# Investigating Ionization in the Intergalactic Medium

Brad Koplitz School of Earth & Space Exploration, Arizona State University,
781 Terrace Mall, Tempe, AZ 85287, USA  Anjali Ramesh School of Earth & Space
Exploration, Arizona State University, 781 Terrace Mall, Tempe, AZ 85287, USA
Sanchayeeta Borthakur School of Earth & Space Exploration, Arizona State
University, 781 Terrace Mall, Tempe, AZ 85287, USA

###### Abstract

The Intergalactic Medium (IGM) contains $>$50% of the baryonic mass of the
Universe, yet the mechanisms responsible for keeping the IGM ionized has not
been fully explained. Hence, we investigate ion abundances from the largest
blind QSO absorption catalog for clouds that show C IV, N V, and O VI
simultaneously. The wavelength range of present UV spectrographs, however,
make it possible to probe C IV and O VI over a small range of redshift
($z\approx 0.12-0.15$). As a result, we only have five IGM absorbing clouds,
yet these provide a powerful and representative tool to probe the IGM
ionization state. We found one cloud to be in collisional ionization
equilibrium while three of five showed signs of being produced by non-
equilibrium processes, specifically conductive interfaces and turbulent mixing
layers. None of the models we explore here were able to reproduce the
ionization state of the remaining system. Energetic processes, such as
galactic feedback from star formation and AGN winds, would be excellent
candidates that can cause such widespread ionization.

Intergalactic medium (813); Quasar absorption line spectroscopy (1317);
Collisional processes (2286); Photoionization (2060)

††facilities: HST (COS); MAST.††software: matplotlib (v3.2.2; Hunter 2007),
astropy (v4.2.1; Astropy Collaboration et al. 2013, 2018), CLOUDY (v.17;
Ferland et al. 2017), numpy (v1.22.0; Harris et al. 2020), scipy (v1.6.2;
Virtanen et al. 2020), pandas (v1.3.5; Reback et al. 2021).

##  1 INTRODUCTION

Most of the baryonic matter in the universe is not contained in stars and
galaxies, but is between galaxies in a dilute, multi-phase, ionized gas called
the Intergalactic Medium (IGM; Meiksin 2009; McQuinn 2016). This reservoir is
thought to regulate the growth of galaxies by facilitating accretion (e.g.,
Kereš et al. 2005; Dekel & Birnboim 2006; Dekel et al. 2009; Hafen et al.
2022; Decataldo et al. 2023) and harboring a large fraction of what gets
ejected through outflows (e.g., Martin 1999; Martin et al. 2010; Steidel et
al. 2010; Peeples et al. 2014; Oppenheimer et al. 2016). Many studies of this
diffuse gas have been done at intermediate redshifts ($z\approx 2-5$) to allow
for the simultaneous detection of multiple Lyman transitions of hydrogen as
well as metals such as C IV or O VI (e.g, Bergeron et al. 1994; Jannuzi et al.
1998; Lopez et al. 1999; Richter et al. 2004; Simcoe et al. 2004; Adelberger
et al. 2005; Chen et al. 2005; Danforth & Shull 2008; Turner et al. 2014;
Morrison et al. 2021; Borthakur 2022).

Metals, in particular, are an import tracer of this diffuse gas as 70% of
Ly$\alpha$ forest absorbers are found to have accompanying metal lines (Simcoe
et al., 2004) and can be present even when the Lyman series is weak (Danforth
et al., 2016). The IGM at $z\approx 2-3$ is also known to be enriched with
carbon and oxygen (e.g., Davé et al. 1998; Aracil et al. 2004; Pieri et al.
2006) and is thought to have retained it and other metals to the present day
(e.g., Richter et al. 2004; Aguirre et al. 2008; Danforth & Shull 2008; Tripp
et al. 2008; Muzahid et al. 2012; Danforth et al. 2016). Additionally, metal
absorption lines are often unsaturated, allowing for more components within a
single cloud to be detected (e.g., Chen & Mulchaey 2009; Danforth et al. 2016;
Pachat et al. 2017; Sankar et al. 2020; Ahoranta et al. 2021). This makes
metals a key window into the ionization processes that govern the IGM.

Furthermore, analyzing metals has revealed the multiphase nature of the IGM
(e.g., Heckman et al. 2002; Savage et al. 2005; Narayanan et al. 2009; Shull
et al. 2012; Ahoranta et al. 2021; Haislmaier et al. 2021), with a cool
$T\approx 10^{4.5}$ K phase and a warm-hot phase, known as the warm ionized
IGM or WHIM, at $T\approx 10^{5}-10^{6}$ K. The presence of the WHIM could
indicate that collisions are likely a dominate ionization process in the IGM
since many of these processes produces radiatively cooling gas at the
intermediate temperatures O VI is found at (e.g., Begelman & Fabian 1990;
Heckman et al. 2002; Gnat et al. 2010; Kwak et al. 2015; Ji et al. 2019),
although multiple studies have assumed photoionization equilibrium (PIE) or
collisional ionization equilibrium (CIE) to estimate IGM masses and densities
(e.g., Lehner et al. 2007; Sobacchi & Mesinger 2013).

In the largest, most complete IGM survey to date, Danforth et al. (2016)
identified 5138 extragalactic absorption features along 82 QSO sight lines
taken with the Cosmic Origins Spectrograph (COS) aboard the Hubble Space
Telescope (HST; Osterman et al. 2011; Green et al. 2012) with signal-to-noise
ratios above 15. They grouped features found at similar redshifts into 2611
absorbing systems. 16% of these had at least one metal line, with O VI being
most frequently detected, similar to what other studies have found in the
local universe (e.g., Danforth & Shull 2008; Tilton et al. 2012). The number
of absorbers per unit $z$ ($\frac{d\mathcal{N}}{dz}$) of O VI and H I were
found to increase with $z$; however, this was not seen in N V, C III, or Si
III which does not seem to evolve (Danforth et al., 2016).

Despite the progress that has been made, it is still unclear whether this
diffuse gas is in ionization equilibrium $-$ photoionization or collisional
ionization $-$ or if non-equilibrium processes are needed to explain
observations. To that end, we have analyzed the absorption features from
Danforth et al. (2016) which allow us to constrain the physical processes
driving ionization in the IGM in the context of equilibrium and non-
equilibrium interactions. The rest of this paper is outlined as follows:
Section 2 details how our sample was selected as well as the measurements used
in our analysis. Section 3 presents the analysis and the results from it.
Finally, we summarize in Section 4 and discuss future directions.

##  2 Sample & Measurements

###  2.1 Sample Selection

The presence of high ionization transitions can be used to constrain the
physical nature of the IGM. In particular, warm-hot metals such as C IV
($\lambda\lambda 1548,1550$ Å; 64.5 eV), N V ($\lambda\lambda 1238,1242$ Å;
97.9 eV), and O VI ($\lambda\lambda 1032,1038$ Å; 138.1 eV), trace the
energies required to ionize the IGM and provide a unique window into some of
the commonly observed non-equilibrium processes that are most likely
responsible for the ionization state of the IGM. To that end, we consider all
absorbing clouds (i.e., sets of absorption features which are aligned in
velocity space) from Danforth et al. (2016)’s Mikulski Archive for Space
Telescopes (MAST) catalog111https://archive.stsci.edu/prepds/igm/ and examine
those with 3$\sigma$ detections of these three ions. While it is possible that
absorption features are kinematically aligned by chance and not actually
associated with one another, the likelihood is very small when multiple ions
are found at similar velocities. As a result, we assume all kinematically
aligned ions originate from a single absorbing cloud.

The selection criteria yield five clouds along four sight lines that showed
absorption in all three transitions. Each of these clouds were best fit by a
single Voigt profile or component. The sight line PG 1216+069 contained two
clouds with a velocity separation of $\sim$75 km s-1. Although the Danforth et
al. (2016) identified them as independent, these could be associated with a
larger structure.

Even though only five absorbing clouds from the Danforth et al. (2016) catalog
are included in our complete sample, these are representative of the IGM. The
small number can be attributed to the narrow $z$ range within which C IV and O
VI can be simultaneously observed using COS. At minimum, $z$ needs to be
$\gtrsim$0.094 to have at least one O VI feature within the G130M grating, and
$z\gtrsim 0.100$ to observe the stronger $\lambda 1032$ Å line. Meanwhile, any
cloud with $z\gtrsim 0.153$ will have both C IV features shifted out of the
G160M grating. Only 341 clouds were within the allowable range, limiting the
number that could have been included. Most of these only contained Ly$\alpha$
with no associated metal features (246 of 341). O VI, the most frequently
detected metal, appeared in 44 individual clouds ($\mathcal{N}$), implying
$\frac{d\mathcal{N}}{dz}(\mathrm{O\ VI})\approx 9$, which is comparable to
what Danforth & Shull (2008) found for O VI ($15^{+3}_{-2}$). The fact that
the 82 QSOs from Danforth et al. (2016) are randomly distributed across the
sky indicates that these absorbers are likely more prevalent than the small
sample size would imply.

###  2.2 Voigt Profile Measurements

We fit Voigt profiles to absorption features to determine the column density
($N$), Doppler width ($b$), and relative velocity or velocity centroid
($v_{\mathrm{obs}}$) of the absorbing gas and check for consistency with the
analysis from Danforth et al. (2016). In addition, Danforth et al. (2016) fit
each absorption profile individually whereas we fit doublets, such as C IV and
O VI, simultaneously. Before fitting, we normalized the continuum within
$\pm$600 km s${}^{-1}\ $of the cloud’s $z$, which we refer to as
$z_{\mathrm{sys}}$, and center the features such that $v_{\mathrm{obs}}$ is
always near 0 km s-1. The absorption features were fit using a reduced
$\chi^{2}$ algorithm (Sembach & Savage, 1992). Once complete, these Voigt
profile parameters can be used to constrain the ionization processes happening
in the IGM. We show the spectrum and associated fits of this analysis for the
4 sightlines in Appendix.

We present the absorber properties used in our analysis in Table 1. The
$v_{\mathrm{obs}}$ returned by our fits are not reported since each was
centered on $z_{\mathrm{sys}}$ before fitting. If the best-fit Doppler width
($b$) of an absorber is narrower than 5 km s-1, we fix $b$ to 5 km s${}^{-1}\
$and report the resulting column density ($N$). This is motivated by the COS
line spread function since it is unable to discern between $b$ values $\leq 5$
km s${}^{-1}\ $at a signal to noise ratio between 10 and 20. This is a
limiation of the data. It is possible for these clouds to be associated with
extremely low Doppler width and much higher column density. Most of our
measurements were consistent with those from Danforth et al. (2016) and so we
adopt our values. We also fit the Si IV ($\lambda\lambda 1393,1402$ Å; 45.1
eV) features of our clouds as they provide an additional probe of the
ionization mechanisms; however, we do not include the ion in our selection
criteria as it may not completely trace the WHIM as indicated by its low
ionization potential (Cen & Ostriker, 1999; Davé et al., 2001; Cen & Ostriker,
2006; McQuinn, 2016). The lower ions Si II ($\lambda\lambda 1260,1193,1190$ Å;
16.4 eV), C II ($\lambda 1334$ Å; 24.4 eV), and Si III ($\lambda 1260$ Å; 33.5
eV) were also fit when detected. If no absorption is present, we measure twice
the error of the rest-frame equivalent width within a 100 km s${}^{-1}\
$window that does not contain intervening absorption. This is then converted
to an upper limit on $N$, assuming we are in the linear portion of the curve
of growth. Neither absorbing cloud along the sightline PG 1216+069 contained
Si II or C II, so we report the same upper limits for both clouds.

Any Voigt profile analysis is limited by the resolution of the spectrograph
used. The COS instrument currently has the highest spectral resolution at
rest-frame FUV wavelengths that could observe the QSOs in the Danforth et al.
(2016) sample. However, it is important to note that it is a medium resolution
instrument ($R$ $\approx$ 20,000; FWHM $\approx$ 15 km s-1). Thus, it is
possible that multiple narrow clouds at similar velocities could appear as a
single, wider component. We assume that the measurements presented here are
dominated by the largest absorbing cloud with the understanding that higher
resolution observations may reveal a more complicated picture.

Table 1: Summary of Measurements

QSO | $z_{\mathrm{sys}}$ | Ion | log$(N/\mathrm{cm}^{-2})$ | $b\left[\mathrm{km\ s}^{-1}\right]$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
(1) PG 1116+215 | 0.13853 | H I | $15.30\pm 0.03$ | $28.9\pm 1.0$
Si II | $12.93\pm 0.04$ | $\leq 5^{\star}$
Si III | $13.73\pm 1.31$ | $5.8\pm 1.9$
C II | $13.98\pm 0.13$ | $8.6\pm 1.3$
Si IV | $12.83\pm 0.20$ | $5.9\pm 4.2$
C IV | $13.40\pm 0.09$ | $\leq 5^{\star}$
N V | $12.78\pm 0.06$ | $15.0\pm 1.2$
O VI | $13.82\pm 0.02$ | $30.8\pm 1.1$
(2) PG 1216+069 | 0.12360 | H I | $14.63\pm 0.05$ | $22.8\pm 1.1$
Si II |  $<$ 11.88 | —
Si III | $12.53\pm 0.21$ | $7.3\pm 2.9$
C II |  $<$ 12.93 | —
Si IV | $12.71\pm 0.11$ | $19.5\pm 2.6$
C IV | $14.24\pm 0.24$ | $10.4\pm 1.3$
N V | $13.42\pm 0.09$ | $\leq 5^{\star}$
O VI | $14.19\pm 0.05$ | $20.2\pm 1.4$
(3) PG 1216+069 | 0.12389 | H I | $14.86\pm 0.06$ | $24.96\pm 1.1$
Si II |  $<$ 11.88 | —
Si III | $12.51\pm 0.10$ | $\leq 5^{\star}$
C II |  $<$ 12.93 | —
Si IV | $12.84\pm 0.12$ | $\leq 5^{\star}$
C IV | $13.99\pm 0.15$ | $11.0\pm 1.3$
N V | $13.04\pm 0.12$ | $10.4\pm 13.6$
O VI | $14.19\pm 0.06$ | $32.0\pm 1.4$
(4) PG 1424+240 | 0.14713 | H I | $15.56\pm 0.12$ | $27.5\pm 1.1$
Si II |  $<$ 11.78 | —
Si III | $14.31\pm 1.21$ | $5.4\pm 1.7$
C II |  $<$ 12.82 | —
Si IV | $12.99\pm 0.16$ | $7.8\pm 2.7$
C IV | $14.24\pm 0.16$ | $14.4\pm 1.3$
N V | $13.16\pm 0.16$ | $33.6\pm 1.9$
O VI | $14.07\pm 0.43$ | $\leq 5^{\star}$
(5) PKS 0637-752 | 0.12288 | H I | $15.36\pm 0.40$ | $30.5\pm 1.2$
Si II |  $<$ 11.74 | —
Si III | $14.05\pm 1.18$ | $8.2\pm 1.7$
C II |  $<$ 12.86 | —
Si IV0 | — | —
{3*C IV | $13.65\pm 0.04$ | $38.8\pm 1.2$
N V | $14.05\pm 0.04$ | $71.9\pm 1.1$
O VI | $14.08\pm 0.03$ | $44.0\pm 1.1$

††footnotetext: Column (1) shows the quasar that the system is towards. Column
(2) is the redshift of the system. Column (3) indicates the ion being fit.
Columns (4) and (5) are the column density and Doppler width of the ion in
units of cm-2 and km s-1, respectively. ⋆ Doppler width was fixed at 5 km
s${}^{-1}\ $due to the absorption line being narrow. See Section 2.2 for
details. It is worth noting that these features are consistent with extremely
narrow ($b<<5$ km s-1) features implying large column densities. Measurements
that are consistent with CIE models (Gnat & Sternberg, 2007) are highlighted
with a bracket. 0 Both transitions fell within a gap in the data, so a fit
could not be completed.

###  2.3 Comparison to Literature Measurements

Many of the clouds we analyze here have been identified by other studies and
fit many of the same transitions as us. We compare our measurements to those
in the literature below.

In the system along PG 1116+215, Sembach et al. (2004), Tripp et al. (2008),
Tilton et al. (2012), and Muzahid et al. (2018) found similar values of log$N$
and $b$ to what we measure for most metals. However, Tilton et al. (2012)
found larger log$N$ values for H I and N V than our fits ($\sim$0.4 and
$\sim$1 dex more, respectively). Muzahid et al. (2018) found log$N$ of H I to
be larger than we report while finding less C IV. The fit from Sembach et al.
(2004) gave a larger log$N$ and smaller $b$ than us. Meanwhile, Tripp et al.
(2008) fit the H I absorption as two components, both of which have narrower
$b$ and larger log$N$ values than we find.

As discussed in Section 2, the small velocity separation of the two absorbing
clouds towards PG 1216+069 could cause them to be treated as one system which
is what Tripp et al. (2008), Chen & Mulchaey (2009), and Tilton et al. (2012)
have done. Tripp et al. (2008) found values for O VI which are consistent with
what we measure. Their H I measurements for the system at $z=0.12360$ are
consistent with what we find, though they find a larger log$N$ and smaller $b$
than we do for system 3 at $z=0.12389$. The O VI features measured by Chen &
Mulchaey (2009) have similar, but inconsistent, log$N$ as we find while their
$b$ value of the system at $z=0.12360$ is narrower. For Si III, Chen &
Mulchaey (2009) set $b=2.4$ km s${}^{-1}\ $and found the corresponding best
fit log$N$, which is consistent with what we find. The H I features of both
clouds were reported as lower limits which is consistent with the system at
$z=0.12360$ but not the system at $z=0.12389$; however, Ly$\beta$ was not
included in their fits which may have impacted this. Tilton et al. (2012)
measured similar log$N$ and $b$ values for O VI in both systems.

For the PG 1424+240 system, Muzahid et al. (2018) measured log$N$ values for C
IV and O VI that are consistent with what we find. However, they measured
larger values of log$N$ for H I, C II, and Si II than we do. Part of the
difference in H I can likely be attributed to how the values are reported.
They report a total log$N$ of H I from a fit with four components while we fit
the feature with two components and only report the value of the single cloud
which is kinematically aligned with the associated metal lines. The
discrepancy for the C II and Si II features can be attributed to how the lines
were identified. Danforth et al. (2016) labeled these as weak H I absorbing
clouds, and so we report an upper limit for these metals where Muzahid et al.
(2018) fit Voigt profiles.

In the PKS 0637$-$752 system, Johnson et al. (2017) fit two components to the
H I, C IV, and O VI features and one component to Si III whereas we fit a
single component to all of the metals features and two components for H I.
This difference caused our $b$ values of C IV and O VI to be larger than what
they find, though we measured similar total log$N$ values for all metals. Our
log$N$ value for H I is consistent with what they find, though we find a
larger $b$ value.

##  3 Results & Discussion

###  3.1 Collisional Ionization Equilibrium Models

To determine whether the absorbing clouds we have identified are in
collisional ionization equilibrium (CIE), we compare the measurements to the
solar metallicity (Asplund et al., 2009) models from Gnat & Sternberg (2007).
PKS 0637$-$752 is the only sight line that is consistent with CIE at a
temperature near 105.3 K, which we show in Figure 1. The remaining four clouds
had less N V than predicted. Gatuzz et al. (2023) looked at the cloud along
the PG 1116+215 sight line and concluded that it was not in CIE, consistent
with what we find here.

Figure 1: Comparing our measurements of the PKS 0637-752 cloud to CIE models
(Gnat & Sternberg, 2007). Measurements are shown as dashed lines with
uncertainties as shaded regions while CIE models are shown as solid lines. The
color indicates the ion being plotted. Black for Si IV, green for C IV, red-
orange for N V, and purple for O VI. Both Si IV lines fell within a gap in the
data and so a measurement could not be made. The black dotted line shows where
the cloud is in CIE at $T\approx 10^{5.3}$ K.

###  3.2 Photoionization Equilibrium Models

The photoionization equilibrium (PIE) code CLOUDY (v.17; Ferland et al. 2017)
allows us to explore whether our measurements can be explained by an incident
radiation field alone. Each model was exposed to a Haardt & Madau (2012)
extragalactic UV background and was iterated until the Ly$\alpha$ column
density was reached. The metallicities of the features in our sample are
assumed to be solar (Asplund et al., 2009). To determine the total H density
(log $n_{\mathrm{H}}$) of the clouds, we varied log ($n_{\mathrm{H}}$/cm-3) in
steps of 0.1 dex from $-7$ to $-2$.

No absorbing cloud was found to be consistent with PIE models, regardless of
the log $n_{\mathrm{H}}$ used. We show the measurements of the absorbing cloud
towards PG 1216+069 at $z=0.12360$ in Figure 2 as an example. This shows that
there is no density consistent with the measured values for all four ions from
these models (log ($n_{\mathrm{H}}$/cm-3) $\approx$ $-$4.1, $-$4.8, $-$4.9,
and $-$5.1 for Si IV, C IV, N V, and O VI respectively). So the absorbing
cloud is not consistent with being in PIE. Changing the metallicity of the
models would not impact this conclusion as this would move all of the models
up or down together concurrently and would not impact their ratios. In four of
five absorbing clouds, C IV and O VI were found in similar amounts, which
CLOUDY was not able to match without needing significantly more N V than we
measured. These show that photoionization is not the dominant ionization
process for most high ions, unlike previously believed (e.g., Narayanan et al.
2009; Muzahid et al. 2011).

Figure 2: Comparing our measurements of the PG 1216+069 absorbing cloud at
$z=0.12360$ (system 2) to PIE models (Ferland et al., 2017). Measurements are
shown as horizontal dashed lines with uncertainties as shaded regions while
PIE models are shown as solid lines. The log $n_{\mathrm{H}}$ values which
best matches our measurements are highlighted with vertical dotted lines. The
color indicates the ion being plotted. Black for Si IV, green for C IV, red-
orange for N V, and purple for O VI.

###  3.3 Non - Equilibrium Models

Since only one cloud is consistent with an equilibrium model, we investigate
our measurements for consistency with non-equilibrium models. Figure 3 shows
the non-equilibrium models as well as the CIE and PIE models previously
discussed. Below, we detail each model while comparing the expected column
density ratios for solar metallicity and relative abundances to our results.

Figure 3: Comparing our measurements to various ionization models. We show our
measurements for the 5 clouds as red stars while the stacked spectra is shown
as a red diamond, which we discuss in Section 3.5. The measurement
uncertainties were always smaller than the size of the points and so are not
shown. Each cloud is label with their corresponding number from Table 1. CIE
models are shown as circles with the color corresponding to the model’s
temperature as shown in the color bar. The black solid ring highlights the
cloud towards PKS 0637-720 which is consistent with being in CIE. PIE models
are shown as a solid blue line. The dashed lines indices where the different
non-equilibrium ionization models are expected to reside with the coloring
indicating the model. Black for SI, green for RC, red-orange for CI, and
purple for TML. These models are discussed in Section 3.3.

Shock ionization (SI) can occur when gas clouds move through an ambient medium
at velocities above the local sound speed. In the IGM, this can be the result
of galactic outflows giving clouds enough energy to escape the galaxy’s dark
matter gravitational potential. When this happens, the temperature gets raised
behind the cloud which causes higher ionization states to be populated, which
Dopita & Sutherland (1996) has modeled using clouds with varying shock
velocities (150 and 500 km s-1) and magnetic parameters (0 $\mu$G cm-3/2
$\leq$ $B_{0}~{}n_{0}^{-3/2}$ $\leq$ 4 $\mu$G cm-3/2). These models are shown
in Figure 3 in black. Most of our systems (four of five) have C IV:O VI ratios
consistent with the predicted values. However, in each case we see
significantly more N V than what is expected based on the models.

Radiative cooling (RC) via recombination can produce warm-hot ions as the gas
temperature decreases from $>$106 K. By cooling gas under a variety of
conditions, Edgar & Chevalier (1986) was able to predict the column density
ratios one would see if RC is the dominate ionization method. The RC models
for flow velocities of 100 km s${}^{-1}\ $are shown in green in Figure 3.
These ratios over predict the amount of O VI we see in all of the clouds. In
two clouds, we find nearly an order of magnitude more C IV than expected from
the models. As a result, we conclude that RC is not a prominent ionization
mechanism in these absorbing clouds.

Conductive interfaces (CI) occur when media at different temperatures come
into contact with each other; for example, when a cool cloud from a galaxy’s
interstellar medium gets ejected into the warm-hot IGM. Collisions at the
contact surface will transfer energy (and temperature) to the colder gas,
producing the warm-hot metals we are interested in. Borkowski et al. (1990)
modeled the contact surface between hot 106 K gas and cooler interstellar
clouds, with the results being shown in Figure 3 in red-orange. These models
vary the angle of the magnetic field orientation between 0$\degree$ and
85$\degree$ as well as interface ages between $10^{5}$ and $10^{7}$ yr. This
age range suggests that these interfaces die out rather rapidly. Star
formation driven galactic winds in starburst galaxies are known to produce
high-ionization transitions such as C IV and O VI in the outer CGM and IGM
(Adelberger et al., 2005; Borthakur et al., 2013; Heckman et al., 2017;
Méndez-Hernández et al., 2022; Banerjee et al., 2023), although the non-
equilibrium nature of those systems are not fully explored. Only system 3 (the
cloud at $z=0.12389$ towards PG 1216+069) is consistent with the predicted
ratios in both panels of Figure 3. Meanwhile, system 1 (the cloud towards PG
1116+215) and system 2 (the cloud at $z=0.12360$ towards PG 1216+069) are only
consistent with the left and right panels, respectively, though system 2 is
near the boarder in the left panel. The PKS 0637$-$752 cloud is in CIE and has
more N V than the models predict, suggesting it may have once had a CI that
has since dissipated as the cloud reached CIE. Given that the ionization
potential of Si IV allows it to be produced by the colder IGM gas phases than
C IV or the hotter ions, we believe that the C IV measurements are a more
robust tracer of the WHIM phase in the IGM. As a result, we conclude that the
ionization of systems 1, 2, and 3 can largely be explained by CIs resulting
from energetic processes impacting the IGM.

When a turbulent hot gas comes into contact with a colder medium, Kelvin-
Helmholtz instabilities can form, causing a mixing of different gases which
are referred to as Turbulent Mixing Layers (TMLs). These layers are at high
enough temperatures to produce the highly ionized species we analyze here.
Slavin et al. (1993) expanded upon Begelman & Fabian (1990) to produce a model
of TMLs over a range of temperatures and gas velocities. These models, which
we show in purple in Figure 3, correspond to entrainment velocities between 25
and 100 km s${}^{-1}\ $as well as temperatures between $10^{5.0}$ and
$10^{5.5}$ K. These models stand out amongst those we analyze because of the
large C IV:O VI and Si IV:O VI ratios predicted. The cloud along PG 1216+069
at $z=0.12360$ (system 2) is near the boundary of TML in the left panel,
suggesting this is a major contributor to its ionization state in addition to
CIs.

The cloud towards PG 1424+240 (system 4) does not match the predicted values
for any of the models we explore here. Though it is near the boarder of CIs
and TMLs in both panels of Figure 3. This could indicate that these mechanisms
are playing some role in the observed ionization state; however, there could
other processes impacting these ionic ratios. The uncertainty of the O VI
log$N$ may also play a part in its positional inaccuracy on Figure 3 as
discussed in Section 2.2.

To summarize these results, most of the clouds we analyze are consistent with
CIs when comparing C IV measurements. Of which, one also match the predicted
ratios for TMLs. This is similar to what was found for Milky Way HVCs (Fox et
al., 2005). The models we investigate here are not able to reproduce the
observed ratios of one system. These results indicate that the IGM may be
predominantly ionized through non-equilibrium processes. More work is needed
with a larger sample, however, to be able to draw more definitive conclusions.

These results could change if the relative abundances are largely different
from solar values. Though this is unlikely to be the case in the absorbing
clouds we investigate here given the low redshifts they reside at and no study
has found strong evidence for non-solar relative abundances in the IGM or
outer CGM. Additionally, Si IV can be produced by multiple gas phases, not
just the warm-hot phase. This means the placement in the right panel of Figure
3 can be thought of as upper limits in the $y$-axis. The multi-phase nature of
Si IV does not impact the conclusions drawn here, given that the vast majority
of the ionic abundances would need to be produced by cooler gas phases to
change our interpretations.

###  3.4 Possible Sources of Ionization

With it being apparent from the above results that non-equilibrium processes
are the dominate way in determining the ionization state of the IGM, it is
important to look for the sources driving the non-equilibrium processes.
Galaxies with large outflows are capable of driving clouds out of the galaxy
to the CGM and IGM (Oppenheimer et al., 2012; Somerville & Davé, 2015). This
is why using a galaxy’s virial radius as boundary between the CGM and IGM is
often insufficient or misleading, given that processes which occur at or near
this border are likely to persist to further radii with little changing (e.g.,
Nelson et al. 2019). To determine whether or not there are galaxies near the
sight lines that could be responsible for the ionization processes we infer in
Section 3.1, we have performed a literature review of the absorbing clouds in
our sample with the results summarized below and in Table 2. The Sloan Digital
Sky Survey and other large spectroscopic surveys are shallow at these
redshifts. Thus, deeper individual surveys are needed to identify nearby
galaxies.

The sight lines PG 1116+215, PG 1216+069, and PG 1424+240 were found to pass
within 140 kpc (Tripp et al., 2008; Muzahid et al., 2018; Scott et al., 2021),
94 kpc (Chen & Mulchaey, 2009; Scott et al., 2021), and 132 kpc (Scott et al.,
2021), respectively, of galaxies at similar redshifts as these clouds. The
virial radii of these galaxies, as shown in Table 2, suggest that the sight
lines PG 1116+215 and PG 1216+06 probe the boundary between the outer CGM and
IGM. Given that CIs are shortly lived processes, these galaxies are likely
responsible for sourcing the observed ionization in these clouds. The sight
line PG 1424+240 is the only cloud located outside the nearby galaxy’s virial
radius. This suggests that the absorbing cloud escaped the gravitational
potential of the galaxy and interacted with the ambient IGM to produce the
observed ionization.

The PKS 0637$-$752 sight line stands out as it is only $\sim$16 kpc from a
galaxy at the same redshift as the cloud in our sample, putting it into the
inner CGM of this star-forming dwarf galaxy (M${}_{\star}\approx 10^{7.9}$ M⊙;
Johnson et al. 2017). It is especially note worthy that this is the only cloud
found to be in CIE at a temperature of $\sim$105.3 K. This temperature is
close to the virial temperature of this galaxy (T $\approx 10^{5.1}$ K),
assuming a stellar mass to halo mass conversion of Kravtsov (2013) and virial
temperature estimation as described in Wang & Abel (2008). In this case, we
are most likely observing the virialized halo of this galaxy, although heated
gas from feedback processes cannot be ruled out.

Table 2: Summary of Possible Ionization Sources

Sight line | $z_{\mathrm{abs}}$ | $z_{\mathrm{gal}}$ | Rvir | $\rho$ | Process(es)
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
PG 1116+215 | 0.13853 | 0.138 | 172 | 140 | CI
PG 1216+069 | 0.12360,0.12389 | 0.124 | 125 | 94 | CI, TML
PG 1424+240 | 0.14713 | 0.15 | 81 | 132 | —
PKS 0637$-$752 | 0.12288 | 0.1229 | 70 | 16 | CIE

††footnotetext: Column (1) indicates the sight line. Columns (2) and (3) are
the redshift of the absorbing cloud and the galaxy, respectively. Column (4)
is the virial radius of the foreground galaxy in units of kpc. Column (5) is
the projected distance between the QSO and galaxy in units of kpc. Column (6)
shows the ionization processes consistent with the log$N$ ratios in Table 1.
Acronyms refer to Conductive Interfaces (CI), Turbulent Mixing Layers (TML),
and Collisional Ionization Equilibrium (CIE).

###  3.5 Stacked Spectra

While analyzing individual absorbing clouds can tell us about the processes
taking place within these few examples, stacking the spectra of many IGM
clouds allows us to better quantify the average strength of the lines. We mean
stacked the 341 IGM clouds, both detections and nondetections, in Danforth et
al. (2016) with a redshift that allows C IV and O VI to be observed, with the
results being shown in Figure 4. The spectra was centered on the $z$ of the
Ly$\alpha$ before stacking. After stacking all IGM clouds, we normalized the
continuum using polynomials of 2nd or 3rd order. The errors from individual
spectra were added in quadrature to obtain the stacked spectra errors, which
we show in the bottom panel of Figure 4.

The log$N$ and $v_{\mathrm{obs}}$ of the stacked spectra were constrained
using the Adaptive Optical Depth (AOD) method (Savage & Sembach, 1991; Lehner
et al., 2020), which we were able to use since the features included in the
stacks were in the linear region of the curve of growth. The IGM line lists
published by Danforth et al. (2016) contain a flag indicating whether or not
the line is saturated, which we use to confirm that the clouds included in the
stacks are in the linear region of the curve of growth. The AOD method uses
the normalized flux in velocity space to estimate the apparent optical depth
in each pixel such that
$\uptau_{a}(v)=\mathrm{ln}\left[F_{\mathrm{cont}}(v)/F_{\mathrm{obs}}(v)\right]$.
Since we are in the linear region of the curve of growth, the apparent column
density in each pixel can be found by assuming the absorber is unsaturated
such that $N_{a}(v)=3.768\times 10^{14}\
\uptau_{a}(v)/(f\lambda[\mathrm{\AA}])$ $\left[\mathrm{cm}^{-2}\ (\mathrm{km\
s^{-1}})^{-1}\right]$. This gives $N$ by integrating over the velocity range
of the absorber. We define the width of the stacks as half the velocity width
where the normalized flux is 1$\sigma$ below the continuum ($V_{1\sigma}$)
after binning the spectra by 4 pixels. These results are presented in Table 3.

Figure 4: Stacked spectra for the expected location of Si IV (left) C IV
(center left), N V (center right), and O VI (right) transitions for all IGM
clouds in Danforth et al. (2016). The upper panels show the stacked flux while
the bottom panels show the correspond to the associated stacked error. The
grey region indicates the range used to calculate equivalent width, which we
show in the bottom right of the upper panels in units of mÅ.

Weak absorption was detected in the C IV and O VI spectra at $>$5.5$\sigma$.
No features were measured in Si IV or N V, even with the higher signal-to-
noise ratios. As as result, we report the upper limits for these ions as twice
their associated uncertainties within $\pm$50 km s-1. While it appears by eye
that narrow features are present in Si IV and N V, they were not detected at
2$\sigma$. It may not be surprising that we do not detect Si IV or N V in the
stack given that the strength of an absorption line is dictated by the product
of the elemental abundance, the fraction of the element in the ionization
state, and oscillator strength of the line. For the strongest transitions of C
IV and O VI, the product of their abundances and oscillator strengths are
similar while the product for Si IV and N V are an order of magnitude smaller
(see Table 4 from Morton et al. (1988)). Thus, Si IV / Si and N V / N would
need to be $\gtrsim$10 times C IV / C and O VI / O for each ion to produce
profiles of similar strengths.

The AOD method reveals that, on average, C IV and O VI are kinematically
aligned with their associated Ly$\alpha$ absorbers, with
$\left|v_{\mathrm{obs}}\right|<20$ km s${}^{-1}\ $in both stacks. From the
individual absorping clouds we have analyzed, we believe this trend would
continue in Si IV and N V if the absorption was stronger.

We constrain the distribution of $v_{\mathrm{obs}}$ of the clouds that went
into the stacks by measuring $V_{1\sigma}$ for the combined features. The
widths of C IV and O VI suggest hotter ions are more likely to differ
kinematically from their Ly$\alpha$ absorbers. We hypothesize that these are
caused by gas kinematics responsible for non-equilibrium processes producing
the coronal transitions.

As seen in Figure 3, the column density ratios of the stacked spectra are
consistent with only SI models. In addition, these ratios are inconsistent
with PIE and CIE models, suggesting that the ionization of the IGM is
frequently driven by non-equilibrium processes. This shows that while SIs do
not play a major role in the individually absorbing clouds we have analyzed,
they may play a part in the overall ionization of the IGM. However, it is
worth noting that the stacked spectra are not suitable for ratio studies and
these results may suffer from issues with averaging multiple populations as
one.

Table 3: Summary of Stacked Spectra

Ion | $W\left[\mathrm{m\AA}\right]$ | log$\left(N/\mathrm{cm}^{-2}\right)$ | $v_{\mathrm{obs}}\left[\mathrm{km\ s}^{-1}\right]$ | $V_{1\sigma}\left[\mathrm{km\ s}^{-1}\right]$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
Si IV |  $<$1.92 |  $<$11.34 | — | —
C IV | $13.01\pm 2.47$ | $12.51\pm 0.05$ | $-0.8\pm 30.58$ | $43.0\pm 21.5$
N V |  $<$1.15 |  $<$11.73 | — | —
O VI | $16.88\pm 1.43$ | $13.13\pm 0.04$ | $-18.5\pm 12.91$ | $85.9\pm 21.5$

††footnotetext: Column (1) shows the ion being stacked. Column (2) is the
equivalent width of the stacked spectra. Columns (3) and (4) are the column
density and velocity centroid of the stacked spectra measured through the AOD
method in units of cm-2 and km s-1, respectively. Column (5) indicates the
1$\sigma$ width of the stacked spectra in units of km s-1.

##  4 Summary

In this work, we explore the ionization processes responsible for the
ionization of the WHIM phase of the IGM in the local universe by analyzing
absorbing clouds with C IV, N V, and O VI detected at $\geq$3$\sigma$ from
Danforth et al. (2016). Our results are summarized as follows:

  1. 1.

Only one cloud was found to be in equilibrium (system 5), specifically CIE
with a temperature near 105.3 K. This sight line is also only $\sim$16 kpc
from a star-forming dwarf galaxy, hinting that the higher densities found
closer to galaxies allow the diffuse halo gas to cool faster than it does
further away, where the densities are lower.

  2. 2.

None of the clouds were found to be consistent with PIE models, even though
the IGM is frequently assumed to be in PIE. Each absorbing cloud required
multiple densities to reproduce the observed abundances. In particular, these
models were not able to reproduce the amount of C IV and O VI we see without
needing significantly more N V than was found.

  3. 3.

We compare our observations to four non-equilibrium models: SI, RC, CI, and
TML. No system was found to be consistent with the expected ratios from SI or
RC models. The system towards PG 1216+069 at $z=0.12389$ (system 3) is
consistent with the expected values for CIs when comparing to C IV and Si IV;
however, the system towards PG 1116+215 (system 1) and the one at $z=0.12360$
towards PG 1216+069 (system 2) only match when comparing to C IV and Si IV,
respectively. System 2 is also on the boarder of the predicted ratios of the
TML models in the left panel of Figure 3, suggesting they are likely
contributing to its ionization state in addition of CIs.

  4. 4.

The ionization models we explore here cannot reproduce the ratios of the cloud
along the PG 1424+240 sight line (system 4). We note that this system is near
the boarder of CIs and TMLs in both panels of Figure 3, which could indicate
that these mechanisms are playing some role in the observed ionization state.
However, other processes may also be in play since the feature is narrow well
beyond the COS line spread function. The true uncertainty in column density is
large.

  5. 5.

Stacking the spectra of all absorbing clouds within the redshifts that allows
for the simultaneous detection of C IV and O VI revealed faint absorption
features at $>$5.5$\sigma$ in C IV and O VI. However, Si IV and N V were not
detected at 2$\sigma$. The column density ratios of these stacks are
consistent with SI models, suggesting that SIs may be another prominent
ionization mechanisms in the IGM, even if not in the individual clouds
presented here.

From these it is clear that the IGM in the local universe is ionized primarily
by non-equilibrium processes, in particular in the outer CGM and IGM. Further
investigation of these three coronal lines (C IV, N V, and O VI) in star
forming galaxies, with HST programs such as COS-MAGIC (HST-GO-17093), will
shed more light on the processes which drive the ionization of the CGM and
IGM.

We thank the anonymous referee for the helpful comments which improved the
paper. B.K. and S.B. were supported by NASA/STScI through grants HST-GO-17093
and NSF grant 2108159. B.K. would like to thank Tyler McCabe for their
fruitful discussions regarding this project. B.K., A.R., and S.B. acknowledge
the native people and the land that Arizona State University’s campuses are
located in the Salt River Valley. The ancestral territories of Indigenous
peoples, including the Akimel O’odham (Pima) and Pee Posh (Maricopa) Indian
Communities, whose care and keeping of these lands allows us to be here today.

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Figure A1: Normalized spectrum of the absorbing cloud along the sightline PG
1116+215. Each panel was centered on the Ly$\alpha$ velocity of the cloud
before fitting. The normalized flux and uncertainties are shown in black and
red, respectively. The associated Voigt profile fits are shown in green.
Intervening absorbers near $v_{\mathrm{sys}}$ that are different from the ion
being plotted are labeled in blue. Figure A2: Same as Figure A1, but for the
absorbing clouds along the sightline PG 1216+069. Figure A3: Same as Figure
A1, but for the absorbing cloud along the sightline PG 1424+240. Figure A4:
Same as Figure A1, but for the absorbing cloud along the sightline PKS
0637-752.


# Triangle singularity in the
$J/\psi\to\phi\pi^{+}a_{0}^{-}(\pi^{-}\eta),\;\phi\pi^{-}a_{0}^{+}(\pi^{+}\eta)$
decays

C. W. Xiao<EMAIL_ADDRESS>Department of Physics, Guangxi Normal
University, Guilin 541004, China  Guangxi Key Laboratory of Nuclear Physics
and Technology, Guangxi Normal University, Guilin 541004, China  School of
Physics, Central South University, Changsha 410083, China  J. M. Dias
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China  L.
R. Dai<EMAIL_ADDRESS>School of Science, Huzhou University, Huzhou
313000, Zhejiang, China  W. H. Liang<EMAIL_ADDRESS>Department of
Physics, Guangxi Normal University, Guilin 541004, China  Guangxi Key
Laboratory of Nuclear Physics and Technology, Guangxi Normal University,
Guilin 541004, China  E. Oset<EMAIL_ADDRESS>Department of Physics, Guangxi
Normal University, Guilin 541004, China  Departamento de Física Teórica and
IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación
de Paterna, Aptdo. 22085, 46071 Valencia, Spain

###### Abstract

We study the $J/\psi\to\phi\pi^{+}a_{0}(980)^{-}(a_{0}^{-}\to\pi^{-}\eta)$
decay, evaluating the double mass distribution in terms of the $\pi^{-}\eta$
and $\pi^{+}a_{0}$ invariant masses. We show that the $\pi^{-}\eta$ mass
distribution exhibits the typical cusp structure of the $a_{0}(980)$ seen in
recent high statistics experiments, and the $\pi^{+}a_{0}$ spectrum shows
clearly a peak around $M_{\rm inv}(\pi^{+}a_{0})=1420\,{\rm MeV}$,
corresponding to a triangle singularity. When integrating over the two
invariant masses we find a branching ratio for this decay of the order of
$10^{-5}$, which is easily accessible in present laboratories.

##  I Introduction

Triangle singularities (TS), introduced in Refs. karplus ; landau , correspond
to processes in which there is a triangle diagram in the amplitude, where the
three intermediate particles can be placed simultaneously on shell,
representing a reaction that can occur at the classical level norton , in
which case the amplitude becomes infinite in the limit of zero width of the
intermediate particles. The issue has experienced a revival in recent years,
one of the reasons, among many, being the fact that after early claims by the
COMPASS Collaboration of the “$a_{1}(1420)$” discovery adolf , it was soon
explained as a consequence of a triangle singularity in which the
$a_{1}(1260)$ resonance decays into $K^{*}\bar{K},\,K^{*}\to\pi K$, and then
$K\bar{K}$ fuse to give the $f_{0}(980)$ resonance, the $\pi f_{0}(980)$ being
the observed decay mode qzhao ; misha ; acetidai ; Alexeev . Earlier than
that, there was an interpretation of the isospin violating decay
$\eta(1405)\to f_{0}(980)\pi^{0}$ beseta  also in terms of a triangle
singularity wuzou ; fcaliang ; wuwu , and more recently the interpretation of
the $pp\to\pi^{+}d$ reaction serre ; said  in terms of a triangle singularity
ikenomolina . More examples were given in the interpretation of the
$J/\psi\to\eta\pi^{0}\phi$ reaction in Ref. jingsakaiguo , and in the enhanced
isospin violation in $D_{s}^{+}\to\pi^{+}\pi^{0}a_{0}(980)[f_{0}(980)]$ or
$\bar{B}_{s}^{0}\to J/\psi\pi^{0}a_{0}(980)[f_{0}(980)]$ sakailiang ;
sakailiaxie . A long list of reactions showing effects of triangle
singularities can be seen in Table 1 of the review paper reviewts . Also, a
reformulation of the TS has been provided in Ref. bayaracetiguo , which is at
the same time pedagogical and practical.

The issue has emerged once more due to the recent BESIII paper besrecent ,
improving considerably on earlier measurements of the
$J/\psi\to\eta\pi^{0}\phi$ reaction, where the ideas of Ref. jingsakaiguo
could be tested. Indeed, that reaction develops a TS in the $\pi^{0}\phi$ mass
distribution, peaking around $M(\phi\pi^{0})\sim 1400$ MeV, where a clear
signal is seen in Ref. besrecent . Yet, the interpretation of the peak is
delicate, because, as discussed in Ref. jingsakaiguo , the peak is masked by a
tree level contribution which has also a singular behaviour. Indeed, the TS
mechanism proceeds via the diagrams of Figs. 1 (a) and (b), but the same final
state is reached by the tree level diagrams of Figs. 1 (c) and (d).

Figure 1: Mechanisms in $J/\psi\to\eta\pi^{0}\phi\;(\phi\to K\bar{K})$: (a),
(b) TS; (c), (d) the tree level.

Then, Schmid theorem schmid  comes into play because if the triangle diagrams
develop a TS they can be reabsorbed into the tree level diagrams with a simple
change of phase, and the decay width is not affected by the TS diagrams. A
thorough review of this issue was done in Ref. vinischmid , showing that the
theorem holds in the limit of the $K^{*}$ width (in the present case) going to
zero, and when there are no inelastic channels. The message of Ref. vinischmid
is that all diagrams must be calculated, and that normally the tree level
diagrams are mostly responsible for the mass distributions, with the effects
of the TS diagrams being diluted in these distributions.

In view of these problems we look now at a related reaction where a TS
develops, which is not masked by the effects of tree level and Schmid theorem.
The reaction is
$J/\psi\to\phi\pi^{-}a_{0}(980)^{+}(\pi^{+}\eta),\;\phi\pi^{+}a_{0}(980)^{-}(\pi^{-}\eta)$.
The mechanism is similar to that in Figs. 1(a) and 1(b), with $\eta$ replaced
by $\phi$, but the intermediate $K\bar{K}$ produce the $a_{0}(980)$ resonance
which decays to $\pi\eta$. Then, the tree level diagrams with $K\bar{K}$
production do not interfere with the triangle diagram, which also develops a
singularity in the $\pi a_{0}(980)$ mass distribution. It is easy to see where
we should expect the singularity applying Eq. (18) of Ref. bayaracetiguo
(with $m_{a_{0}}$ slightly larger than $2\,m_{K}$), and one finds that a
singularity should appear at $M_{\rm inv}(\pi a_{0})\sim 1417$ MeV. The
purpose of the present work is to do a detailed study of the reaction and make
a realistic prediction of the shape and size of the $\pi a_{0}(980)$ mass
distribution in that reaction.

##  II Formalism

Figure 2: Triangle diagrams for $J/\psi\to\phi\pi^{+}a_{0}^{-}$ decay (a) and
$J/\psi\to\phi\pi^{-}a_{0}^{+}$ decay (b). Figs. (c) and (d) illustrate the
processes of (a) and (b) respectively, with a clear depiction of the decay
channel of $a_{0}^{-}$ and $a_{0}^{+}$. In Fig. (a), the momenta of the
particles are shown, where $P=p_{J/\psi}-p_{\phi}$.

We look at the diagrams of Fig. 2. The diagrams of Figs. 2(a) and 2(b) show
the coalescence processes where the $a_{0}^{-},\,a_{0}^{+}$ are produced,
independent on the decay channel in which the $a_{0}$ resonances are observed.
The diagrams of Figs. 2(c) and 2(d) show the explicit reaction, with four body
final state when the $a_{0}^{-/+}$ decay into $\pi^{-/+}\eta$, the only
sizeable decay channel. We shall consider this final state, but it is
practical to consider first the coalescence processes, with only three
particles in the final state. We consider the reactions
$J/\psi\to\phi\pi^{-}a_{0}^{+}$ and $J/\psi\to\phi\pi^{+}a_{0}^{-}$ as
different reactions and will concentrate on the first one. The same
distributions would be obtained for the second reaction.

In order to be able to determine absolute rates for the
$J/\psi\to\phi\pi^{-}a_{0}^{+}\;(J/\psi\to\phi\pi a_{0})$ reaction we need
information on the $J/\psi\to\phi K^{*-}K^{+}$ reaction, which we take from
experiment. On the other hand, the dynamics of $K^{*}\to K\pi$ and
$K\bar{K}\to a_{0}\to\pi\eta$ are well known, and for the $K\bar{K}\to
a_{0}\to\pi\eta$ amplitudes we shall use the chiral unitary approach npa ;
kaiser ; markushin ; juan .

###  II.1 The $J/\psi\to\phi K^{*}\bar{K}$ reaction

In the Particle Data Group (PDG) pdg , we have the branching ratio,

$\text{Br}(J/\psi\to\phi K^{*}(892)\bar{K}+c.c.)=(2.18\pm 0.23)\times
10^{-3}.$ (1)

However, we are only interested in $J/\psi\to\phi K^{*-}K^{+}$. It is easy to
see the rate for this particular channel using isospin and $C$ parity
arguments. With the isospin convention for the multiplets $(K^{+},\,K^{0})$,
$(\bar{K}^{0},\,-K^{-})$, $(K^{*+},\,K^{*0})$, $(\bar{K}^{*0},\,-K^{*-})$, the
$K^{*}\bar{K}$ isospin zero states demanded in the reaction of Eq. (1) are
given by

$\displaystyle|K^{*}\bar{K},\;I=0\rangle=-\frac{1}{\sqrt{2}}\;\left(K^{*+}K^{-}+K^{*0}\bar{K}^{0}\right),$
(2)
$\displaystyle|\bar{K}^{*}K,\;I=0\rangle=\frac{1}{\sqrt{2}}\;\left(\bar{K}^{*0}K^{0}+K^{*-}K^{+}\right).$

Knowing that $CK^{*+}=-K^{*-}$, $CK^{+}=K^{-}$ etc, the right combination for
$J/\psi\to\phi K^{*}\bar{K}+c.c.$ is given by

$J/\psi\to\phi\left(K^{*+}K^{-}+K^{*0}\bar{K}^{0}-K^{*-}K^{+}-\bar{K}^{*0}K^{0}\right),$
(3)

which means that the branching ratio of $J/\psi\to\phi K^{*+}K^{-}$ will be
one fourth of the one in Eq. (1),

$\text{Br}(J/\psi\to\phi K^{*+}K^{-})=(0.55\pm 0.06)\times 10^{-3}.$ (4)

Furthermore, the structure of the amplitude in $S$-wave is given by

$V_{J/\psi,\;\phi
K^{*+}K^{-}}=C\;\vec{\epsilon}_{J/\psi}\cdot\left(\vec{\epsilon}_{\phi}\times\vec{\epsilon}_{K^{*}}\right).$
(5)

We can determine $C$ from the rate of Eq. (4) using

$\dfrac{d\Gamma_{J/\psi\to\phi K^{*+}K^{-}}}{dM_{\rm
inv}(K^{*+}K^{-})}=\dfrac{1}{(2\pi)^{3}}\;\dfrac{1}{4M_{J/\psi}^{2}}\;p_{\phi}\,\tilde{p}_{K^{-}}\,\overline{\sum}\sum|t|^{2},$
(6)

with

$\displaystyle
p_{\phi}=\dfrac{\lambda^{1/2}(M_{J/\psi}^{2},\,M_{\phi}^{2},\,M_{\rm
inv}^{2}(K^{*+}K^{-}))}{2M_{J/\psi}},$ (7)
$\displaystyle\tilde{p}_{K^{-}}=\dfrac{\lambda^{1/2}(M_{\rm
inv}^{2}(K^{*+}K^{-}),\,M_{K^{*+}}^{2},\,M_{K^{-}}^{2})}{2M_{\rm
inv}(K^{*+}K^{-})},$

and $\overline{\sum}\sum|t|^{2}$ stands for the average and sum over the
polarizations of $J/\psi$, $\phi$, and $K^{*}$ mesons, with $t$ defined in Eq.
(5). Therefore,

$\overline{\sum}\sum|t|^{2}=2\,C^{2}.$ (8)

Then we find

$\dfrac{C^{2}}{\Gamma_{J/\psi}}=\dfrac{\text{Br}(J/\psi\to\phi
K^{*+}K^{-})}{\displaystyle\int\dfrac{2}{(2\pi)^{3}}\;\dfrac{1}{4M_{J/\psi}^{2}}\,p_{\phi}\,\tilde{p}_{K^{-}}\;dM_{\rm
inv}(K^{*+}K^{-})},$ (9)

from where we obtain

$\dfrac{C^{2}}{\Gamma_{J/\psi}}=1.381\times 10^{-2}\quad(\text{MeV}^{-1}),$
(10)

which we will use to evaluate the strength of the triangle mechanism.

###  II.2 The $a_{0}^{-}\to K^{-}K^{0}$ coupling and $K^{*}\to K\pi$ vertex

The $K^{*+}\to K^{0}\pi^{+}$ coupling is easily obtained from the standard
Lagrangian

${\cal L}=-ig\langle[P,\;\partial_{\mu}P]V^{\mu}\rangle,$ (11)

with $P$ and $V$ representing the SU(3) $q\bar{q}$ matrix written in terms of
pseudoscalar or vector mesons, respectively Ikeno:2019grj . The coupling $g$
is defined as $g=\frac{M_{V}}{2f}$, where $M_{V}=800$ MeV and $f=93$ MeV. This
yields the vertex

$-it=-ig\epsilon_{j}(K^{*})(2k+q)^{j},$ (12)

which is evaluated in the frame where we take
$\vec{P}=\vec{p}_{J/\psi}-\vec{p}_{\phi}=0$. In this frame, we can neglect the
$\epsilon^{0}$ component of the $K^{*}$ sakairamos .

The coupling of $a_{0}\to K^{+}K^{-}$ needed in the evaluation of the diagram
of Fig. 2(a) without further decay of $a_{0}\to\pi^{-}\eta$ can be accounted
for in the following way. It is clear that if we evaluate the
$J/\psi\to\phi\pi^{+}\pi^{-}\eta$ decay we would only need the
$K^{-}K^{0}\to\pi^{-}\eta$ amplitude. The coalescence decay
$J/\psi\to\phi\pi^{+}a_{0}^{-}$ should also be able to be calculated using the
$K\bar{K}$ amplitudes and this is formally done as discussed below.

Assuming that, close to the peak of the $a_{0}(980)$,

$t_{K^{-}K^{0},\,K^{-}K^{0}}(M_{\rm
inv})=\frac{g^{2}_{a_{0},\,K^{-}K^{0}}}{M^{2}_{\rm inv}-m^{2}_{a_{0}}+iM_{\rm
inv}\Gamma_{a_{0}}},$ (13)

then, using Cauchy’s integration we find immediately

$g^{2}_{a_{0},\,K^{-}K^{0}}=-\frac{1}{\pi}\int dM^{2}_{\rm
inv}\;\text{Im}\,t_{K^{-}K^{0},\,K^{-}K^{0}}(M_{\rm inv}),$ (14)

which is also trivially obtained in the limit of $\Gamma_{a_{0}}\to 0$ using

$\text{Im}\,\left[\left(M^{2}_{\rm
inv}-m^{2}_{a_{0}}\right)+i\epsilon\right]^{-1}=-i\pi\delta\left(M^{2}_{\rm
inv}-m^{2}_{a_{0}}\right).$ (15)

In the coalescence process we will use Eq. (14) in the evaluation of the
$|t_{\rm TS}|^{2}$ of the triangle diagram. We will have

$\dfrac{d\Gamma_{J/\psi,\to\phi\pi^{+}a_{0}^{-}}}{dM_{\rm
inv}(\pi^{+}a_{0}^{-})}=\dfrac{1}{(2\pi)^{3}}\;\dfrac{1}{4M_{J/\psi}^{2}}\;p_{\phi}\;\tilde{p}_{\pi^{+}}\;\overline{\sum}|t_{\rm
TS}|^{2},$ (16)

where

$\displaystyle
p_{\phi}=\dfrac{\lambda^{1/2}(M_{J/\psi}^{2},\,m_{\phi}^{2},\,M_{\rm
inv}^{2}(\pi^{+}a_{0}^{-}))}{2M_{J/\psi}},$ (17)
$\displaystyle\tilde{p}_{\pi^{+}}=\dfrac{\lambda^{1/2}(M_{\rm
inv}^{2}(\pi^{+}a_{0}^{-}),\,m_{\pi^{+}}^{2},\,m_{a_{0}^{-}}^{2})}{2M_{\rm
inv}(\pi^{+}a_{0}^{-})},$ (18)

with $t_{\rm TS}$ corresponding to the amplitude for the process of Fig. 2(a).
Since $|t_{\rm TS}|^{2}$ contains $g^{2}_{a_{0},\,K^{-}K^{0}}$, given in Eq.
(14), we can undo the $dM^{2}_{\rm inv}$ integration in Eq. (16) and write

$\displaystyle\dfrac{d^{2}\Gamma_{J/\psi,\to\phi\pi^{+}a_{0}^{-}(\pi^{-}\eta)}}{dM_{\rm
inv}(\pi^{-}\eta)\;dM_{\rm inv}(\pi^{+}a_{0}^{-})}=$
$\displaystyle-\dfrac{1}{\pi}\;2M_{\rm
inv}(\pi^{-}\eta)\;\text{Im}\,t_{K^{-}K^{0},\,K^{-}K^{0}}(M_{\rm
inv}(\pi^{-}\eta))$ (19)
$\displaystyle\times\dfrac{1}{(2\pi)^{3}}\;\dfrac{1}{4M_{J/\psi}^{2}}\;p_{\phi}\;\tilde{p}_{\pi^{+}}^{\prime}\overline{\sum}|\tilde{t}_{\rm
TS}|^{2},$

where $\tilde{p}_{\pi^{+}}^{\prime}$ is given by Eq. (18) substituting
$m_{a_{0}^{-}}$ by $M_{\rm inv}(\pi^{-}\eta)$, and $|\tilde{t}_{\rm TS}|^{2}$
is the magnitude $|t_{\rm TS}|^{2}$ where we remove
$g^{2}_{a_{0},\,K^{-}K^{0}}$.

Eq. (19) can be immediately reinterpreted. Indeed, in the evaluation of
$|t_{\rm TS}|^{2}$ for the triangle diagram of Fig. 2(c), we will need
$|t_{K^{-}K^{0},\;\pi^{-}\eta}|^{2}$ along with the extra phase space for
$\pi^{-}\eta$ with respect to Fig. 2(a). But
$|t_{K^{-}K^{0},\;\pi^{-}\eta}|^{2}$ times the phase space for
$a_{0}^{-}\to\pi^{-}\eta$ decay is what is given by
$\text{Im}\,t_{K^{-}K^{0},\;K^{-}K^{0}}$ via the optical theorem. The
derivation done, however, has served to go from the differential mass
distribution in the three body final state to the one of the four body in a
simple and intuitive way.

###  II.3 The triangle amplitude

In Fig. 2(a), we show the momenta of the particles. Since we are concerned
about the triangle singularity, occurring when the intermediate particles are
all on shell, we can simplify the calculation (see Ref. bayaracetiguo ) taking
the positive energy part of the propagators, which is the one that can go on
shell. Hence for a $K^{-}$ propagator we would write

$\dfrac{1}{q^{2}-m_{K}^{2}+i\epsilon}=\dfrac{1}{2\omega(\vec{q}\,)}\;\left(\dfrac{1}{q^{0}-\omega_{K}(\vec{q}\,)+i\epsilon}-\dfrac{1}{q^{0}+\omega_{K}(\vec{q}\,)-i\epsilon}\right),$
(20)

with $\omega_{K}(\vec{q}\,)=\sqrt{\vec{q}\;^{2}+m_{K}^{2}}$, and with $q^{0}$
positive only the first term of the former equation can go on shell, and we
shall then keep this term alone. This simplifies the expression for the loop
amplitude which reads, removing the $g_{a_{0},\,K^{-}K^{0}}$ vertex:

$\displaystyle-i\tilde{t}_{\rm TS}=$
$\displaystyle-i\,C\int\dfrac{d^{4}q}{(2\pi)^{4}}\;\epsilon_{ijl}\;\epsilon_{i}(J/\psi)\;\epsilon_{j}(\phi)\;\epsilon_{l}(K^{*})\;(-i)g\;\epsilon_{m}(K^{*})\;(2k+q)_{m}$
(21)
$\displaystyle\times(-i)\;\dfrac{1}{2\omega_{K^{-}}(\vec{q}\,)}\;\dfrac{1}{2\omega_{K^{0}}(\vec{q}+\vec{k})}\;\dfrac{1}{2\omega_{K^{*+}}(\vec{q}\,)}\;\dfrac{i}{q^{0}-\omega_{K^{-}}(\vec{q}\,)+i\epsilon}$
$\displaystyle\times\frac{i}{P^{0}-q^{0}-\omega_{K^{*+}}(\vec{q}\,)+i\frac{\Gamma_{K^{*}}}{2}}\;\dfrac{i}{P^{0}-q^{0}-k^{0}-\omega_{K^{0}}(\vec{q}+\vec{k}\,)+i\epsilon},$

with $\omega_{K^{-}}(\vec{q}\,)=\sqrt{\vec{q}\;^{2}+m_{K}^{2}}$,
$\omega_{K^{*+}}(\vec{q}\,)=\sqrt{\vec{q}\;^{2}+m_{K^{*}}^{2}}$, where we have
explicitly taken into account the $K^{*}$ width in the $K^{*}$ propagator, and
$P^{0}$, $k^{0}$ are given by

$\displaystyle P^{0}=M_{\rm inv}(\pi^{+}a_{0}^{-}),$ (22) $\displaystyle
k^{0}=\dfrac{P^{02}+m^{2}_{\pi^{+}}-M^{2}_{\rm inv}(\pi^{-}\eta)}{2P^{0}}.$

We sum over the $K^{*}$ polarizations
$\sum\epsilon_{l}(K^{*})\epsilon_{m}(K^{*})=\delta_{lm}$, and integrating
analytically over $q^{0}$ in Eq. (21) using Cauchy’s residues, we obtain

$\displaystyle\tilde{t}_{\rm TS}=$ $\displaystyle
g\,C\,\epsilon_{ijl}\;\epsilon_{i}(J/\psi)\;\epsilon_{j}(\phi)\;\int\dfrac{d^{3}q}{(2\pi)^{3}}\;(2k+q)_{l}\;\dfrac{1}{2\omega_{K^{-}}(\vec{q}\,)}\;\dfrac{1}{2\omega_{K^{*+}}(\vec{q}\,)}\;\dfrac{1}{2\omega_{K^{0}}(\vec{q}+\vec{k}\,)}$
(23)
$\displaystyle\times\dfrac{i}{P^{0}-\omega_{K^{-}}(\vec{q}\,)-\omega_{K^{*+}}(\vec{q}\,)+i\frac{\Gamma_{K^{*}}}{2}}\;\dfrac{i}{P^{0}-k^{0}-\omega_{K^{-}}(\vec{q}\,)-\omega_{K^{0}}(\vec{q}+\vec{k}\,)+i\epsilon}.$

Next, considering that

$\int d^{3}q\,F(\vec{q},\vec{k})\,q_{l}=k_{l}\,\int
d^{3}q\;F(\vec{q},\vec{k})\;\dfrac{\vec{q}\cdot\vec{k}}{\vec{k}^{\,2}},$ (24)

we finally write $\tilde{t}_{\rm TS}$ as

$\tilde{t}_{\rm
TS}=g\,C\,\epsilon_{ijl}\;\epsilon_{i}(J/\psi)\;\epsilon_{j}(\phi)\;k_{l}\;\tilde{t}^{\prime}_{\rm
TS},$ (25)

with

$\displaystyle\tilde{t}^{\prime}_{\rm TS}=$
$\displaystyle\int\dfrac{d^{3}q}{(2\pi)^{3}}\;\theta(q_{max}-|\vec{q}\;^{*}|\,)\;\left(2+\dfrac{\vec{q}\cdot\vec{k}}{\vec{k}^{\,2}}\right)\;\dfrac{1}{2\omega_{K^{-}}(\vec{q}\,)}\,\dfrac{1}{2\omega_{K^{*+}}(\vec{q}\,)}\,\dfrac{1}{2\omega_{K^{0}}(\vec{q}+\vec{k}\,)}$
(26)
$\displaystyle\times\dfrac{i}{P^{0}-\omega_{K^{-}}(\vec{q}\,)-\omega_{K^{*+}}(\vec{q}\,)+i\dfrac{\Gamma_{K^{*}}}{2}}\;\dfrac{i}{P^{0}-k^{0}-\omega_{K^{-}}(\vec{q}\,)-\omega_{K^{0}}(\vec{q}+\vec{k}\,)+i\epsilon},$

and

$\overline{\sum}\left|\tilde{t}_{\rm
TS}\right|^{2}=\frac{2}{3}\,\vec{k}^{\,2}\,g^{2}\,C^{2}\;\left|\tilde{t}^{\prime}_{\rm
TS}\right|^{2}.$ (27)

In Eq. (26) we have introduced the factor $\theta(q_{max}-|\vec{q}\;^{*}|)$,
where $\vec{q}\;^{*}$ is the $K^{-}$ momentum in the $\pi^{-}\eta$ rest frame
given by bayaracetiguo

$\vec{q}\;^{*}=\left[\left(\frac{E_{a_{0}}}{M_{\rm
inv}(\pi^{-}\eta)}-1\right)\frac{\vec{q}\cdot\vec{k}}{\vec{k}^{\,2}}+\frac{q^{0}}{M_{\rm
inv}(\pi^{-}\eta)}\right]\vec{k}+\vec{q}\;,$ (28)

where $E_{a_{0}}=\sqrt{M^{2}_{\rm inv}+\vec{k}^{\,2}}$,
$q^{0}=\sqrt{m^{2}_{K}+\vec{q}^{\;2}}$. This factor is demanded because in the
cutoff regularization to obtain the $K\bar{K}$ scattering we have
$t(\vec{q},\vec{q}^{\;\prime})=t\,\theta(q_{max}-|\vec{q}\,|)\,\theta(q_{max}-|\vec{q}\;^{\prime}|)$,
with $q_{max}$ the cutoff momentum used in the regularization of the
$G_{K\bar{K}}$ function danijuan .

###  II.4 $K\bar{K}$ amplitudes

We need to calculate the $t_{K^{-}K^{0},\;K^{-}K^{0}}$ amplitude, for which we
use the chiral unitary approach. In this case, the $t_{K^{-}K^{0},K^{-}K^{0}}$
amplitude is one of the $T$-matrix elements, that is extracted by solving the
Bethe-Salpeter equation in coupled-channels,

$T=[1-VG]^{-1}V,$ (29)

with $V$ the kernel encoding the $V_{ij}$ amplitudes from $i$\- to
$j$-channel. The channels considered are: $K\bar{K}$, $\pi\eta$, $\pi\pi$, and
$\eta\eta$. The relevant $V_{ij}$ amplitudes among all the channels in our
case are taken from Ref. lianglin , using explicitly the $\eta-\eta^{\prime}$
mixing of Ref. bramon  (Eq. (A.4) of Ref. lianglin ). Furthermore, $G$ is a
diagonal matrix with its elements $G_{l}$ corresponding to the loop function
for the $l$-th channel. We take the $G_{l}$ loop function regularized by means
of a cut-off $q_{\rm max}$ in the three-monentum,

$G_{l}=\int_{|\vec{q}\,|<q_{\rm
max}}\dfrac{d^{3}q}{(2\pi)^{3}}\;\dfrac{\omega_{1}+\omega_{2}}{2\omega_{1}\,\omega_{2}}\;\dfrac{1}{s-(\omega_{1}+\omega_{2})^{2}+i\epsilon}.$
(30)

Since, only the zero charged components are considered in Ref. lianglin , we
make use of the fact that $K^{-}K^{0}$ is the $I=1,\,I_{3}=-1$ component of
$K\bar{K}$ and write,

$t_{K^{-}K^{0},\;K^{-}K^{0}}=\frac{1}{2}\left(t_{K^{0}\bar{K}^{0},\;K^{0}\bar{K}^{0}}+t_{K^{+}K^{-},\;K^{+}K^{-}}-2\,t_{K^{0}\bar{K}^{0},\;K^{+}K^{-}}\right).$
(31)

##  III Results

In Fig. 3, we show the factor $A\equiv[-\dfrac{2}{\pi}\;M_{\rm
inv}(\pi^{-}\eta)\;{\rm Im}\,t_{K^{-}K^{0},K^{-}K^{0}}(M_{\rm
inv}(\pi^{-}\eta))]$, which is involved in the double differential decay width
of Eq. (19).

Figure 3: Factor $A\equiv\big{[}-\dfrac{2}{\pi}\;M_{\rm
inv}(\pi^{-}\eta)\;{\rm Im}\,t_{K^{-}K^{0},K^{-}K^{0}}(M_{\rm
inv}(\pi^{-}\eta))\big{]}$ as a function of $M_{\rm inv}(\pi^{-}\eta)$.

We can see a cusp-like structure around $M_{\rm
inv}(\pi^{-}\eta)=m_{a_{0}}=980\,\textrm{ MeV}$, reflecting the spectral
function of the $a_{0}(980)$.

Next, we discuss the amplitude $\tilde{t}^{\prime}_{\rm TS}$ of Eq. (26) and
$\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ of Eq. (19), which are
functions of both $M_{\rm inv}(\pi^{-}\eta)$ and $M_{\rm inv}(\pi^{+}a_{0})$.
We will present the results in three cases: 1) fixing $M_{\rm
inv}(\pi^{-}\eta)=m_{a_{0}}=980\textrm{ MeV}$; 2) fixing $M_{\rm
inv}(\pi^{+}a_{0})=1416\,\textrm{ MeV}$, where the TS occurs in the triangle
loops of Fig. 2; and 3) integrating $\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ over $M_{\rm
inv}(\pi^{-}\eta)$.

###  III.1 Fixing $M_{\rm inv}(\pi^{-}\eta)=m_{a_{0}}=980\textrm{ MeV}$

In Fig. 4, we show results for ${\rm Re}\,\tilde{t}^{\prime}_{\rm TS}$, ${\rm
Im}\,\tilde{t}^{\prime}_{\rm TS}$ and $|\tilde{t}^{\prime}_{\rm TS}|$ as a
function of $M_{\rm inv}(\pi^{+}a_{0})$ while keeping $M_{\rm
inv}(\pi^{-}\eta)$ fixed.

Figure 4: $\tilde{t}^{\prime}_{\rm TS}$ given by Eq. (26) as a function of
$M_{\rm inv}(\pi^{+}a_{0})$ when fixing $M_{\rm inv}(\pi^{-}\eta)=m_{a_{0}}$.

The structure of the amplitude exhibits features typical of triangle
singularities observed in other cases (see Fig. 5 of Ref. sakairamos , Fig. 4
of Ref. PavaoSakai , Fig. 3 of Ref. sakailiang , Fig. 4 of Ref. OsetRoca ,
Fig. 5 of Ref. DaiYu , Fig. 4 of Ref. LiangChen , Fig. 19 of Ref. GuoRev ). It
has the imaginary part peaking around the $M_{\rm inv}(\pi^{+}a_{0})$, as
provided by Eq. (18) of Ref. bayaracetiguo , and the real part changes sign
around the peak of the imaginary part. It resembles much the structure of a
resonance, and hence the danger to identify these peaks as genuine resonances,
but as we can see, the origin of this structure simply comes from the triangle
diagram and is exclusively tied to the combination of masses and invariant
masses of the particles involved. It does not have its origin in the
interaction of quarks or the interaction of hadrons. This is why it is
referred to as a kinematical singularity. The modulus of this amplitude,
$|\tilde{t}^{\prime}_{\rm TS}|$, has a clear peak that should manifest in the
studied reaction.

In Fig. 5, we show the double differential decay width normalized to the
$J/\psi$ width as a function of $M_{\rm inv}(\pi^{+}a_{0})$, while fixing
$M_{\rm inv}(\pi^{-}\eta)=m_{a_{0}}=980\textrm{ MeV}$.

Figure 5: $\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ as a function of $M_{\rm
inv}(\pi^{+}a_{0})$ when fixing $M_{\rm inv}(\pi^{-}\eta)=m_{a_{0}}$.

We see a clear peak around $M_{\rm inv}(\pi^{+}a_{0})=1440\textrm{ MeV}$,
coming from $|\tilde{t}^{\prime}_{\rm TS}|$ in Eq. (19). This is what one
would observe in a devoted experiment with a bin of $1\textrm{ MeV}$ for
$M_{\rm inv}(\pi^{+}a_{0})$, and $1\textrm{ MeV}$ for $M_{\rm
inv}(\pi^{-}\eta)$. Obviously, the accumulation of events in bigger bins
increases the statistics, as we shall see.

###  III.2 Fixing $M_{\rm inv}(\pi^{+}a_{0})=1416\,\textrm{ MeV}$

In Fig. 6, we now set $M_{\rm inv}(\pi^{+}a_{0})=1416\,\textrm{ MeV}$ and plot
$\tilde{t}^{\prime}_{\rm TS}$ as a function of $M_{\rm inv}(\pi^{-}\eta)$.

Figure 6: $\tilde{t}^{\prime}_{\rm TS}$ given by Eq. (26) as a function of
$M_{\rm inv}(\pi^{-}\eta)$ when fixing $M_{\rm inv}(\pi^{+}a_{0})=1416\textrm{
MeV}$.

Once more, we display the real and imaginary parts of the amplitude, as well
as its modulus. We see again that the imaginary part and the modulus delineate
the shape of the $a_{0}(980)$ resonance. The real part changes sign at the
peak of the $a_{0}(980)$, reflecting a typical resonance behaviour. It is
interesting to see that even if the $a_{0}(980)$ appears as cusp,
corresponding to a nearly missed bound state, or virtual state, it still
exhibits the typical shape of a resonance amplitude. Such kind of behaviours
for nearly missed bound states can be seen in other cases. For instance, in
the $pd\to{{}^{3}\rm He}\,\eta$ reaction He3eta  (see Fig. 8 of that work),
the amplitude exhibits a resonance structure with a peak very close to and
just below the ${{}^{3}\rm He}\,\eta$ threshold. However, there is no pole
below threshold, and technically, no bound state.

In Fig. 7, we show again $\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ as a function of $M_{\rm
inv}(\pi^{-}\eta)$, fixing now $M_{\rm inv}(\pi^{+}a_{0})$ at the peak of the
TS amplitude.

Figure 7: $\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ as a function of $M_{\rm
inv}(\pi^{-}\eta)$ when fixing $M_{\rm inv}(\pi^{+}a_{0})=1416\textrm{ MeV}$.

This comes from Eq. (19) and contains $|\tilde{t}^{\prime}_{\rm TS}|^{2}$
along with the phase space. The shape of the $a_{0}(980)$ resonance shows up
as a clear cusp structure, as seen in recent experiments with high resolution
Rubin ; midhalo ; besnew .

###  III.3 integrating $\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ over $M_{\rm
inv}(\pi^{-}\eta)$

Fig. 8 shows $\frac{1}{\Gamma_{J/\psi}}\,\frac{{\rm
d}^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm d}M_{\rm
inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ as a function of $M_{\rm
inv}(\pi^{+}a_{0})$, when integrating over $M_{\rm inv}(\pi^{-}\eta)$ in the
ranges of $m_{a_{0}}\pm 10\textrm{ MeV}$, $m_{a_{0}}\pm 20\textrm{ MeV}$,
$m_{a_{0}}\pm 50\textrm{ MeV}$ and $m_{a_{0}}\pm 100\textrm{ MeV}$,
respectively.

Figure 8:
$\frac{1}{\Gamma_{J/\psi}}\,\frac{d^{2}\Gamma_{J/\psi\to\phi\pi^{+}a_{0}(980)}}{{\rm
d}M_{\rm inv}(\pi^{-}\eta)\;{\rm d}M_{\rm inv}(\pi^{+}a_{0})}$ as a function
of $M_{\rm inv}(\pi^{+}a_{0})$ when integrating over $M_{\rm
inv}(\pi^{-}\eta)$ in the ranges: $m_{a_{0}}\pm 10\textrm{ MeV}$,
$m_{a_{0}}\pm 20\textrm{ MeV}$, $m_{a_{0}}\pm 50\textrm{ MeV}$ and
$m_{a_{0}}\pm 100\textrm{ MeV}$.

In all the cases, we observe a peak corresponding to the TS. By looking at
Fig. 8, we can see that integrating the double mass distribution over $M_{\rm
inv}(\pi^{-}\eta)$ within the range $m_{a_{0}}\pm 100\textrm{ MeV}$ accounts
for the whole strength of the $a_{0}(980)$ resonance. And we interpret this as
indicative of what should be observed in the experiments. The shape of the TS
is clearly observed.

For the case where $M_{\rm
inv}(\pi^{-}\eta)\in[m_{a_{0}}-100,m_{a_{0}}+100]\textrm{ MeV}$, integrating
over $M_{\rm inv}(\pi^{+}a_{0})$ in the range
$[m_{\pi^{+}}+m_{a_{0}},M_{J/\psi}-m_{\phi}]$ gives the branching ratio

${\rm Br}(J/\psi\to\phi\pi^{+}a_{0})=1.07\times 10^{-5}.$ (32)

This estimate is realistic, as the only unknown magnitude required to evaluate
the diagrams of Figs. 2(a) and 2(c) is the $J/\psi\to\phi K^{*+}K^{-}$
amplitude, which we obtained from the experimental data. This branching ratio
is not small, given the copious production of $J/\psi$ at BESIII, which allows
to detect decays with branching fractions as small as $10^{-7}$ pdg . In view
of this, we can only encourage the measurement of this reaction, which could
also serve to clarify issues on the $J/\psi\to\pi^{0}\eta\phi\,(\phi\to
K\bar{K})$ reaction besrecent  and its interpretation in Ref. jingsakaiguo  in
terms of a TS.

##  IV Conclusions

We have conducted a study of the
$J/\psi\to\phi\pi^{+}a_{0}(980)^{-}\;(a_{0}^{-}\to\pi^{-}\eta)$ decay, showing
that it develops a triangle singularity at $M_{\rm inv}(\pi^{+}a_{0})$ of
about $1420\,\rm MeV$. The reaction proposed is motivated by the recent
measurement at BESIII of the $J/\psi\to\eta\pi^{0}\phi\;(\phi\to K\bar{K})$
reaction besrecent , that according to the work of Ref. jingsakaiguo  also
develops a triangle singularity, however, blurred by the tree level competing
mechanisms and their interconnection with the Coleman Norton theorem. In the
reaction proposed there is no tree level competing mechanism and then the TS
appearing can be clearly interpreted. We evaluate the mass distributions in
terms of $M_{\rm inv}(\pi^{-}\eta)$ and $M_{\rm inv}(\pi^{+}a_{0})$. In the
$\pi^{-}\eta$ mass distribution we see a clear cusp structure, as observed in
recent high statistics experiments, and in the $\pi^{+}a_{0}$ mass
distribution we observe the TS peak around $M_{\rm
inv}(\pi^{+}a_{0})=1420\,\rm MeV$. By taking information for the needed
$J/\psi\to\eta K^{*}\bar{K}$ amplitude from experiment, we are able to
determine absolute rates for the reaction. Integrating the double mass
distribution in the range of the $a_{0}(980)$ mass and in the range of the
$\pi^{+}a_{0}$ mass distribution, we predict a branching ratio for the
reaction of the order of $10^{-5}$. Given the present rates of $J/\psi$
production at BESIII, where branching ratios of $10^{-7}$ can be measured, we
advocate for the measurement of this reaction, that apart from showing a new
example of a TS can also shed light on the interpretation of the recent BESIII
measurements of the $J/\psi\to\eta\pi^{0}\phi\;(\phi\to K\bar{K})$ reaction.

###### Acknowledgements.

J.M. Dias would like to express gratitude to Guangxi Normal University for the
warm hospitality, as part of this work was conducted there. This work is
partly supported by the National Natural Science Foundation of China under
Grant Nos. 11975083, 12365019 and 11975009, and by the Central Government
Guidance Funds for Local Scientific and Technological Development, China (No.
Guike ZY22096024). This work is also supported partly by the Natural Science
Foundation of Changsha under Grant No. kq2208257 and the Natural Science
Foundation of Hunan province under Grant No. 2023JJ30647 and the Natural
Science Foundation of Guangxi province under Grant No. 2023JJA110076 (CWX).
J.M. Dias acknowledges the support from the Chinese Academy of Sciences under
Grant No. XDB34030000, and No. YSBR-101; by the National Key R&D program of
Chinese under Grant No. 2023YFA1606703; by NSFC under Grant No. 12125507, No.
12361141819, and No. 12047503. This project has received funding from the
European Union Horizon 2020 research and innovation programme under the
program 2020-INFRAIA-2018-1, grant agreement No. 824093 of the STRONG-2020
project.

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