1}^{\prime})\to\gamma ABC$ can be parametrized as

$\displaystyle M_{J/\psi\to\gamma(f_{1}+f_{1}^{\prime})\to\gamma
ABC}=M_{J/\psi\gamma f_{1}}\left(\frac{i}{D_{f_{1}}}M_{f_{1}\to
ABC}+\frac{g_{f}^{\prime}}{g_{f}}\frac{i}{D_{f_{1}^{\prime}}}M_{f_{1}^{\prime}\to
ABC}\right),$ (78)

where $M_{f_{1}\to ABC}$ and $M_{f_{1}^{\prime}\to ABC}$ are the decay
amplitudes for $f_{1}$ and $f_{1}^{\prime}$, respectively, and will be
explicitly calculated in the next subsection.

For the convenience of calculation, we define $M_{f_{1}^{\prime}\to
ABC}=\sum_{i}M_{f_{1}^{\prime}\to ABC}^{(i)}$ and $M_{f_{1}^{\prime}\to
ABC}^{(i)}=\chi_{i}M_{f_{1}\to ABC}^{(i)}$, where $i$ denotes the $i$-th
intermediate state. Then, the spectrum can be expressed as the following form:

$\displaystyle\frac{d\Gamma}{d\sqrt{s}}=\frac{2s}{\pi}\Gamma_{J/\psi\to\gamma
f_{1}}\frac{1}{2\sqrt{s}}\int
d\Phi_{ABC}\frac{1}{3}|\sum_{i}\left(\frac{i}{D_{f_{1}}}+\frac{\chi_{i}g_{f}^{\prime}}{g_{f}}\frac{i}{D_{f_{1}^{\prime}}}\right)M_{f_{1}\to
ABC}^{(i)}|^{2},$ (79)

where the $\Phi_{ABC}$ is the phase space for the three-body $ABC$ in
$J/\psi\to\gamma ABC$; Functions $D_{f_{1}}$ and $D_{f_{1}^{\prime}}$ are the
inverse propagators of $f_{1}$ and $f_{1}^{\prime}$, respectively, i.e.

$\displaystyle D_{f_{1}}$ $\displaystyle=$ $\displaystyle
s-m_{f_{1}}^{2}+im_{f_{1}}\Gamma_{f_{1}},$ (80) $\displaystyle
D_{f_{1}^{\prime}}$ $\displaystyle=$ $\displaystyle
s-m_{f_{1}^{\prime}}^{2}+im_{f_{1}^{\prime}}\Gamma_{f_{1}^{\prime}}(s),$ (81)

where $\Gamma_{f_{1}}=22.7\ \text{MeV}$ and
$\Gamma_{f_{1}^{\prime}}=\Gamma_{f_{1}^{\prime}\to
K\bar{K}\pi}+\Gamma_{f_{1}^{\prime}\to a_{0}\pi\to\eta\pi\pi}$, are adopted.

Different from the production of $f_{1}/f_{1}^{\prime}$ the production of
$a_{1}$ in the $J/\psi$ radiative decays will be suppressed due to the
isospin. To produce $a_{1}$ in the $J/\psi$ radiative decays it needs to go
through the final state photon radiation. This is relatively suppressed by the
additional gluon exchange in $c\bar{c}$ annihilations. In $e^{+}e^{-}$
annihilations the ideal process to probe the production of $a_{1}$ is via
$\chi_{c1}\to\pi a_{1}(1260)/a_{1}(1420)\to 4\pi$, where $\chi_{c1}$ can be
copiously produced through $\psi(3686)\to\gamma\chi_{c1}$. For the production
of $\chi_{c1}\to\pi a_{1}$ it can be described by the following Lagrangian:

$L_{\chi
AP}=ig_{a}\epsilon_{\mu\nu\rho\sigma}\partial^{\mu}\chi_{c1}^{\nu}\partial^{\rho}A^{\sigma}P\
,$ (82)

where $g_{a}$ is the coupling strength for producing a pair of
$(q\bar{q})_{1^{++}}$ and $(q\bar{q})_{0^{-+}}$ in the $\chi_{c1}$ decays.

In principle, $g_{a}$ can be determined by the partial decay width for
$\chi_{c1}\to\pi a_{1}(1260)$. Note that this quantity is correlated with the
measurement of the exclusive process with the $a_{1}$ decays into final stable
particles, e.g. $a_{1}\to 3\pi$. At this moment, the data are not available.
However, for the purpose of determining the lineshape as a characteristic
feature $g_{a}$ can be treated as an overall parameter for the production
strength. So we need only consider the $3\pi$ spectrum in the analysis which
can be described as

$\displaystyle\frac{d\Gamma}{d\sqrt{s}}\sim\frac{2s}{\pi}\frac{\Gamma_{a_{1}\to
ABC}(s)}{(s-m_{a_{1}(1260)})^{2}+s\Gamma_{a_{1}}(s)},$ (83)

where the $\Gamma_{a_{1}\to ABC}$ represents the energy-dependent width for
$a_{1}^{-}\to\rho^{0}\pi^{-}\to\pi^{+}\pi^{-}\pi^{0}$ or $a_{1}^{-}\to
f_{0}\pi^{-}\to\pi^{+}\pi^{-}\pi^{0}$. The energy-dependent total width
$\Gamma_{a_{1}}(s)$ is estimated by

$\Gamma_{a_{1}}(s)\simeq\Gamma_{a_{1}\to\rho\pi\
(S-\text{wave})}/B.R.(a_{1}\to\rho\pi\ (S-\text{wave}))\ .$ (84)

Note that both the total width and partial decay width
$\Gamma_{a_{1}\to\rho\pi\ (S-\text{wave})}$ are far from well established for
$a_{1}(1260)$ Patrignani:2016xqp , although the early measurement by the CLEO
Collaboration suggested $B.R.(a_{1}\to\rho\pi\ (S-\text{wave}))\simeq 0.6$
Asner:1999kj . This will prevent us from extracting the absolute value for the
partial width for $\chi_{c1}\to\pi a_{1}(1260)$, but will not affect the study
of the lineshape since both the total width and partial decay width are not
expected to be sensitive to the invariant mass spectrum of the $3\pi$ in the
range of 1.2$\sim$ 1.4 GeV. Moreover, one notices that the coupling for
$a_{1}\to\rho\pi$ can be related to that for $a_{1}\to K^{*}\bar{K}$ with the
SU(3) flavor symmetry. This allows us to estimate the relative significance of
the TS mechanism in the $3\pi$ spectrum in comparison with the tree process of
$a_{1}\to\rho\pi\to 3\pi$. This will be shown in the study of $a_{1}\to 3\pi$
later.

###  III.3 Decay mechanisms for $f_{1}$ and $f_{1}^{\prime}$ into
$\eta\pi\pi$, $K\bar{K}\pi$, and $3\pi$

The decay mechanisms for $f_{1}$ and $f_{1}^{\prime}$ into $\eta\pi\pi$,
$K\bar{K}\pi$, and $3\pi$ are grouped into two types. One is the tree-level
transitions and the other one is loop transitions via triangle rescatterings.
We first illustrated the mechanisms by Feynman diagrams in Figs. 1 and 2 for
the decay channels into $\eta\pi\pi$ and $K\bar{K}\pi$. The isospin-violating
decay of $f_{1}/f_{1}^{\prime}\to 3\pi$ is illustrated by Fig. 3 and will be
discussed later.

For both decay channels of $\eta\pi\pi$ and $K\bar{K}\pi$ the tree-level
couplings are originated from the quark model axial vector meson couplings to
$a_{0}(980)\pi$ in a $P$-wave and to $K^{*}\bar{K}+c.c.$ in an $S$-wave. The
strong $S$-wave $K^{*}\bar{K}$ interaction will have two major physical
consequences. One is to dress the bare states and the masses of the physical
states correspond to the experimentally measured values. The second one is
that the physical couplings for $f_{1}$ ($f_{1}^{\prime}$) to $a_{0}\pi$ will
be renormalized by the triangle transitions. In the $f_{1}^{\prime}$ decays
the kinematics of the triangle transitions are located within the physical
range of the TS mechanism. Thus, one realizes that the inclusion of the TS
contributions is necessary. It implies that the physical couplings defined for
$f_{1}\ (f_{1}^{\prime})\to a_{0}(980)\pi$ should include both tree-level
couplings and loop contributions. They will be constrained by experimental
data in the combined analysis.

Figure 1: Diagrams for $f_{1}(f_{1}^{\prime})\to a_{0}\pi\to\eta\pi\pi$.

Figure 2: Diagrams for $f_{1}(f_{1}^{\prime})\to K\bar{K}\pi$.

The following typical Lagrangians are needed in the transition amplitudes for
Figs. 1 and 2:

(i) Following the convention of Eq. (36) the Lagrangian for the $f_{1}\
(f_{1}^{\prime})$ coupling to the vector ($V$) and pseudoscalar ($P$) mesons
is expressed as

$\displaystyle L_{1}=ig_{AVP}Tr[A^{\mu}[V_{\mu},P]].$ (85)

where $A$ denotes the axial vector matrix on the SU(3) basis, i.e.

$\displaystyle
A^{\mu}=\left(\begin{array}[]{ccc}\frac{\cos{\alpha_{f}}f_{1}^{\prime}+\sin{\alpha_{f}}f_{1}}{\sqrt{2}}+\frac{a_{1}^{0}}{\sqrt{2}}&a_{1}^{+}&K_{1A}^{+}\\\
a_{1}^{-}&\frac{\cos{\alpha_{f}}f_{1}^{\prime}+\sin{\alpha_{f}}f_{1}}{\sqrt{2}}-\frac{a_{1}^{0}}{\sqrt{2}}&K_{1A}^{0}\\\
K_{1A}^{-}&\bar{K}_{1A}^{0}&-\sin{\alpha_{f}}f_{1}^{\prime}+\cos{\alpha_{f}}f_{1}\end{array}\right).$
(89)

The resulting Lagrangians for $f_{1}(f_{1}^{\prime})\to K^{*}\bar{K}$ are

$\displaystyle
L_{f_{1}}=ig_{AVP}\frac{-2\cos{\alpha_{f}}+\sqrt{2}\sin{\alpha_{f}}}{2}f_{1}^{\mu}(K^{*0}_{\mu}\bar{K}^{0}-\bar{K}^{*0}_{\mu}K^{0}+K^{*+}_{\mu}K^{-}-K^{*-}_{\mu}K^{+}),$
(90) $\displaystyle
L_{f_{1}^{\prime}}=ig_{AVP}\frac{\sqrt{2}\cos{\alpha_{f}}+2\sin{\alpha_{f}}}{2}f_{1}^{\prime\mu}(K^{*0}_{\mu}\bar{K}^{0}-\bar{K}^{*0}_{\mu}K^{0}+K^{*+}_{\mu}K^{-}-K^{*-}_{\mu}K^{+})\
.$ (91)

This allows us to define the leading order physical couplings for
$f_{1}/f_{1}^{\prime}$ to $K^{*}\bar{K}$, i.e.

$\displaystyle g_{f_{1}K^{*0}\bar{K}^{0}}$ $\displaystyle\equiv$
$\displaystyle ig_{AVP}\frac{-2\cos{\alpha_{f}}+\sqrt{2}\sin{\alpha_{f}}}{2}\
,$ $\displaystyle g_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}$ $\displaystyle\equiv$
$\displaystyle ig_{AVP}\frac{\sqrt{2}\cos{\alpha_{f}}+2\sin{\alpha_{f}}}{2}.$
(92)

With $\theta_{K_{1}}=34^{\circ}$ adopted as the input, the mixing angle has a
value of $\alpha_{f}=90^{\circ}-5.52^{\circ}=84.48^{\circ}$.

(ii) The Lagrangians for $f_{1}\ (f_{1}^{\prime})$ coupling to $a_{0}\pi$ have
the following form:

$\displaystyle L_{f_{1}a_{0}\pi}$ $\displaystyle=$ $\displaystyle
g_{ASP}\sin{\alpha_{f}}f_{1}^{\mu}(\pi\partial_{\mu}a_{0}-a_{0}\partial_{\mu}\pi)\
,$ (93) $\displaystyle L_{f_{1}^{\prime}a_{0}\pi}$ $\displaystyle=$
$\displaystyle
g_{ASP}\cos{\alpha_{f}}f_{1}^{\prime\mu}(\pi\partial_{\mu}a_{0}-a_{0}\partial_{\mu}\pi)\
.$ (94)

With $\alpha_{f}=84.48^{\circ}$ the coupling $g_{f_{1}^{\prime}a_{0}\pi}$ is
significantly suppressed with respect of $g_{f_{1}a_{0}\pi}$ by a factor of
$\cot{\alpha_{f}}\simeq 0.1$. In the numerical calculations we find that Fig.
2 (a) can be neglected in $f_{1}^{\prime}\to K\bar{K}\pi$. The TS effect and
the intermediate $a_{0}(980)$ resonance is able to influence the $K\bar{K}$
spectrum in $f_{1}^{\prime}\to K\bar{K}\pi$ at the lower end. But the
significance would depend on the relative phase between the two remained
amplitudes (i.e. Fig. 2 (b) and (c)), which arises from the coupling
$g_{a_{0}K\bar{K}}$. This phase can be obtained in
$J/\psi\to\gamma\eta(1405)\to\gamma K\bar{K}\pi$ as shown in Ref. Du:2019idk .

As follows, we provide the transition amplitudes for each decay processes in
Figs. 1 and 2.

####  III.3.1 $f_{1}^{\prime}/f_{1}\to\eta\pi\pi$

In this subsection we discuss the detailed transition mechanisms for
$f_{1}^{\prime}\to\eta\pi\pi$. Then, the corresponding formulas for
$f_{1}\to\eta\pi\pi$ can be obtained by a simple replacement of the vertex
coupling constants.

For $f_{1}^{\prime}\to\eta\pi\pi$ the amplitudes of the tree (Fig. 1 (a)) and
triangle diagram (Fig. 1 (b)) have the following expressions, respectively,

$\displaystyle M_{\eta\pi\pi}^{tree}$ $\displaystyle=$
$\displaystyle\epsilon_{f_{1}^{\prime}\mu}\frac{1}{\sqrt{2}}ig_{f^{\prime}a_{0}\pi}ig_{a_{0}\eta\pi^{0}}\frac{i}{D_{a_{0}}}i(p_{a}^{\mu}+p_{d}^{\mu}-p_{b}^{\mu})+(b\leftrightarrow
d).$ (95)

and

$\displaystyle M_{\eta\pi\pi}^{loop}$ $\displaystyle=$
$\displaystyle\epsilon_{f_{1}^{\prime}\mu}\frac{1}{\sqrt{2}}ig_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}ig_{a_{0}K^{0}\bar{K}^{0}}ig_{a_{0}\eta\pi^{0}}2(\hat{I}^{(n)\mu}+\hat{I}^{(c)\mu})\frac{i}{D_{a_{0}}}+(b\leftrightarrow
d)$ (96) $\displaystyle=$
$\displaystyle\epsilon_{f_{1}^{\prime}\mu}\frac{1}{\sqrt{2}}ig_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}ig_{a_{0}K^{0}\bar{K}^{0}}ig_{a_{0}\eta\pi^{0}}4\hat{I}^{\mu}\frac{i}{D_{a_{0}}}+(b\leftrightarrow
d),$

where $\hat{I}^{(n)\mu}$ and $\hat{I}^{(c)\mu}$ are the loop functions defined
for the triangle diagrams in Figs. 1 and 2. The superscripts $n$ and $c$
denote the charge-neutral or charged intermediate mesons. We will give the
expression of $\hat{I}^{\mu}$ later in this section. The difference between
$\hat{I}^{(n)\mu}$ and $\hat{I}^{(c)\mu}$ is due to the slightly different
masses between the charged and neutral kaons, or between the charged and
neutral $K^{*}$. We will see later that this gives a novel source of isospin
breaking via the TS mechanism. In the above equation the $\eta$ momentum is
labelled by $p_{a}$, and the momenta for the rest two pions by $p_{b}$ and
$p_{d}$, respectively. We also note that for $f^{\prime}\to\eta\pi\pi$, Fig. 1
(a) is insignificant compared to the triangle amplitude for $f^{\prime}\to
a_{0}\pi\to\eta\pi^{0}\pi^{0}$ (Fig. 1 (b)).

The $a_{0}(980)$ resonance is described by a unitary propagator

$\displaystyle\frac{i}{D_{a_{0}(k^{2})}}$ $\displaystyle\equiv$
$\displaystyle\frac{i}{k^{2}-m_{a_{0}}^{2}-i\sum_{ab}g_{a_{0}ab}^{2}\Pi_{ab(k^{2})}},$
(97)

where

$\displaystyle\Pi_{ab(k^{2})}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{((q-k)^{2}-m_{a}^{2})(q^{2}-m_{b}^{2})},$
(98)

and $ab\in\\{\eta\pi,K^{0}\bar{K}^{0},K^{+}K^{-}\\}$, $g_{a_{0}ab}$ is the
coupling constant of $a_{0}(980)\to a+b$ and $k$ is the four-vector momentum
of $a_{0}$. We mention in advance that a similar treatment will also be
adopted for $f_{0}(980)$ in this work, for which the propagator is

$\displaystyle\frac{i}{D_{f_{0}(s)}}=\frac{i}{s-m_{f_{0}}^{2}-i\sum_{ab}g_{f_{0}ab}^{2}\Pi_{ab}}$
(99)

with $ab=\\{K^{+}K^{-},K^{0}\bar{K}^{0},\pi^{+}\pi^{-},\pi^{0}\pi^{0}\\}$.

For the decays of $f_{1}$, the expression is similar and we just need to
replace $g_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}$ and $g_{f_{1}^{\prime}a_{0}\pi}$
by $g_{f_{1}K^{*0}\bar{K}^{0}}$ and $g_{f_{1}a_{0}\pi}$, respectively.

####  III.3.2 $f_{1}^{\prime}\to K\bar{K}\pi$

For $f_{1}^{\prime}\to K\bar{K}\pi$ we denote the momenta of $K$, $\bar{K}$
and $\pi$ as $p_{a}$, $p_{d}$ and $p_{b}$, respectively. Thus, the transition
amplitudes for the processes of Fig. 2 can be obtained as follows:

$\displaystyle M_{K\bar{K}\pi}^{tree-1}$ $\displaystyle=$
$\displaystyle\epsilon_{f_{1}^{\prime}\mu}ig_{f_{1}^{\prime}a_{0}\pi}ig_{a_{0}K^{0}\bar{K}^{0}}\frac{i}{D_{a_{0}}}i(p_{a}^{\mu}+p_{d}^{\mu}-p_{b}^{\mu}),$
(100) $\displaystyle M_{K\bar{K}\pi}^{tree-2}$ $\displaystyle=$ $\displaystyle
ig_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi}\epsilon_{f_{1}^{\prime}\mu}\frac{i}{D_{K^{*}}}(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}})i(p_{a}^{\mu}-p_{b}^{\mu})+(a\rightarrow
d),$ (101)

with $q=p_{a}+p_{d}$, and

$\displaystyle M_{K\bar{K}\pi}^{loop}$ $\displaystyle=$
$\displaystyle\epsilon_{f_{1}^{\prime}\mu}ig_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}ig_{a_{0}K^{0}\bar{K}^{0}}ig_{a_{0}K^{0}\bar{K}^{0}}2(\hat{I}^{(n)\mu}+\hat{I}^{(c)\mu})\frac{i}{D_{a_{0}}}$
(102) $\displaystyle=$
$\displaystyle\epsilon_{f_{1}^{\prime}\mu}ig_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}ig_{a_{0}K^{0}\bar{K}^{0}}ig_{a_{0}K^{0}\bar{K}^{0}}4\hat{I}^{\mu}\frac{i}{D_{a_{0}}}.$

The total amplitude is

$\displaystyle M_{f^{\prime}\to
K\bar{K}\pi}=M_{K\bar{K}\pi}^{tree-1}+M_{K\bar{K}\pi}^{tree-2}+M_{K\bar{K}\pi}^{loop}.$
(103)

Note that the contribution of $M_{K\bar{K}\pi}^{tree-1}$ is much smaller than
the other two terms due to the small $f_{1}^{\prime}a_{0}\pi$ coupling. So,
the sum of Fig. 2 (b) and (c) can be a reasonable approximation for
$f^{\prime}\to K\bar{K}\pi$.

####  III.3.3 $f_{1}^{\prime}\to f_{0}(980)\pi\to\pi^{+}\pi^{-}\pi^{0}$

The isospin-violating process $f_{1}^{\prime}\to
f_{0}(980)\pi\to\pi^{+}\pi^{-}\pi^{0}$ can go through the $a_{0}-f_{0}$ mixing
via a tree-level $a_{0}$ production (Fig. 3 (a)) and the triangle mechanism
(Figs. 3 (b) and (c)). This is very similar to the case of the isospin-
violating decay of $\eta(1405/1475)\to 3\pi$ Wu:2011yx ; Wu:2012pg ;
Du:2019idk . In Ref. Wu:2012pg  the process $f_{1}^{\prime}\to 3\pi$ in
$J/\psi\to\gamma+3\pi$ was first investigated which contains the contributions
from Figs. 3 (a) and (b). It was also found that Fig. 3 (b) dominates over
(a), i.e. the isospin-breaking effects are mainly from the TS mechanism. An
improvement of the study of $\eta(1405/1475)\to 3\pi$ by Ref. Du:2019idk
suggests that the contributions from Fig. 3 (c) (the initial state will be
$\eta(1405/1475)$ in this case) is also non-negligible. Due to the TS
mechanism the production of $a_{0}$ will be strongly enhanced by the triangle
diagram. Thus, the $a_{0}-f_{0}$ mixing will also be enhanced. In this work we
will include the mechanism of Fig. 3 (c) as a complete and self-consistent
treatment of the isospin-violating decays of $f_{1}$ and $f_{1}^{\prime}$.

Figure 3: Diagrams for $f_{1}^{\prime}\to
f_{0}(980)\pi\to\pi^{+}\pi^{-}\pi^{0}$ channel.

The decay amplitudes of Figs. 3 (a), (b) and (c) read as follows,
respectively,

$M_{f_{1}\to
3\pi}^{treemix}=ig_{f_{1}a_{0}\pi}ig_{a_{0}K^{+}K^{-}}ig_{f_{0}\pi^{+}\pi^{-}}\frac{i}{D_{a_{0}}}(\hat{\mathcal{I}}^{(c)}-\hat{\mathcal{I}}^{(n)})\frac{i}{D_{f_{0}}}\
,$ (104)

and

$\displaystyle M_{f_{1}\to
3\pi}^{loop}=2ig_{f_{1}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}ig_{f_{0}K^{0}\bar{K}^{0}}ig_{f_{0}\pi^{+}\pi^{-}}\epsilon_{f\mu}(\hat{I}^{(n)\mu}-\hat{I}^{(c)\mu})\frac{i}{D_{f_{0}}},$
(105)

and

$M_{f_{1}\to
3\pi}^{loop+mix}=2ig_{f_{1}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}(ig_{a_{0}K^{0}\bar{K}^{0}})^{2}ig_{f_{0}K^{0}\bar{K}^{0}}ig_{f_{0}\pi^{+}\pi^{-}}\epsilon_{f\mu}(\hat{I}^{(n)\mu}+\hat{I}^{(c)\mu})\frac{i}{D_{a_{0}}}(\hat{\mathcal{I}}^{(n)}-\hat{\mathcal{I}}^{(c)})\frac{i}{D_{f_{0}}}.$
(106)

In Eqs. (104) and (106) functions $\hat{\mathcal{I}}^{(n)}$ and
$\hat{\mathcal{I}}^{(c)}$ represent the two-point loop functions with the
neutral $K^{0}\bar{K}^{0}$ and charged $K^{+}K^{-}$ as intermediate particles,
respectively, for the $a_{0}-f_{0}$ mixing, i.e.

$\hat{\mathcal{I}}^{(n/c)}=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{i^{2}}{(q^{2}-m_{K})[(p-q)^{2}-m_{K}^{2}]}\
,$ (107)

where the superscripts “n” and “c” denote the neutral and charged kaon pairs
in the loop function. One can see that these two loop functions cancel each
other due to a sign difference between the products of
$g_{a_{0}K^{0}\bar{K}^{0}}g_{f_{0}K^{0}\bar{K}^{0}}$ and
$g_{a_{0}K^{+}\bar{K}^{-}}g_{f_{0}K^{+}\bar{K}^{-}}$ in the SU(3) flavor
symmetry. Thus, the isospin breaking effects are given by the small non-
vanishing part caused by the mass difference between the charged and neutral
kaon pairs. The propagators of $a_{0}$ and $f_{0}$ have been given in Eqs.
(97) and (99), respectively.

One qualitative feature of the axial vector productions in the $J/\psi$
radiative decays is that the preferred mixing angle $\alpha_{f}\sim
90^{\circ}$ implies that Fig. 3 (a) should be insignificant in comparison with
Fig. 3 (b), and the latter should be the main contribution to the isospin
violation in $J/\psi\to\gamma f_{1}^{\prime}\to\gamma+3\pi$.

###  III.4 $a_{1}(1260)$ and $a_{1}(1420)$ decays into $3\pi$

The strong couplings of the axial vector mesons $f_{1}$ and $f_{1}^{\prime}$
to $K^{*}\bar{K}+c.c.$ and the presence of the TS mechanism in this kinematic
region imply that the $S$-wave isovector coupling between $K^{*}\bar{K}+c.c.$
can produce observable effects in the $3\pi$ spectrum. Experimental evidence
is provided by the observation of $a_{1}(1420)$ at COMPASS in $\pi^{-}p\to
a_{1}(1420)^{\pm}\pi^{\mp}n\to\pi^{+}\pi^{-}\pi^{0}n$ Adolph:2015pws . As a
natural consequence of the TS mechanism, it can be accounted for by the strong
coupling of the nearby $a_{1}(1260)$ to the $K^{*}\bar{K}+c.c.$ threshold
Ketzer:2015tqa ; Aceti:2016yeb . In the invariant mass spectrum of $3\pi$ the
non-vanishing coupling would be strong enough to create the peak of
$a_{1}(1420)$ in the TS kinematic region.

A crucial issue about the role played by the TS mechanism is its relative
strength to the $a_{1}(1260)$ signal in the exclusive decay channel, i.e.
$a_{1}\to 3\pi$. In the picture of TS mechanism for $a_{1}(1420)$ enhancement,
the off-shell coupling of $a_{1}(1260)$ to the intermediate
$K^{*}\bar{K}+c.c.$ via the triangle loops is actually constrained. Therefore,
a combined analysis of $a_{1}(1260)\to f_{0}\pi^{-}\to\pi^{+}\pi^{-}\pi^{0}$
with the production of $a_{1}(1420)$ will help clarify the nature of
$a_{1}(1420)$.

Figure 4: Diagrams for $a_{1}\to\rho^{0}\pi^{-}\to\pi^{+}\pi^{-}\pi^{-}$
(a),$a_{1}^{-}\to f_{0}(980)\pi^{-}\to\pi^{+}\pi^{-}\pi^{-}$ via the tree (b)
and the triangle loop (c) diagrams.

The transition of $a_{1}\to 3\pi$ is illustrated by Fig. 4 which includes two
tree diagrams and a triangle process. In Fig. 4 (a) the transition can go
through the intermediate $\rho\pi$ and the corresponding amplitude is

$M_{a_{1}\to\rho\pi\to
3\pi}^{tree}=\frac{1}{\sqrt{2}}ig_{a_{1}^{-}\rho^{0}\pi^{-}}ig_{\rho^{0}\pi^{+}\pi^{-}}\epsilon_{a_{1}}^{\mu}\frac{i(-g_{\mu\nu}+\frac{p_{c\mu}p_{c\nu}}{p_{c}^{2}})}{D_{\rho}}i(p_{a}^{\nu}-p_{d}^{\nu})+(b\leftrightarrow
d),$ (108)

where $p_{a}$ and $p_{d}$ denote the momenta of $\pi^{+}$ and $\pi^{-}$
emitted from the $\rho$ meson with momentum $p_{c}=p_{a}+p_{d}$, respectively.
The coupling $g_{a_{1}^{-}\rho^{0}\pi^{-}}$ can be related to
$g_{f_{1}^{\prime}K^{*0}\bar{K}^{0}}$ in the SU(3) flavor symmetry via Eq.
(85). This term plays a role as a background which can be separated out by a
partial wave analysis. Moreover, since it is an $S$-wave-dominant decay, it
will not interfere with the transition via the $f_{0}\pi$ channel. We will see
later that the intermediate $\rho\pi$ behaves differently from the $f_{0}\pi$.

Figure 4 (b) describes the tree-level transition via the intermediate
$f_{0}\pi$ and the Lagrangian reads,

$L_{a_{1}^{-}f_{0}\pi^{-}}=g_{ASP}\sin{\alpha_{f_{0}}}a_{1}^{-\mu}(\pi^{+}\partial_{\mu}f_{0}-f_{0}\partial_{\mu}\pi^{+}),$
(109)

The bare coupling between $a_{1}^{-}$ and $f_{0}\pi^{-}$ is assumed to be
proportional to the $n\bar{n}$ component of the $f_{0}$ meson, which can be
described by the mixing angle $\alpha_{f_{0}}$ between $n\bar{n}$ and
$s\bar{s}$ in the scalar meson sector. In this case, the bare coupling
strength can be related to the bare coupling between $f_{1}$ and $a_{0}\pi$,
i.e.$g_{f_{1}a_{0}\pi}$. Therefore, the relative phase between the tree and
the triangle amplitude of $f_{0}\pi$ channel can be fixed. The corresponding
amplitude is

$M_{a_{1}\to f_{0}\pi\to
3\pi}^{tree}=\frac{1}{\sqrt{2}}ig_{a_{1}^{-}f_{0}\pi^{-}}\frac{i}{D_{f_{0}}}ig_{f_{0}\pi^{+}\pi^{-}}i\epsilon_{a_{1}}\cdot(p_{a}+p_{d}-p_{b})+(b\leftrightarrow
d)\ .$ (110)

Figure 4 (c) describes the transition via the triangle loop and the amplitudes
has the following form,

$\displaystyle M_{a_{1}\to 3\pi}^{loop}$ $\displaystyle=$ $\displaystyle
2ig_{a_{1}K^{*0}\bar{K}^{0}}ig_{K^{*0}K^{0}\pi^{0}}ig_{f_{0}K^{0}\bar{K}^{0}}ig_{f_{0}\pi^{+}\pi^{-}}\epsilon_{a\mu}(\hat{I}^{(n)\mu}+\hat{I}^{(c)\mu})\frac{i}{D_{f_{0}}}+(b\leftrightarrow
d).$ (111)

In Eqs. (110) and (111) the momenta of $\pi^{+}$ and $\pi^{-}$ emitted from
$f_{0}$ are denoted by $p_{a}$ and $p_{d}$, and momentum of the $\pi^{-}$
which recoils against $f_{0}$ is $p_{b}$. In Eq. (111) the neutral and charged
loop amplitudes are in a constructive phase to be compared with the loop
amplitude of Eq. (105) for the isospin-violating decay of $f_{1}\to
f_{0}\pi\to 3\pi$. Note that the combined effect from Eqs. (110) and (111) is
to dress up the bare $a_{1}f_{0}\pi$ coupling with the triangle loop
transition. Similar to $f_{1}\to a_{0}\pi$ the sum of these two amplitudes
will define the physical coupling which can be extracted from the experimental
data for $a_{1}\to f_{0}\pi$.

###  III.5 Triangle loop function in the axial vector meson decays with
$J^{PC}=1^{++}$

In Figs. 1, 2, 3 and 4 the triangle transitions are through the same
rescattering processes. The kinematic conditions for the TS mechanism have
been discussed in detail in Refs. Wu:2011yx ; Wu:2012pg ; Liu:2015taa ;
Du:2019idk . But as pointed out in Ref. Du:2019idk , for different types of
vertex couplings in the triangle transitions the TS amplitudes have different
structures that leads to different TS effects in these processes. This feature
can be seen again for the axial vector decays here. We first consider the
triangle amplitude for the axial vector mesons with $C=+1$. The kinematic
variables are defined in Fig. 5. The loop function is defined as

$\displaystyle\hat{I}^{\mu}=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}})(q_{\nu}-2p_{b\nu})F{(q^{2})}}{(q^{2}-m_{1}^{2}+im_{1}\Gamma_{1})((q-p_{b})^{2}-m_{2}^{2})((p_{0}-q)^{2}-m_{3}^{2})},$
(112)

where $\\{m_{1},m_{2},m_{3}\\}=\\{m_{K^{*}},m_{K},m_{\bar{K}}\\}$ and the
$\Gamma_{1}=50$ MeV is the width of $K^{*}$. The function $F(q^{2})$ is the
form factor, i.e.

$F(q^{2})=\prod_{i=\\{1,2,3\\}}\frac{\Lambda_{i}^{2}-m_{i}^{2}}{\Lambda_{i}^{2}-p_{i(q)}^{2}},$
(113)

where $p_{i(q)}$ denotes the four-vector momentum of the intermediate meson of
mass $m_{i}$ which can be expressed as a function of the integration variable
$q$; The momenta of the external particles are denoted by $p_{a}$, $p_{d}$ and
$p_{b}$, respectively. The cut-off energy is defined by $\Lambda_{i}\equiv
m_{i}+\beta\Lambda_{QCD}$ ($\Lambda_{QCD}=250\ \text{MeV}$). Though the
original integral without the form factor does converge, it happens often that
the dispersive part of the loop amplitude is overestimated when the
interacting hadrons are treated as fundamental particles. The introduction of
the form factor will then cut off the unphysical ultra-violet contributions in
the dispersive part of the loop amplitude.

Figure 5: Conventions for momenta assignments.

##  IV Production and decays of axial vector mesons with $J^{PC}=1^{+-}$

###  IV.1 Productions of the $C=-1$ axial vector mesons in charmonium decays

The axial vector mesons with $C=-1$ can be produced in association with a
pseudoscalar meson in the $J/\psi$ hadronic decays, e.g. $J/\psi\to
b_{1}^{+}\pi^{-}+c.c.$ and $h_{1}\eta$. Then the axial vector mesons can decay
into a vector plus a pseudoscalar via either a tree process or a triangle loop
transition. There are special advantages with the $b_{1}(1235)$ and
$h_{1}(1170)/h_{1}(1415)$ (i.e. $h_{1}/h_{1}^{\prime}$) decays into $\phi\pi$.
Since $b_{1}(1235)$ is an isovector it does not contain the strangeness in the
constituents. Therefore, its decays into $\phi\pi$ will be suppressed at the
tree level. This provides an ideal place for investigating the TS mechanism
which would produce unique signals for the TS mechanism. It is also possible
that the production of $J/\psi\to K^{*}\bar{K}\pi+c.c.$ gives access to the
quantum number of $(I,\ J^{P(C)})=(1,\ 1^{+(-)})$ directly via the
$K^{*}\bar{K}+c.c.$ scatterings. Such a possibility has been studied recently
by Ref. Jing:2019cbw . For $J/\psi\to h_{1}\eta$ and $h_{1}^{\prime}\eta$ with
$h_{1}/h_{1}^{\prime}$ decays into $\phi\pi$, the decays of
$h_{1}/h_{1}^{\prime}$ violate isospin. Similar to the isospin-violating
decays of $f_{1}\to 3\pi$ or $\eta(1405)\to 3\pi$, this channel is ideal for
probing the role played by the TS mechanism as the leading isospin-breaking
mechanism.

For the purpose of detecting the TS mechanism the three-body decays of the
axial vector mesons is of interest. For instance, it would be interesting to
examine the $K\bar{K}$ invariant mass spectrum in $J/\psi\to
b_{1}^{+}\pi^{-}+c.c.\to K\bar{K}\pi^{+}\pi^{-}$ since it covers the physical
region of the TS. Although the intermediate $\phi$ meson will account for the
main cross sections for $b_{1}\to K\bar{K}\pi$, the impact of the TS mechanism
may influence the lineshape of the $K\bar{K}$ spectrum even near the $\phi$
peak. In the following subsections we investigate two correlated processes,
i.e. $J/\psi\to b_{1}^{+}\pi^{-}+c.c.\to K\bar{K}\pi^{+}\pi^{-}$ and
$J/\psi\to h_{1}^{\prime}\eta\to K\bar{K}\pi\eta$ in order to disentangle the
role played by the TS mechanism.

The interactions between $J/\psi\to b_{1}\pi$ and $J/\psi\to
h_{1}^{\prime}\eta$ can be parametrized by the following Lagrangian in the
SU(3) symmetry,

$L_{\psi BP}=g_{\psi BP}\psi^{\mu}Tr[B_{\mu}P],$ (114)

from which the interaction reads

$\displaystyle L_{\psi b_{1}\pi}$ $\displaystyle=$ $\displaystyle g_{\psi
BP}\psi^{\mu}(b_{1\mu}^{+}\pi^{-}+b_{1\mu}^{-}\pi^{+}),$ (115) $\displaystyle
L_{\psi h_{1}^{\prime}\eta}$ $\displaystyle=$ $\displaystyle g_{\psi
BP}\cos{(\alpha_{h}-\alpha_{P})}\psi^{\mu}h_{1\mu}^{\prime}\eta,$ (116)
$\displaystyle L_{\psi h_{1}^{\prime}\eta^{\prime}}$ $\displaystyle=$
$\displaystyle-g_{\psi
BP}\sin{(\alpha_{h}-\alpha_{P})}\psi^{\mu}h_{1\mu}^{\prime}\eta^{\prime}\ ,$
(117) $\displaystyle L_{\psi h_{1}\eta}$ $\displaystyle=$ $\displaystyle
g_{\psi BP}\sin{(\alpha_{h}-\alpha_{P})}\psi^{\mu}h_{1\mu}\eta\ ,$ (118)
$\displaystyle L_{\psi h_{1}\eta^{\prime}}$ $\displaystyle=$ $\displaystyle
g_{\psi BP}\cos{(\alpha_{h}-\alpha_{P})}\psi^{\mu}h_{1\mu}\eta^{\prime}\ .$
(119)

The $\alpha_{P}$ is the mixing angle between $I^{G}J^{PC}=0^{+}0^{-+}$
$n\bar{n}$ and $s\bar{s}$ states as defined in Eq. (67). The typical value of
$\alpha_{P}$ is within a range of $38^{\circ}\sim 42^{\circ}$ and we adopt
$\alpha_{P}=40^{\circ}$ in this study. With the mixing angle
$\alpha_{h}=91.77^{\circ}$, the coupling constants in the decay channels of
Eqs. (115)-(119) are compatible with each other. For the purpose of
identifying the impact from the TS mechanism, we will focus on the production
of $h_{1}^{\prime}$ recoiling $\eta$ in the $J/\psi$ decays. This will benefit
from a relatively larger phase space factor than $J/\psi\to
h_{1}^{\prime}\eta^{\prime}$.

The coupling strength $g_{\psi BP}$ can be fixed by the branching ratio of
$J/\psi\to b_{1}^{\pm}\pi^{\mp}=(3\pm 0.5)\times 10^{-3}$ Patrignani:2016xqp ,
which leads to $g_{\psi BP}=4.35\times 10^{-3}$ GeV. With $g_{\psi BP}$ and
Eq. (117), one obtains the branching ratio of $J/\psi\to
h_{1}^{\prime}\eta^{\prime}$, i.e. $B.R.(J/\psi\to
h_{1}^{\prime}\eta^{\prime})=5.9\times 10^{-4}$. This value is approximately 3
times larger than the experimental measurement of the combined branching ratio
$Br^{exp}(J/\psi\to h_{1}^{\prime}\eta^{\prime}\to
K^{*}\bar{K}\eta^{\prime}+c.c.)=5.9\times 10^{-4}$ by BESIII Ablikim:2018ctf .
It implies that the decay of $h_{1}^{\prime}\to K^{*}\bar{K}+c.c.$ is about
$1/3\sim 1/2$ of the total width of $h_{1}^{\prime}$ which is a reasonable
expectation.

###  IV.2 $J/\psi\to b_{1}^{+}\pi^{-}+c.c.\to K\bar{K}\pi^{+}\pi^{-}$

In $J/\psi\to b_{1}^{+}\pi^{-}+c.c.\to K\bar{K}\pi^{+}\pi^{-}$ the $b_{1}$
decay can go through the processes illustrated in Fig. 6, where (a) shows the
tree-level transition via the intermediate $K^{*}\bar{K}+c.c.$ and (b) is the
loop transition via the intermediate $\phi\pi$. Note that the tree-level decay
of $b_{1}\to\phi\pi$ will be suppressed by the OZI rule. Thus, the triangle
loop transition of Fig. 6 (b) actually provides a mechanism for evading the
OZI rule. Meanwhile, the tree-level transition of Fig. 6 (a) behaves as a
background contribution to Fig. 6 (b). We include this tree-level transition
in the analysis of the $b_{1}^{+}\to K\bar{K}\pi^{+}$ in order to have a
realistic description of the $K\bar{K}$ invariant mass spectrum with the
presence of the TS mechanism.

The amplitude for $J/\psi\to b_{1}^{+}\pi^{-}+c.c.\to\phi\pi^{+}\pi^{-}$ with
the processes considered in Fig. 6(b) can then be expressed as

$M_{J/\psi\to b_{1}\pi\to\phi\pi^{+}\pi^{-}}=g_{\psi
BP}\epsilon_{\psi\mu}\epsilon^{*}_{\phi\rho}\left[\frac{i(-g^{\mu\nu}+\frac{p_{c}^{\mu}p_{c}^{\nu}}{s_{c}})}{D_{b_{1}^{+}}(s_{c})}J^{+\rho}_{\nu}+\frac{i(-g^{\mu\nu}+\frac{p_{ab}^{\mu}p_{ab}^{\nu}}{s_{ab}})}{D_{b_{1}^{-}}(s_{ab})}J^{-\rho}_{\nu}\right]\
,$ (120)

where $p_{a}$ and $p_{c}$ are the momenta of $\phi$ and $b_{1}^{+}$,
respectively; $p_{b}$ is the recoiled $\pi^{-}$ momentum and
$p_{ab}=p_{a}+p_{b}$ correspond to the $b_{1}^{-}$ momentum in the charge
conjugation channel, and $s_{c}\equiv p_{c}^{2}$ and $s_{ab}\equiv
p_{ab}^{2}$; $J^{\pm\rho}_{\nu}$ denotes the current for
$b_{1}^{\pm}\to\phi\pi^{\pm}$; $g_{\psi BP}$ is the coupling constant for
$J/\psi\to b_{1}\pi$ introduced in Eq. (115). In the above equation the
inverse propagator for $b_{1}$ is

$D_{b_{1}}(s)=s-m_{b_{1}}^{2}+im_{b_{1}}\Gamma_{b_{1}}(s),$ (121)

where we have assumed an $S$-wave energy dependence of $\Gamma_{b_{1}}$, i.e.

$\Gamma_{b_{1}}(s)=\frac{m_{b_{1}}|\vec{p}_{\omega(s)}|}{\sqrt{s}|\vec{p}_{\omega(m_{b_{1}})}|}\Gamma_{b_{1}}(m_{b_{1}}^{2}),$
(122)

with $\Gamma_{b_{1}}(m_{b_{1}}^{2})=0.142$ GeV Patrignani:2016xqp .

For $b_{1}\to K\bar{K}\pi$, the amplitude is parametrized as the following
form

$M_{b_{1}^{+}\to
K^{+}K^{-}\pi^{+}}=\epsilon_{b_{1}\mu}(\lambda_{a}p_{a}^{\mu}+\lambda_{b}p_{b}^{\mu}+\lambda_{d}p_{d}^{\mu}),$
(123)

where $p_{a}$, $p_{d}$ and $p_{b}$ are momenta of $K^{+}$, $K^{-}$ and
$\pi^{+}$, respectively. The coefficients $\lambda_{i}$ contains contributions
from Fig. 6 (a) and (b).

For the convenience of calculations we provide the amplitudes for
$b_{1}^{+}\to K^{+}K^{-}\pi^{+}$ as follows for these two processes in Fig. 6.
The amplitude of $b_{1}^{+}\to K^{+}\bar{K}^{*0}\to K^{+}K^{-}\pi^{+}$ (Fig. 6
(a)) reads

$M_{b_{1}^{+}\to
K^{+}K^{-}\pi^{+}}^{tree}=ig_{b_{1}^{+}\bar{K}^{*0}K^{+}}ig_{\bar{K}^{0}K^{-}\pi^{+}}\epsilon_{b_{1}\mu}\frac{i(-g^{\mu\nu}+\frac{p_{ab}^{\mu}p_{ab}^{\nu}}{s_{ab}})}{D_{\bar{K}^{*0}}}i(p_{a}-p_{b})_{\nu},$
(124)

where $p_{ab}=p_{a}+p_{b}$ is the momentum of $\bar{K}^{*0}$.

Similarly, the amplitude of $b_{1}^{+}\to\phi\pi^{+}\to K^{+}K^{-}\pi^{+}$
(Fig. 6 (b)) can be obtained

$\displaystyle M_{b_{1}^{+}\to K^{+}K^{-}\pi^{+}}^{tri}$ $\displaystyle=$
$\displaystyle
2ig_{b_{1}^{+}\bar{K}^{*0}K^{+}}ig_{\bar{K}^{*0}K^{-}\pi^{+}}ig_{\phi
K^{+}K^{-}}ig_{\phi
K^{+}K^{-}}\epsilon_{b_{1}\mu}\tilde{I}^{\mu\alpha}\frac{i(-g^{\alpha\beta}+\frac{p_{c}^{\alpha}p_{c}^{\beta}}{p_{c}^{2}})}{p_{c}^{2}-m_{\phi}^{2}+im_{\phi}\Gamma_{\phi}}i(p_{a}-p_{d})_{\beta}$
(125) $\displaystyle=$ $\displaystyle
ig_{b_{1}^{+}\bar{K}^{*0}K^{+}}ig_{\bar{K}^{*0}K^{-}\pi^{+}}ig_{\phi
K^{+}K^{-}}ig_{\phi
K^{+}K^{-}}\epsilon_{b_{1}\mu}(\tilde{I}_{n}^{\mu\alpha}+\tilde{I}_{c}^{\mu\alpha})\frac{i(-g^{\alpha\beta}+\frac{p_{c}^{\alpha}p_{c}^{\beta}}{p_{c}^{2}})}{p_{c}^{2}-m_{\phi}^{2}+im_{\phi}\Gamma_{\phi}}i(p_{a}-p_{d})_{\beta}\
,$

where the momenta of the $\pi^{+}$, $\phi$, $K^{+}$ and $K^{-}$ mesons are
labeled by $p_{b}$, $p_{c}$, $p_{a}$ and $p_{d}$, respectively;
$\tilde{I}_{\mu\alpha}$ is the typical loop function for the triangle loop,
i.e.

$\displaystyle\tilde{I}^{\mu\alpha}\equiv-i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}})(q-2p_{b})_{\nu}(p_{c}+2p_{b}-2q)_{\alpha}F(q^{2})}{(q-m_{1}^{2}+im_{1}\Gamma_{1})((q-p_{b})^{2}-m_{2}^{2})((p_{0}-q)^{2}-m_{3}^{2})},$
(126)

which is a tensor integral and will be contracted by the polarization vectors
of $b_{1}$ and $\phi$. Considering the slightly different masses between the
charged and neutral intermediate states due to the isospin symmetry breaking,
we note the charged and neutral loop integral functions by
$\tilde{I}_{c}^{\mu\alpha}$ and $\tilde{I}_{n}^{\mu\alpha}$, respectively.

In these amplitudes the unknown coupling $g_{b_{1}^{+}\bar{K}^{*0}K^{+}}$ can
be calculated by two ways based on the SU(3) symmetry. On the one hand, it can
be connected to the decay of $b_{1}^{+}\to\omega\pi^{+}$ where
$g_{b_{1}^{+}\omega\pi^{+}}$ can be extracted by assuming that
$b_{1}\to\omega\pi$ exhausts its total width. The partial width for
$b_{1}^{+}\to\omega\pi^{+}$ reads

$\Gamma(b_{1}^{+}\to\omega\pi^{+})=\frac{|\vec{p}_{\omega}|}{8\pi
m_{b_{1}}^{2}}\frac{g_{b_{1}^{+}\omega\pi^{+}}^{2}}{3}(2+\frac{(m_{b_{1}}^{2}+m_{\omega}^{2}-m_{\pi}^{2})^{2}}{4m_{b_{1}}^{2}m_{\omega}^{2}}),$
(127)

where the $|\vec{p}_{\omega}|$ refers to the momentum of $\omega$ in the rest
frame of $b_{1}$.

On the other hand, $g_{b_{1}^{+}\bar{K}^{*0}K^{+}}$ can be calculated by
$g_{BVP}$, which is obtained by calculating $h_{1}\to\rho\pi$. In the SU(3)
limit, these two methods should give the same result. The numerical results
show that there is about $10\%$ discrepancy between these two methods, which
is acceptable. The discrepancy may results from the following aspects:

  * •

The SU(3) flavour symmetry only approximately holds.

  * •

The estimation based on $g_{h_{1}\rho\pi}$ is correlated with the
determination of the $h_{1}$-$h_{1}^{\prime}$ mixing. Thus, the mixing angle
may introduce uncertainties to the coupling constant.

  * •

The $b_{1}\to\omega\pi$ process contains non-negligible $D$-wave
contributions, which is not considered in the above estimation.

However, we regard that the $10\%$ discrepancy is an indication of self-
consistency in the determination of the $h_{1}$-$h_{1}^{\prime}$ mixing angle.
So we adopt $g_{BVP}$ extracted from $h_{1}\to\rho\pi$ in the final numerical
calculations.

Figure 6: Diagrams of $b_{1}^{+}\to K^{+}K^{-}\pi^{+}$ via $K^{*+}\bar{K}^{0}$
channel (a) and $\phi\pi^{+}$ channel (b).

###  IV.3 $h_{1}^{\prime}\to\phi\pi\to K\bar{K}\pi$

The triangle loop transition in Fig. 7 provides an isospin breaking mechanism
for $h_{1}^{\prime}\to\phi\pi\to K\bar{K}\pi$. The loop integral has the same
form as Eq. (126), except that the charged and neutral loops must cancel due
to the isospin breaking. Similar to the case of the isospin-breaking decay of
$f_{1}^{\prime}\to 3\pi$, the unequal masses between the intermediate charged
and neutral $K$ (or $K^{*}$) mesons will lead to residue amplitudes after the
cancellation.

The production of $h_{1}^{\prime}$ in $J/\psi\to h_{1}^{\prime}\eta$ can be
described the same way as that for $J/\psi\to b_{1}\pi$ in the previous
subsection. Here, we consider the decay of $h_{1}^{\prime}\to\phi\pi$ for
which the amplitude can be written as:

$\displaystyle M_{h_{1}^{\prime}\to K^{+}K^{-}\pi^{0}}^{tri}$ $\displaystyle=$
$\displaystyle
ig_{h_{1}^{\prime}K^{+}\bar{K}^{*-}}ig_{\bar{K}^{*-}K^{-}\pi^{0}}ig_{\phi
K^{+}K^{-}}ig_{\phi
K^{+}K^{-}}\epsilon_{h_{1}^{\prime}\mu}(\tilde{I}_{n}^{\mu\alpha}-\tilde{I}_{c}^{\mu\alpha})\frac{i(-g^{\alpha\beta}+\frac{p_{c}^{\alpha}p_{c}^{\beta}}{p_{c}^{2}})}{p_{c}^{2}-m_{\phi}^{2}+im_{\phi}\Gamma_{\phi}}i(p_{a}-p_{d})_{\beta}\
,$ (128)

which is to be compared with Eq. (125) and the loop function,
$\tilde{I}^{\mu\alpha}$ is the same as Eq. (126). The inverse propagator for
$h_{1}^{\prime}$ is

$D_{h_{1}^{\prime}}(s)=s-m_{h_{1}^{\prime}}^{2}+im_{h_{1}^{\prime}}\Gamma_{h_{1}^{\prime}}(s),$
(129)

with

$\Gamma_{h_{1}^{\prime}}(s)\simeq\Gamma_{h_{1}^{\prime}\to K^{*}\bar{K}\to
K\bar{K}\pi}(s)\ ,$ (130)

where we have assumed that the $K^{*}\bar{K}$ channel is dominant in the
$h_{1}^{\prime}$ decays. As shown by Eq. (74), this is a reasonable
approximation concerning the poor experimental status of $h_{1}$ and
$h_{1}^{\prime}$.

Figure 7: Triangle diagram of $h_{1}^{\prime}\to\phi\pi\to K\bar{K}\pi$. The
charged and neutral loops are destructive.

##  V Results and discussions

###  V.1 Numerical results for the $J^{PC}=1^{++}$ states

####  V.1.1 Parameters

The parameters in this study include the vertex coupling constants and a cut-
off energy for the loop integrals. For the positive $C$-parity channel the key
parameters are $g_{f_{1}^{\prime}K^{*}\bar{K}}$ and
$g_{f_{1}^{\prime}a_{0}\pi}$. They will determine the relative strengths
between the $K\bar{K}\pi$ and $\eta\pi\pi$ channel and the sum of these two
partial widths will nearly saturate the total width. Also, by assuming that
both couplings, $g_{f_{1}^{\prime}a_{0}\pi}$ and $g_{a_{1}f_{0}\pi}$, are via
the non-strange component in the wave functions of $f_{1}^{\prime}$ and
$a_{1}$, they can be expressed by the bare $g_{ASP}$ coupling.

At this moment the experimental data do not allow a precise determination of
these parameters. Our strategy for the parameter determinations is as follows:
Since the $K\bar{K}\pi$ channel is dominant in the $f_{1}^{\prime}$ decay we
assume that this channel accounts for a partial width of 50 MeV such that we
can describe the available data best. This determines
$g_{f_{1}^{\prime}K^{*}\bar{K}}=2.17$. In the SU(3) flavor symmetry the
coupling $g_{f_{1}K^{*}\bar{K}}=1.24$ and $g_{a_{1}K^{*}\bar{K}}=2.04$ can
then be determined. Notice that the couplings $g_{f_{0}K^{+}K^{-}}$ and
$g_{a_{0}K\bar{K}}=2.24$ are relatively well established quantities, and
$g_{K^{*}K\pi}=3.2$ can be determined by experimental data for $K^{*}\to
K\pi$. In Table 3 the coupling constants which appear in the $C=+1$ axial
vector meson decays are listed.

Table 3: Coupling constants adopted in the $1^{++}$ sector.  Coupling constant | Value
---|---
$g_{f_{0}K^{+}K^{-}}$ |  $5.92\pm 0.13$ GeV Aloisio:2002bt
$g_{f_{0}\pi^{+}\pi^{-}}$ |  $2.96\pm 0.12$ GeV Aloisio:2002bt
$g_{a_{0}K^{+}K^{-}}$ |  $2.24\pm 0.11$ GeV Aloisio:2002bsa
$g_{a_{0}\eta\pi}$ |  $3.02\pm 0.35$ GeV Aloisio:2002bsa
$g_{AVP}$ |  $2.04$ GeV
$g_{f_{1}a_{0}\pi}$ | $2.93e^{2.26\pi i}$
$g_{f_{1}^{\prime}a_{0}\pi}$ | $0.282e^{2.26\pi i}$
$g_{f_{1}K^{*}\bar{K}}$ |  $1.24$ GeV
$g_{f_{1}^{\prime}K^{*}\bar{K}}$ |  $2.17$ GeV
$g_{a_{1}K^{*}\bar{K}}$ |  $2.04$ GeV
$g_{a_{1}f_{0}\pi}$ | $2.93e^{2.26\pi i}$

####  V.1.2 Partial decay widths

In Tables 4 and 5 we list the calculated partial decay widths for
$f_{1}/f_{1}^{\prime}$ and $a_{1}$, respectively, and compare them with the
available data. For the decays of $f_{1}/f_{1}^{\prime}$, it shows that the
partial decay widths for both states are consistent with the experimental
measurements Patrignani:2016xqp . Some specific points can be learned:

  * •

The decay of $f_{1}\to\eta\pi\pi$ is a dominant channel due to the large tree
diagram contribution, i.e. $f_{1}\to a_{0}\pi\to\eta\pi\pi$. In contrast, the
triangle loop contribution is larger than the tree process in
$f_{1}^{\prime}\to\eta\pi\pi$. The reason is because of the different
couplings in these two channels, i.e.
$g_{f_{1}a_{0}\pi}>>g_{f_{1}^{\prime}a_{0}\pi}$ and
$g_{f_{1}K^{*}\bar{K}}<<g_{f_{1}^{\prime}K^{*}\bar{K}}$ (see Table 3).

  * •

The decay of $f_{1}\to K\bar{K}\pi$ is suppressed by the phase space. In
contrast, this is the dominant channel for $f_{1}^{\prime}$. The main
contribution is from the tree diagram transition of $f_{1}^{\prime}\to
K^{*}\bar{K}+c.c.\to K\bar{K}\pi$. Note that the mass of $f_{1}^{\prime}$ is
located within the kinematic of the TS condition. As a consequence, the
contributions from the triangle diagram are larger than the tree process in
the transitions involving $a_{0}\pi$. This also enhances the isospin-breaking
effects in $f_{1}^{\prime}\to 3\pi$ than the tree-level $a_{0}-f_{0}$ mixing
contribution.

  * •

The isospin-breaking decays, in $f_{1}$ and $f_{1}^{\prime}\to 3\pi$, can be
illustrated by the following ratios, respectively,

$\displaystyle R_{f_{1}\to 3\pi}$ $\displaystyle\equiv$
$\displaystyle\frac{B.R.(f_{1}\to
f_{0}\pi\to\pi^{+}\pi^{-}\pi^{0})}{B.R.(f_{1}\to
a_{0}\pi\to\eta\pi^{0}\pi^{0})}=\frac{3\times 0.01}{9.7}=0.31\%\ ,$
$\displaystyle R_{f_{1}^{\prime}\to 3\pi}$ $\displaystyle\equiv$
$\displaystyle\frac{B.R.(f_{1}^{\prime}\to
f_{0}\pi\to\pi^{+}\pi^{-}\pi^{0})}{B.R.(f_{1}^{\prime}\to
a_{0}\pi\to\eta\pi^{0}\pi^{0})}=\frac{3\times 0.16}{10}=4.8\%.$ (131)

It shows that the $a_{0}$-$f_{0}$ mixing and contributions from the triangle
diagram are compatible. Due to the low mass of $f_{1}$, the condition for the
TS mechanism has not been fulfilled. Thus, the isospin-breaking effects in
$f_{1}\to 3\pi$ are rather small. In contrast, the isospin-breaking effects in
$f_{1}^{\prime}\to 3\pi$ are strongly enhanced by the TS mechanism. One
notices that the exclusive partial width of $f_{1}^{\prime}\to 3\pi$ via the
tree-level $a_{0}$-$f_{0}$ mixing is much smaller than that of $f_{1}\to
3\pi$. The reason is that the coupling $g_{f_{1}^{\prime}a_{0}\pi}$ is much
smaller than $g_{f_{1}a_{0}\pi}$ as shown in Tab. 3. It is interesting to
compare the above results with the anomalously large isospin violations in
$\eta(1405/1475)\to 3\pi$ ($17.9\%$ BESIII:2012aa ).

  * •

In Tab. 5 the partial decay widths of $a_{1}^{-}\to\rho\pi\to 3\pi$ and
$a_{1}^{-}\to f_{0}\pi^{-}$ are predicted. The $\rho\pi$ channel contributes
dominantly and the $f_{0}\pi$ channel is very small. This is consistent with
the experimental measurement, and may reflect the molecular nature of $f_{0}$.

We also estimate the errors with the partial decay widths which are from two
sources. One is the uncertainties with the coupling constants, and the other
is due to the cut-off dependence of the loop integrals. For instance, in
$f_{1}^{\prime}\to K\bar{K}\pi$ we consider the errors with $g_{K^{*}K\pi}$,
$g_{a_{0}K\bar{K}}$ and $g_{a_{0}\eta\pi}$ which are extracted from the
experimental data (Table 3) for the first type of the error source. For the
second source we find that the results are insensitive to the cut-off energy.
In Fig. 8 we plot the cut-off dependence of the partial decay width for
$f_{1}^{\prime}\to a_{0}\pi\to\eta\pi\pi$ as a demonstration. It shows that
the partial decay width varies rather slowly in terms of the cut-off
parameter.

Table 4:  Calculated partial widths of $f_{1}$ and $f_{1}^{\prime}$, in unit of MeV. The corresponding experiment values are shown in round brackets. | $f_{1}$ | $f_{1}^{\prime}$
---|---|---
$a_{0}\pi\to\eta\pi\pi$ (Fig. 1 (a): tree) | $7.5\pm 1.7$ | $(1.7\pm 0.4)\times 10^{-1}$
$a_{0}\pi\to\eta\pi\pi$ (Fig. 1 (b): tri.) | $(2.0\pm 0.6)\times 10^{-1}$ | $9.2\pm 3.0$
Partial width of $f_{1}/f_{1}^{\prime}\to\eta\pi\pi$ |  $9.7\pm 2.4$ ($8.6\pm 1.3$ Patrignani:2016xqp ) | $10\pm 3.5$
$a_{0}\pi\to K\bar{K}\pi$ (Fig. 2 (a): tree) | $(3.7\pm 0.4)\times 10^{-1}$ | $(1.7\pm 0.2)\times 10^{-2}$
$a_{0}\pi\to K\bar{K}\pi$ (Fig. 2 (c): tri.) | $(2.4\pm 0.5)\times 10^{-2}$ | $1.2\pm 0.2$
$a_{0}\pi\to K\bar{K}\pi$ (Fig. 2 (a)+(c) ) | $(5.0\pm 0.5)\times 10^{-1}$ | $1.0\pm 0.1$
$K^{*}\bar{K}\to K\bar{K}\pi$ (Fig. 2 (b): tree) |  $(3.0\pm 0.1)\times 10^{-1}$ (not seen Patrignani:2016xqp ) | $49\pm 1$
Partial width of $f_{1}/f_{1}^{\prime}\to K\bar{K}\pi$ |  $1.47\pm 0.20$ ($1.95^{+0.37}_{-0.39}$ Barberis:1998by , $2.1\pm 0.2$ Patrignani:2016xqp ) | $52\pm 3$
$f_{0}\pi\to 3\pi$ (Fig. 3 (a): mixing) | $(3.6\pm 1.2)\times 10^{-3}$ | $(9.0\pm 2.9)\times 10^{-5}$
$f_{0}\pi\to 3\pi$ (Fig. 3 (b): tri.) | $(1.0\pm 0.1)\times 10^{-3}$ | $(1.1\pm 0.1)\times 10^{-1}$
$f_{0}\pi\to 3\pi$ (Fig. 3 (c): mixing via tri.) | $(1.4\pm 0.4)\times 10^{-4}$ | $(8.0\pm 2.6)\times 10^{-3}$
Partial width of $f_{1}/f_{1}^{\prime}\to f_{0}\pi\to 3\pi$ | $(1.0\pm 0.3)\times 10^{-2}$ | $(1.6\pm 0.3)\times 10^{-1}$
Table 5: Predicted partial decay widths of $a_{1}$ in comparison with experimental data from CLEO Collaboration Asner:1999kj .  Channel | Width
---|---
$a_{1}^{-}\to\rho\pi\to 3\pi$ |  $274$ MeV ($221\pm 5^{+17}_{15}$ MeV) Asner:1999kj
$a_{1}^{-}\to f_{0}\pi^{-}$ (Fig. 4(c): tri.) |  $0.033$ MeV
Figure 8: Sensitivity of $\Gamma(f_{1}^{\prime}\to a_{0}\pi\to\eta\pi\pi)$ to
the cut-off parameter $\beta$.

####  V.1.3 Invariant mass spectra

To proceed, we discuss the results for the invariant mass spectra in different
transition processes.

(i) $J/\psi\to\gamma f_{1}/f_{1}^{\prime}\to\gamma\eta\pi\pi$

In Fig. 9 the $\eta\pi\pi$ invariant mass spectrum (solid line) in
$J/\psi\to\gamma f_{1}/f_{1}^{\prime}\to\gamma\eta\pi\pi$ is plotted. As a
comparison, the exclusive contributions from the tree-level process
$f_{1}/f_{1}^{\prime}\to a_{0}\pi$ (dashed line) and the triangle loop
transition (dot-dashed line) are also presented. It is interesting to see that
the signals for $f_{1}^{\prime}$ are rather small. This is mainly due to the
relatively small production coupling for $J/\psi\to\gamma f_{1}^{\prime}$. One
notices that with $\alpha_{f}=84.48^{\circ}$ the production coupling ratio in
Eq. (77) gives
${g_{f}^{\prime}}/{g_{f}}={(\sqrt{2}\cos{\alpha_{f}}-\sin{\alpha_{f}})}/{(\sqrt{2}\sin{\alpha_{f}}+\cos{\alpha_{f}})}\simeq-0.57$
which suggests that the production of $f_{1}$ will be relatively enhanced.
However, if we only compare the partial decay widths between $f_{1}$ and
$f_{1}^{\prime}\to\eta\pi\pi$, they are actually comparable with each other as
shown in Table 4.

Figure 9: The $\eta\pi\pi$ invariant mass spectrum in $J/\psi\to\gamma
f_{1}/f_{1}^{\prime}\to\gamma\eta\pi\pi$. The blue solid, red dashed and green
dot-dashed lines are contributions from the full, tree level and triangle
amplitudes, respectively.

(ii) $J/\psi\to\gamma f_{1}/f_{1}^{\prime}\to\gamma K\bar{K}\pi$

In Fig. 10 the invariant mass spectrum of $K\bar{K}\pi$ in $J/\psi\to\gamma
f_{1}/f_{1}^{\prime}\to\gamma K\bar{K}\pi$ is plotted where the signals for
both $f_{1}$ and $f_{1}^{\prime}$ are clear. Recall that the production of
$f_{1}^{\prime}$ in the $J/\psi$ radiative decays will be relatively
suppressed. The strong signals of $f_{1}^{\prime}$ are driven by its strong
coupling to $K^{*}\bar{K}$ which leads to the dominant contributions from the
tree diagram of Fig. 2 (b), i.e. $f_{1}^{\prime}\to K^{*}\bar{K}\to
K\bar{K}\pi$. The exclusive contributions are shown by the dotted line in Fig.
10. In contrast, the intermediate $a_{0}\pi$ contributions are negligible.

In Fig. 11 the invariant mass spectra of $K\bar{K}$ (left panel) and $K\pi$
(right panel) in $J/\psi\to\gamma f_{1}^{\prime}\to\gamma K\bar{K}\pi$ are
illustrated by the solid lines. The dashed lines denote the exclusive
contributions from the tree diagram of $f_{1}^{\prime}\to K^{*}\bar{K}\to
K\bar{K}\pi$. The difference between the solid and dashed lines indicates the
effects from the intermediate $a_{0}\pi$ contributions. In particular, for the
$f_{1}^{\prime}$ decays into $K\bar{K}\pi$ the TS contribution is larger than
the tree-level $a_{0}\pi$ contribution. Therefore, the difference between the
solid and dashed lines is largely due to the TS mechanism. Meanwhile, the
strong threshold enhancement at the $K\bar{K}$ threshold is due to the
$a_{0}(980)$ pole structure. In contrast, in Fig. 11 (b) the signal for
$K^{*}$ is evident.

Figure 10: $K\bar{K}\pi$ spectrum of $J/\psi\to\gamma
f_{1}/f_{1}^{\prime}\to\gamma K\bar{K}\pi$. The full spectrum is represented
by the blue solid line. The blue dotted line stands for the tree-level
$K^{*}\bar{K}$ contribution. The red short dashed and green dot-dashed lines
are contributions from the tree-level and triangle amplitudes of the
$a_{0}\pi$ channel, respectively, and the purple long dashed is the sum of
both. For a better display, the latter three contributions are multiplied by a
factor of 2.

Figure 11: Invariant mass spectra of $K\bar{K}$ (left) and $K\pi$ (right) in
$J/\psi\to\gamma f_{1}^{\prime}\to\gamma K\bar{K}\pi$. The blue solid lines
denote the full calculation results, and the red dashed lines denote the
contribution from the tree-level $K^{*}\bar{K}$ channel.

(iii) $J/\psi\to\gamma f_{1}/f_{1}^{\prime}\to\gamma 3\pi$

In this process the decays of $f_{1}/f_{1}^{\prime}$ into $3\pi$ violate
isospin symmetry. For the processes in Fig. 3 the isospin-violating mechanisms
actually imply an abnormal lineshape with the $\pi\pi$ invariant mass spectrum
due to the cancellations outside the kinematic regions between the thresholds
of the charged and neutral $K\bar{K}$ pairs.

The $\pi^{+}\pi^{-}$ invariant mass spectrum is shown in Fig. 12, where we can
see the cancellation between charged and neutral loop amplitudes produces a
narrow $f_{0}(980)$ peak with a width of $\sim(2m_{K^{0}}-2m_{K^{+}})\simeq
10$ MeV. In general, this is the signature for the $a_{0}$-$f_{0}$ mixing
mechanism for the isospin breaking effects. This narrow structure also becomes
the signature for the TS mechanism if significantly large isospin breaking
effects are observed within the TS kinematic region Wu:2011yx ; Wu:2012pg ;
Du:2019idk .

Some special features arise from the process of $J/\psi\to\gamma
f_{1}/f_{1}^{\prime}\to\gamma 3\pi$. Since the tree diagram (Fig. 3 (a)) is
relatively suppressed by the $f_{1}-f_{1}^{\prime}$ mixing the isospin
violation of $f_{1}^{\prime}$ via the tree-level $a_{0}-f_{0}$ mixing is also
suppressed. As shown in Table 4 the TS mechanism has the largest contribution
to the isospin breaking effects. In addition, the TS mechanism can enhance the
production of $a_{0}$ which will further enhance the $a_{0}-f_{0}$ mixing
contributions. This scenario is similar to the case of the isospin violation
of $\eta(1405/1475)\to 3\pi$ Du:2019idk , but have been overlooked in other
studies Wu:2011yx ; Wu:2012pg ; Aceti:2012dj ; Achasov:2015uua . One can see
that although this amplitude is small, its interferences cannot be neglected
(see Table 4). Combining together the isospin-breaking contributions from
those three processes in Fig. 3, we see that the isospin breaking effects are
about $4.8\%$ (see Eq. (• ‣ V.1.2)). This value is significantly larger than
the usual ones of about $1\sim 2\%$ from the pure $a_{0}-f_{0}$ mixing, and
has indicated a signature of the TS mechanism.

We will also see later that the TS mechanism will put a constraint on the
$a_{1}(1420)\to 3\pi$. With the same initial energy around 1.4 GeV, the
charged and neutral loop amplitudes will constructively add to each other.

Figure 12: Invariant mass spectrum of $\pi^{+}\pi^{-}$ in
$f_{1}^{\prime}\to\pi^{+}\pi^{-}\pi^{0}$. The dot-dashed, short dashed and
long dashed lines correspond to the contributions from Fig. 3 (a), (b) and
(c), respectively.

(iv) $\chi_{c1}\to a_{1}\pi\to 4\pi$

Figure 13: The invariant mass spectrum of $\pi^{+}\pi^{-}\pi^{-}$ in
$\chi_{c1}\to a_{1}(1260)\pi\to 4\pi$. The black dotted, red thick solid, blue
dashed and orange thin solid lines represent the contributions from the tree
diagram of the $\rho^{0}\pi^{-}$ channel, triangle diagram of the $f_{0}\pi$
channel, tree diagram of the $f_{0}\pi$ channel, and a coherent sum of the
$f_{0}\pi$ channels, respectively. The blue dashed and orange thin solid line
are calculated assuming $\alpha_{f0}=\pm\frac{\pi}{2}$. Note that the
$\rho\pi$ contributions are scaled down by a factor of $10^{-2}$, which means
that the contributions from the intermediate $f_{0}\pi$ channels are small.

As shown in subsection III.4 the decay of $\chi_{c1}\to 4\pi$ gives access to
the production of $a_{1}(1260)$ in its decays into $3\pi$. Note that
$a_{1}(1260)$ is a broad state, i.e. $\Gamma_{a_{1}}=250\sim 600$ MeV
Patrignani:2016xqp , it is necessary to investigate the $3\pi$ invariant mass
spectrum in a relatively broad mass region. This will bring the kinematics
into the physical region of the TS condition. Therefore, a systematic study of
the $a_{1}(1260)$ production in a broad mass region via the $3\pi$ invariant
mass spectrum will provide a test of the nature of $a_{1}(1420)$ which could
be an enhancement produced by the TS mechanism.

The corresponding processes are illustrated in Fig. 4. In Fig. 13 the $3\pi$
invariant mass spectrum is plotted. One can see that the dominant decay of
$a_{1}(1260)$ is the $\rho\pi$ channel, which accounts for about $60\%$ of the
total width Patrignani:2016xqp . This helps determine the coupling of
$a_{1}\to\rho\pi$. As shown by the dashed line, the intensity of
$f_{0}\pi^{-}$ channel relative to the $\rho^{0}\pi^{-}$ channel ranges from
$(0.4\sim 1)\%$. This is consistent with the COMPASS observation
Adolph:2015pws . Given the broad width of $a_{1}(1260)$ and its strong
coupling to the nearby $K^{*}\bar{K}+c.c.$ threshold, the TS mechanism can
produce a significant enhancement at about 1.4 GeV (see the solid line). This
structure, noted as $a_{1}(1420)$, can be regarded as a natural consequence of
the TS mechanism and is not necessarily interpreted as a genuine resonance
state. The orange thin solid line in Fig. 13 shows the combined contributions
from $a_{1}(1260)\to f_{0}\pi$. The lineshape indicates the interference
between the amplitudes of the tree-level and TS transitions, and is consistent
with the partial wave analysis in experiment Adolph:2015pws . The observation
of $a_{1}(1420)$ hence can be regarded as a signature for the TS mechanism
which drives the rich phenomena observed near the $K^{*}\bar{K}+c.c.$
threshold in various processes.

###  V.2 Numerical results for the $J^{PC}=1^{+-}$ states

####  V.2.1 Parameters

Parameters in the calculations of the $C=-1$ axial vectors include its
production coupling in $J/\psi$ decays into a pseudoscalar and $C=-1$ axial
vector, i.e. $g_{\psi BP}$. As discussed in Subsection IV.1, this quantity can
be determined by the experimental data for $J/\psi\to b_{1}^{\pm}\pi^{\mp}$
Patrignani:2016xqp . The experimental data for $h_{1}\to\rho\pi$ and
$b_{1}^{+}\to\omega\pi^{+}$ can connect all the couplings among the SU(3)
multiplets together as discussed in subsections IV.1 and IV.2. In Table 6 all
the vertex couplings are listed.

Table 6: Coupling constants for the $1^{+-}$ sector. Coupling constant | Value
---|---
$g_{BVP}$ |  $3.03$ GeV
$g_{h_{1}\rho\pi}$ |  $4.28$ GeV
$g_{h_{1}^{\prime}\rho\pi}$ |  $0.13$ GeV
$g_{h_{1}^{\prime}K^{*}\bar{K}}$ |  $3.09$ GeV
$g_{b_{1}K^{*}\bar{K}}$ |  $3.03$ GeV
$g_{b_{1}\omega\pi}$ |  $4.28$ GeV
$g_{\psi BP}$ |  $4.35\times 10^{-3}$ GeV

####  V.2.2 Invariant mass spectra

(i) $J/\psi\to b_{1}\pi\to\phi\pi^{+}\pi^{-}$

Figure 14: Invariant mass spectrum of $K^{+}K^{-}$ in $J/\psi\to
b_{1}^{+}\pi^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$ at $\sqrt{s}=m_{b_{1}}$ (left)
and $\sqrt{s}=1.4$ GeV (right). The gray dashed, red dot-dashed and blue solid
lines represent the contributions from the $K^{+}\bar{K}^{*0}$ , $\phi\pi^{+}$
channels and the coherent sum, respectively.

As discussed earlier the decay of $b_{1}\to\phi\pi$ is an OZI violation
process. The triangle loop actually provides a mechanism to evade the OZI
rule. The OZI-rule violation effects can be indicated by the following
branching ratio fraction:

$\displaystyle R_{b_{1}}\equiv\frac{B.R.(b_{1}^{+}\to\phi\pi^{+}\to
K\bar{K}\pi^{+})}{B.R.(b_{1}^{+}\to\omega\pi^{+})(S-wave)}=\frac{\Gamma(b_{1}^{+}\to\phi\pi^{+}\to
K^{0}\bar{K}^{0}\pi^{+})+\Gamma(b_{1}^{+}\to\phi\pi^{+}\to
K^{+}K^{-}\pi^{+})}{\Gamma(b_{1}^{+}\to\omega\pi^{+})}=1.8\times 10^{-4}.$
(132)

which is consistent with the experimental upper limit $R_{b_{1}}<0.004$.
Recalling that $b_{1}$ is broad and the decay of $b_{1}\to\phi\pi\to
K\bar{K}\pi$ will have the tree-level contribution from $b_{1}\to
K^{*}\bar{K}\to K\bar{K}\pi$ as the background contribution, it should be
interesting to examine the $K\bar{K}$ invariant mass spectrum from the on-
shell mass to the TS kinematic region in the $J/\psi$ decay. In Fig. 14 the
$K^{+}K^{-}$ spectra at $\sqrt{s}=m_{b_{1}}=1.235$ GeV (left) and
$\sqrt{s}=1.40$ GeV (right) are plotted with the cut-off parameter $\beta=2$.
Although the $\phi$ meson production is an OZI-rule violating process the
signal stands clearly out of the smooth background. Comparing these two plots
in Fig. 14 it shows that the lineshape of the $\phi$ meson is distorted at
$\sqrt{s}=1.40$ GeV due to the TS mechanism. Moreover, the cross section at
$\sqrt{s}=1.40$ GeV is much larger than that at $\sqrt{s}=m_{b_{1}}=1.235$
GeV. This is a strong indication of the TS mechanism in this channel.

In Fig. 15 we present the $\phi\pi^{+}$ invariant mass spectrum and the Dalitz
plot for $\phi\pi^{+}\pi^{-}$ which can further disentangle the role played by
the TS mechanism. In Fig. 15 (a), a set of curves which correspond to
different cut-off energies ($\beta\Lambda_{QCD}$), are shown. It is
interesting to see that although the $b_{1}$ signal is sensitive to the cut-
off energies, the TS peak (close to the normal threshold of $K^{*}\bar{K}$)
appears to be stable. This is because, at the kinematic region away from the
on-shell conditions, the loop integrals, which are dominated by the dispersive
part, will become more sensitive to the cut-off energies. In contrast, at the
TS kinematics the internal states are all on-shell or near threshold. The
absorptive part is dominant and the cut-off becomes quite irrelevant. The
dispersive part, when moves away from the on-shell kinematics, will always be
largely cut off. Therefore, the residue amplitude is insensitive to the cut-
off energies.

With a typical cut-off energies $\beta\Lambda_{QCD}=500$ MeV (i.e. $\beta=2$),
we see that the red dotted line in Fig. 15 (a) has a trivial structure at the
mass of $b_{1}$, but a clear TS enhancement at the $K^{*}\bar{K}$ threshold.
This mechanism can produce signature pattern in the Dolitz plot as shown by
Fig. 15 (b). The presence of the TS mechanism actually makes a unique
prediction for the $\phi\pi^{+}$ invariant mass spectrum in $J/\psi\to
b_{1}\pi\to\phi\pi^{+}\pi^{-}$ that can be investigated in experiment.

Figure 15: Invariant mass spectrum of $\phi\pi^{+}$ (left) and Dalitz plot of
$J/\psi\to b_{1}^{+}\pi^{-}+c.c.\to\phi\pi^{+}\pi^{-}$ (right). In the
$\phi\pi$ spectrum (left), the green solid, cyan dashed, red dotted, purple
dot-dashed, and blue thin-solid lines stands for calculations with different
cut-off parameters, i.e. $\beta=4$, $\beta=3$, $\beta=2$, $\beta=1$, and
without the form factor, respectively. The first peak is due to the resonance
of $b_{1}(1235)$ and the second peak is due to the presence of the TS. The
Dalitz plot (right) is calculated with $\beta=2$. The mass of $b_{1}(1235)$
and the location of the TS are marked by the dashed and dotted lines,
respectively.

(ii) $J/\psi\to h_{1}^{\prime}\eta\to\phi\pi\eta\to K\bar{K}\pi\eta$

The decay channel of $J/\psi\to h_{1}^{\prime}\eta\to\phi\pi\eta\to
K\bar{K}\pi\eta$ can also provide signals for the TS mechanism. Although the
mass of $h_{1}^{\prime}$ still bares a relatively large uncertainty, the
signature of the TS mechanism exhibits some model independent feature that can
be identified in experiment. Note that the decay of $h_{1}^{\prime}\to\phi\pi$
is an isospin-violating transition. The TS mechanism actually provides a
leading source for violating the isospin symmetry 555The $\rho^{0}$-$\phi$
mixing is negligibly small. Thus, we do not include its contributions in this
calculation.. As discussed earlier, the charged and neutral triangle loop
amplitudes will cancel out. This process is correlated with the decay of
$J/\psi\to b_{1}\pi\to\phi\pi^{+}\pi^{-}$ where the charged and neutral loop
amplitudes add constructively to each other.

In Fig. 16 (a) we depict the $\phi\pi^{0}$ invariant mass spectrum with (blue
solid line) and without (orange dashed line) the $h_{1}^{\prime}$ propagator.
It shows that without the $h_{1}^{\prime}$ the cross section will be much
smaller and the location of the peak position is determined by the TS
kinematics, i.e. between the $K^{*+}K^{-}$ and $K^{*0}\bar{K}^{0}$ thresholds.
In contrast, the inclusion of the $h_{1}^{\prime}$ pole will significantly
enhance the peak strength. Furthermore, the combined effects from the pole
structure with $m_{h_{1}^{\prime}}=1.423$ GeV Ablikim:2018ctf  and the TS
enhancement will shift the peak position by about 10 MeV towards to the
$K^{*}\bar{K}$ threshold. It should also be noted that the results are
insensitive to the cut-off energies since the dispersive part will cancel out
in this isospin-violating process.

The process $J/\psi\to h_{1}^{\prime}\eta\to\phi\pi\eta$ has also been studied
in Ref. Jing:2019cbw . Different from our approach, in Ref. Jing:2019cbw  no
$h_{1}^{\prime}$ pole is included and the intermediate $K^{*}\bar{K}$ is
introduced by a contact interaction in the $J/\psi$ decays. As shown by Fig.
16 (a) without considering the relative magnitude between the two curves it
may be difficult to distinguish the two cases by about 10 MeV difference. The
Dalitz plot are also shown in Fig. 16 (b), which is in agreement with the
observation of BESIII Ablikim:2018pik .

Taking into account the mass uncertainties with $h_{1}^{\prime}$, we also
investigate the strength of isospin violations in a range of possible masses.
By varying the $h_{1}^{\prime}$ mass we plot the partial widths of
$h_{1}^{\prime}\to\phi\pi\to K\bar{K}\pi^{0}$ in Fig. 17 for three different
cut-off energies. Due to the enhancement of the TS mechanism, the partial
width reaches the maximum near $M_{K\bar{K}\pi}=1.39$ GeV, which is between
the $K^{*0}\bar{K}^{0}$ and $K^{*+}K^{-}$ thresholds. Also, it shows that the
partial width is insensitive to the cut-off energies as shown by the dashed
and dot-dashed lines.

It is interesting to compare the branching ratios between $J/\psi\to
h_{1}^{\prime}\eta\to\phi\pi^{0}\eta$ and $J/\psi\to
b_{1}^{+}\pi^{-}+c.c.\to\phi\pi^{+}\pi^{-}$ with the presence of the TS
mechanism. Recalling that the former is isospin breaking and the later is OZI
evading. With the cut-off parameter $\beta=2$, we find $B.R.(J/\psi\to
h_{1}^{\prime}\eta\to\phi\pi^{0}\eta)=6.3\times 10^{-8}$ and $B.R.(J/\psi\to
b_{1}^{+}\pi^{-}+c.c.\to\phi\pi^{+}\pi^{-})=1.0\times 10^{-5}$. It shows that
the OZI-evading b.r. is much larger than that of the isospin breaking one.
This is an interesting indication of the importance of the non-perturbative
mechanism via meson loop interactions.

Figure 16: The $\phi\pi$ spectrum of $h_{1}^{\prime}$ in $J/\psi\to\eta
h_{1}^{\prime}\to\eta\phi\pi$ and the Dalitz plot of $J/\psi\to\eta
h_{1}^{\prime}\to\eta\phi\pi$ (right). The orange dashed line is calculated
assuming that there is no $h_{1}^{\prime}$ resonance produced in $J/\psi$
decay, and it is normalized to the blue solid one, which is calculated
considering the resonance of $h_{1}^{\prime}$, with the mass
$m_{h_{1}^{\prime}}=1.423$ GeV measured by BESIII Ablikim:2018ctf . Figure 17:
The partial width of $h_{1}^{\prime}\to\phi\pi\to K\bar{K}\pi^{0}$ as a
function of $M_{h_{1}^{\prime}}$. The light blue dot-dashed, blue solid and
purple dashed lines represent the calculations with different cut-off energies
represented by $\beta$, i.e. $\Lambda_{i}=m_{i}+\beta\Lambda_{QCD}$, where
$m_{i}$ is the mass of internal particles of the triangle diagram.

##  VI Conclusions

In this work we present a comprehensive study of the $C=\pm 1$ light axial
vector mesons. We first discuss the mixing angles between $f_{1}$ and
$f_{1}^{\prime}$ (i.e. $f_{1}(1285)$ and $f_{1}(1420)$), and between $h_{1}$
and $h_{1}^{\prime}$ (i.e. $h_{1}(1170)$ and $h_{1}(1415)$). These two angles
can be related to the mixing between $K_{1}(1270)$ and $K_{1}(1400)$ through
the Gell-Mann-Okubo mass relation. We then introduce the intermediate
$K^{*}\bar{K}+c.c.$ meson loop transitions in the description of the
productions and decays of these axial vector mesons. The presence of the
nearby $S$-wave $K^{*}\bar{K}+c.c.$ to which these axial vector mesons have
strong couplings, turns out to be crucial for understanding many puzzling
questions related to their productions and decays. This is because that the
$K^{*}\bar{K}+c.c.$ rescatterings by the kaon exchange satisfy the triangle
singularity condition. Therefore, the TS mechanism can introduce special
interference effects in the exclusive decays of these light axial vector
mesons.

In the $C=+1$ sector the main decay channels of the axial vectors are
$K\bar{K}\pi$ and $\eta\pi\pi$ for $f_{1}$ and $f_{1}^{\prime}$, and $3\pi$
for $a_{1}(1260)$. We show that a combined study of these processes by taking
into account of the TS mechanism provides clear evidences for the assignment
of $f_{1}$ and $f_{1}^{\prime}$ as the mixing states of the flavor singlet and
octet. Although $f_{1}$ can produce an enhancement around 1.4 GeV due to the
TS mechanism, without the inclusion of $f_{1}^{\prime}$ as a genuine state and
the SU(3) partner of $f_{1}$, it is impossible to understand its productions
and decays in both $K\bar{K}\pi$ and $\eta\pi\pi$ channels. We also show that
the TS mechanism accounts for the $a_{1}(1420)$ enhancement which is
originated from the strong $a_{1}(1260)$ coupling to the $K^{*}\bar{K}+c.c.$
threshold. The same mechanism also accounts for the relatively large isospin
breaking effects in $f_{1}^{\prime}\to 3\pi$ except that the charged and
neutral triangle loop amplitudes have a destructive interfering phase. The
combined analysis of these channels provides a self-consistent check of the
underlying dynamics.

In the $C=-1$ sector the experimental information is still sparse. We choose
to investigate two special processes, i.e. $J/\psi\to
b_{1}\pi\to\phi\pi^{+}\pi^{-}$ and $J/\psi\to
h_{1}^{\prime}\eta\to\phi\pi\eta\to K\bar{K}\pi\eta$, in order to highlight
the role played by the TS mechanism. In the first process the decay of
$b_{1}\to\phi\pi$ is the OZI-rule suppressed. Thus, the tree-level transition
is highly suppressed. The triangle diagram actually provides an OZI-evading
mechanism which will produce abnormal lineshapes in both $K\bar{K}$ and
$\phi\pi$ invariant mass spectra. In particular, a clear enhancement at about
1.39 GeV is predicted in the $\phi\pi$ invariant mass spectrum due to the TS
mechanism. The second process is an isospin-breaking decay for
$h_{1}^{\prime}\to\phi\pi$. It shows that the TS mechanism provides a leading
source of the isospin-breaking effects. It is interesting to notice that
whether or not to include the $h_{1}^{\prime}$ pole does not cause drastic
differences in the lineshapes of the $\phi\pi$ invariant mass spectrum.
However, the absolute values of the branching ratios in these two scenarios
will be very different. Without the inclusion of the $h_{1}^{\prime}$ pole,
the branching ratio of $J/\psi\to K^{*}\bar{K}\eta+c.c.\to\phi\pi\eta$ will be
much smaller than that of $J/\psi\to h_{1}^{\prime}\eta\to\phi\pi\eta$. These
predictions can be searched for in experiment at BESIII.

In brief, the productions and decays of these two sets of axial vector mesons
have provided important evidences for the role played by the TS mechanism.
This combined analysis has helped clarify some crucial issues concerning their
identifications and classifications.

###### Acknowledgements.

This work is supported, in part, by the National Natural Science Foundation of
China (Grant Nos. 11425525 and 11521505), 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).

## Appendix

We include here a pedagogic deduction of the Gell-Mann-Okubo relation. With
the definitions of mixing angles in the article, the masses of $K_{1A}$ and
$K_{1B}$ are described by

$\displaystyle
m_{K_{1A}}^{2}=m_{K_{1}(1400)}^{2}\cos^{2}{\theta_{K_{1}}}+m_{K_{1}(1270)}^{2}\sin^{2}{\theta_{K_{1}}}$
$\displaystyle
m_{K_{1B}}^{2}=m_{K_{1}(1400)}^{2}\sin^{2}{\theta_{K_{1}}}+m_{K_{1}(1270)}^{2}\cos^{2}{\theta_{K_{1}}}.$
(133)

Under the bases $\tilde{f}_{1}$ and $\tilde{f}_{8}$, the mass matrix is

$\displaystyle\left(\begin{array}[]{cc}\langle\tilde{f}_{8}|H^{2}|\tilde{f}_{8}\rangle&\langle\tilde{f}_{8}|H^{2}|\tilde{f}_{1}\rangle\\\
\langle\tilde{f}_{1}|H^{2}|\tilde{f}_{8}\rangle&\langle\tilde{f}_{1}|H^{2}|\tilde{f}_{1}\rangle\end{array}\right)=\left(\begin{array}[]{cc}m_{\tilde{f}_{8}}^{2}&m_{\tilde{f}_{18}}^{2}\\\
m_{\tilde{f}_{18}}^{2}&m_{\tilde{f}_{1}}^{2}\end{array}\right).$ (138)

By diagonalizing the above matrix, one obtains the physical masses of $f$ and
$f^{\prime}$:

$\displaystyle\left(\begin{array}[]{cc}m_{f^{\prime}}^{2}&0\\\
0&m_{f}^{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos{\theta_{f}}&-\sin{\theta_{f}}\\\
\sin{\theta_{f}}&\cos{\theta_{f}}\end{array}\right)\left(\begin{array}[]{cc}m_{\tilde{f}_{8}}^{2}&m_{\tilde{f}_{18}}^{2}\\\
m_{\tilde{f}_{18}}^{2}&m_{\tilde{f}_{1}}^{2}\end{array}\right)\left(\begin{array}[]{cc}\cos{\theta_{f}}&\sin{\theta_{f}}\\\
-\sin{\theta_{f}}&\cos{\theta_{f}}\end{array}\right).$ (147)

Then, one has

$\displaystyle m_{\tilde{f}_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[m_{f}^{2}+m_{f^{\prime}}^{2}-(m_{f^{\prime}}^{2}-m_{f}^{2})\cos{2\theta_{f}}]\
,$ (148) $\displaystyle m_{\tilde{f}_{8}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[m_{f}^{2}+m_{f^{\prime}}^{2}+(m_{f^{\prime}}^{2}-m_{f}^{2})\cos{2\theta_{f}}]\
,$ (149) $\displaystyle m_{\tilde{f}_{18}}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(m_{f^{\prime}}^{2}-m_{f}^{2})\sin{2\theta_{f}}.$
(150)

From Eqs. (148), (149) and (150), the following relations can be deduced:

$\displaystyle m_{\tilde{f}_{1}}^{2}+m_{\tilde{f}_{8}}^{2}$ $\displaystyle=$
$\displaystyle m_{f}^{2}+m_{f^{\prime}}^{2}\ ,$ (151) $\displaystyle
m_{\tilde{f}_{18}}^{4}$ $\displaystyle=$ $\displaystyle
m_{\tilde{f}_{1}}^{2}m_{\tilde{f}_{8}}^{2}-m_{f^{\prime}}^{2}m_{f}^{2}\
(m_{\tilde{f}_{18}}^{2}<0)\ ,$ (152) $\displaystyle\tan{\theta}$
$\displaystyle=$
$\displaystyle\frac{m_{\tilde{f}_{18}}^{2}}{m_{f}^{2}-m_{\tilde{f}_{8}}^{2}}=\frac{m_{\tilde{f}_{8}}^{2}-m_{f^{\prime}}^{2}}{m_{\tilde{f}_{18}}^{2}}.$
(153)

The solution to these equations is Eq. (40).

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