# The TESS Triple 9 Catalog: 999 uniformly vetted exoplanet candidates

Luca Cacciapuoti,1,2,3 Veselin B. Kostov,4,5 Marc Kuchner,4 Elisa V.
Quintana,4 Knicole D. Colón,4 Jonathan Brande,6 Susan E. Mullally,7 Quadry
Chance,8 Jessie L. Christiansen,9 John P. Ahlers,10 Marco Z. Di Fraia,11,16
Hugo A. Durantini Luca,12,16 Riccardo M. Ienco,3,16 Francesco Gallo,3,16 Lucas
T. de Lima,13,16 Michiharu Hyogo,14,16 Marc Andrés-Carcasona,15,16 Aline U.
Fornear,16 Julien S. de Lambilly,16 Ryan Salik16,17 John M. Yablonsky16 and
Shaun Wallace18 Sovan Acharya,16,19

1European Southern Observatory, Karl-Schwarzschild-Strasse 2 D-85748 Garching
bei Munchen, Germany
2Fakultat fur Physik, Ludwig-Maximilians-Universitat Munchen, Scheinerstr. 1,
81679 Munchen, Germany
3Department of Physics "Ettore Pancini", University of Naples Federico II,
Naples, Italy
4NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
5SETI Institute,189 Bernardo Ave, Suite 200, Mountain View, CA 94043, USA
6Department of Physics & Astronomy, University of Kansas, 1082 Malott, 1251
Wescoe Hall Dr., Lawrence, KS 66045, USA
7Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
21218, USA
8University of Florida, Gainesville, FL 32611
9Infrared Processing and Analysis Center, Caltech, Pasadena CA 91125, USA
10Exoplanets and Stellar Astrophysics Laboratory, Code 667, NASA Goddard Space
Flight Center (GSFC), Greenbelt, MD 20771, USA
11Oxford Dynamics, G19, Building R71, RAL, Didcot OX11 0QX
12IATE-OAC, Universidad Nacional de Córdoba-CONICET, Laprida 854, X5000 BGR,
Córdoba, Argentina
13Geoscience Department, University of Aveiro. Campus Santiago, 3810-193,
Aveiro
14Meisei University, 2-1-1 Hodokubo, Hino, Tokyo 191-0042, Japan
15Institut de Física d’Altes Energies (IFAE), Barcelona Institute of Science
and Technology, E-08193 Barcelona, Spain
16Citizen Scientist, Planet Patrol Collaboration
17Staples High School, 70 North Ave, Westport, Connecticut 06880
18Department of Computer Science Brown University
19SA Citizen Science Group-Ignited Minds VIPNET Club
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Transiting Exoplanet Survey Satellite (TESS) has detected thousands of
exoplanet candidates since 2018, most of which have yet to be confirmed. A key
step in the confirmation process of these candidates is ruling out false
positives through vetting. Vetting also eases the burden on follow-up
observations, provides input for demographics studies, and facilitates
training machine learning algorithms. Here we present the TESS Triple-9 (TT9)
catalog – a uniformly-vetted catalog containing dispositions for 999 exoplanet
candidates listed on ExoFOP-TESS, known as TESS Objects of Interest (TOIs).
The TT9 was produced using the Discovery And Vetting of Exoplanets pipeline,
DAVE , and utilizing the power of citizen science as part of the Planet Patrol
project. More than 70% of the TOIs listed in the TT9 pass our diagnostic
tests, and are thus marked as true planetary candidates. We flagged 144
candidates as false positives, and identified 146 as potential false
positives. At the time of writing, the TT9 catalog contains $\sim 20\%$ of the
entire ExoFOP-TESS TOIs list, demonstrates the synergy between automated tools
and citizen science, and represents the first stage of our efforts to vet all
TOIs. The DAVE generated results are publicly available on ExoFOP-TESS.

###### keywords:

catalogues, planets and satellites

††pubyear: 2022††pagerange: The TESS Triple 9 Catalog: 999 uniformly vetted
exoplanet candidates–11

##  1 Introduction

A plethora of ground- and space-based exoplanet-hunting efforts have
contributed to the exponential growth of the number of discovered planets
orbiting stars other than our Sun. Prior to 2010, exoplanet discoveries were
mainly based on precise radial velocity curves, e.g. (Mayor & Queloz, 1995),
(Marcy et al., 1998), (Lovis et al., 2005), (Naef et al., 2010). A paradigm
shift in exoplanet discoveries started with NASA’s Kepler mission (Borucki et
al., 2008), launched in 2009. This endeavor enabled the scientific community
to gather a large exoplanet sample via the photometric transit technique. This
mission alone contributed to the confirmation of more than 2,700 exoplanets
and the discovery of as many candidate exoplanets still to be
confirmed111https://exoplanetarchive.ipac.caltech.edu. he wealth of planetary-
like signals discovered by photometric surveys will likely grow and expand
thanks to large archival datasets, and ongoing and future planned survey
missions, e.g. TESS (Ricker et al., 2015) and PLATO (Rauer et al., 2021).

Indeed, since 2018, the Transiting Exoplanet Survey Satellite (TESS) space
mission has been surveying most of the sky and searching for additional
transiting exoplanets. TESS has already identified more than 5,000 TESS Object
of Interest (TOIs), i.e. candidate exoplanets, and is expected to detect
thousands more, e.g. (Barclay et al., 2018). However, only 199 out of the
$\sim 5,000$ TOIs listed on ExoFOP-TESS have been statistically validated
and/or confirmed at the time of writing, according to the NASA Exoplanet
Archive. Given that non-planetary astrophysical sources (most notably
eclipsing binary stars, see e.g. Ciardi et al. (2018)) and/or systematic
effects can often mimic exoplanet transits, revealing the true nature of the
remaining TOIs requires ruling out possible false positive scenarios and/or
follow-up observations.

Ideally, the precision radial velocity (PRV) technique, e.g. (Baranne et al.,
1996), (Pepe et al., 2004), is utilized to confirm the planetary nature of a
specific candidate. This is indeed the method that was used to discover the
first confirmed exoplanet orbiting a Sun-like star: 51 Pegasi b (Mayor &
Queloz, 1995). Specifically, the stellar spectrum is obtained at different
epochs to detect periodic Doppler shifts of its absorption lines due to the
orbit of the star around the center of mass of the star-planet system. The
major downside of the PRV technique is that spectroscopic observations are
time-consuming. To be able to detect a planet around a star with the PRV
method, two conditions should apply. First, the star has to be sufficiently
bright to allow observing its spectrum in a reasonable amount of time and with
a sufficient signal-to-noise ratio (SNR). Secondly, planets with short orbital
periods facilitate the detection since this allows obtaining complete cycle
radial velocity curves in a reasonable amount of observing time. Given the
number of exoplanets an all-sky survey like TESS is expected to discover
(Barclay et al., 2018), it is unfeasible to obtain PRV observations for every
star with a transiting planetary candidate.

Thus, vetting procedures play a critical role in the analysis of exoplanet
candidates. First, vetting is needed to ease the burden of ground-based
follow-up facilities and to validate planets for which no PRV measurements can
be obtained. Secondly, it is fundamental to refine the true population of
known exoplanets that underline demographics and population synthesis efforts.
Finally, vetting serves as a mean to build a knowledge base for upcoming
machine-learning codes that will have the necessary function of validating the
stream of new candidates.

A number of pipelines have been developed to validate photometric signals
caused by transiting exoplanets, such as Robovetter (Coughlin, 2020) and
Autovetter (Jenkins et al., 2014). These codes are based on supervised machine
learning – a single decision tree for the former and a random forest algorithm
for the latter – and have been trained to recognize true planet candidates
based on tens of thousands of human-inspected signals available at the time of
development. Other algorithms, like vespa (Morton, 2015) and triceratops
(Giacalone et al., 2021), compute the Bayesian probability that the signal
might be a false positive based on the number of nearby sources in TESS field
of view, their properties, as well as the shape and strength of the
investigated signal. Yet another approach, pioneered by Astronet (Shallue &
Vanderburg, 2018) for Kepler and exploited by astroramjet (Olmschenk et al.,
2021) for TESS, utilizes a deep learning algorithm known as a Convolutional
Neural Network (CNN). The code is trained to recognize patterns in light
curves and discriminate among classes based on the data alone. Finally, Kostov
et al. (2019a) used the Discovery And Vetting of Exoplanets (DAVE ) pipeline
that tests the candidate signal at both the pixel and light curve levels
mimicking the process that expert human vetters apply. Kostov et al. (2019a)
made use of DAVE to uniformly vet 776 K2 candidates and collected the data and
dispositions in a catalog hosted on the Kepler Threshold Crossing Event Review
Team (TCERT) website. Since then, DAVE has been adapted to TESS data and used
to vet a handful of individual planet candidates, see e.g. Kostov et al.
(2019b), Gilbert et al. (2020).

Here we present the TESS Triple-9 (TT9) catalog of 999 uniformly-vetted
exoplanet candidates detected in TESS data. The candidates were identified by
the community using the SPOC (Jenkins et al., 2016) and QLP (Huang et al.,
2020) pipelines, and are listed on ExoFOP-TESS. To create the catalog, we vet
the signals employing the DAVE pipeline and citizen science. Specifically,
each candidate is analysed through DAVE and inspected by human vetters, and
also utilizing auxiliary information as provided by ExoFOP
TESS222https://exofop.ipac.caltech.edu/tess/. This effort has been possible
also thanks to the Planet Patrol project (Kostov et al., submitted), a NASA-
led citizen science project hosted on Zooniverse through which a group of
citizen scientists interested in validating TESS planets joined our science
team. The main goal of Planet Patrol is to evaluate the reliability of DAVE
results with the help of volunteers. The outcomes of the classifications have
been implemented in DAVE to rule out bad images and compute high-fidelity
photocenter statistics. Soon after the launch of the project, several citizen
scientists expressed interest in further helping the vetting efforts and,
after joining the science team, quickly became proficient vetters. These
"Superusers" have been key to the success of this endeavor. The Superusers and
the science team members will be referred hereafter as "vetters".

This paper is organized as follows. We present our vetting workflow in Section
2; the details on the citizen science project Planet Patrol are described in
Section 3; the TESS Triple-9 catalog is presented in Section 4. Finally, we
wrap up the conclusions in Section 5.

Figure 1: A flow diagram showing the algorithmic process the vetters would go
through during the vetting of the TOIs. The workflow starts with the left
"Centroids" branch and moves on to the "Light Curve" branch if needed, as
explained in Section 2. All of the used abbreviations are shown in Table 1.
Footnote1: centroids are considered reliable if no other field bright variable
source is present. Footnote2: the mentioned features could make it difficult
to interpret the nature of the signal without jeopardizing its true planetary
nature a priori. Footnote3: Given that these secondary eclipses and/or odd-
even differences are not caused by any irregular feature in the light curve
(previously inspected).

##  2 The vetting workflow

For every TOI listed in our TT9 catalog, we performed uniform vetting analysis
following a workflow that is based on the interpretation of DAVE results using
as input the transit ephemeris, duration, and depth as provided by ExoFOP-
TESS. Where available, we also evaluate ancillary information that the
pipeline does not account for such as archival data. The workflow proceeds as
outlined in the next subsections, it is summarized in Fig.1 and it is mainly
based on products generated by the DAVE pipeline that we describe below.

  * •

Photocenter: this module of DAVE generates the in-transit and the two (before
and after the event) out-of-transit images for each transit. Then, it
subtracts the in-transit image from the average out-of-transit image to
produce a difference image. Finally, it measures the center-of-light for each
difference image by fitting the TESS pixel-response-function (PRF) and a
Gaussian point-spread-function (PSF) to the image. This is highlighted in
Fig.2 for the case of TIC 43647325, where a fainter field star is present a
few pixels above and to the right of the target (upper right panel). This
field star is, however, not present in the difference image, demonstrating
that it is not varying in brightness during the detected transit events. The
measured photocenters are on-target (black star; TIC 43647325), confirming
that it is the source of the detected transits. The overall photocenter (red
circle) is calculated by averaging over the individual photocenters (red dots)
corresponding to each detected transit as examined in the previous step (see
Fig.2). See Kostov et al. (2019a) for further details.

  * •

Modelshift: this module of DAVE phase-folds the light curve and convolves it
with a trapezoid model using the transit parameters provided on ExoFOP-TESS,
thus highlighting light curve features that resemble the detected transits but
occur at orbital phases other than zero. The module is designed to highlight
the average of the input primary signals, the average of the odd and even
signals, the most prominent secondary, tertiary and positive features. The
results of the module are summarized in an automatically-generated PDF file
(see Fig. 9). See Kostov et al. (2019a) for further details.

  * •

Lomb-Scargle: DAVE runs a Lomb-Scargle (LS) periodogram (Lomb, 1976),
(Scargle, 1982) and generates a PDF file showing both the light curve phase-
folded on the period of the inspected signal and the best-fit LS period (see
Fig.3). This test is performed to help the vetter check whether the
modulations occur on the same (or half) period of the candidate signal. In
such cases, the vetter might comment for potential beaming, ellipsoidal,
and/or reflections effects that might indicate a binary star system, see
(Morris & Naftilan, 1993), (Faigler & Mazeh, 2011), (Shporer, 2017).
Performing this test is the reason we opt out of applying custom detrending of
the light curves.

  * •

Summary: DAVE produces a summary PDF file that contains the full light curve,
a zoom-in of each transit in the light curve, the photocenter module images
and the LS result.

Besides DAVE results, the vetters consulted stellar catalogs such as SIMBAD
(Wenger et al., 2000) and Gaia EDR3 (Gaia Collaboration et al., 2021) to keep
track of the field stars in TESS images. This is critical to assess whether
known nearby field stars could potentially be the true source of the observed
TESS signals. Accompanying DAVE products and our final dispositions, we kept
track of any noticeable feature for each TOI by means of comments. The details
of these are listed in Table 1.

Abbreviation |  Meaning |  Description
---|---|---
Disposition |  |
PC |  Planetary Candidate |  A TOI that passed all vetting tests.
(p)FP |  False Positive |  A TOI that does not (fully) pass the vetting tests.
Comments |  |
(p)SS |  Significant Secondary |  A statistically significant secondary is highlighted by the Modelshift module. A secondary eclipse is typical of an eclipsing binary star.
(p)CO |  Centroid Offset |  A significant photocenter shift is detected. This indicates that the target star is not the source of the investigated signal.
UC |  Unreliable Centroids |  A centroid image is considered unreliable if stray light, bright field stars or low SNR cause the difference image to be too noisy for proper photocenter measurements.
(p)OED |  Odd-Even Difference |  A statistically significant difference between odd and even eclipses is highlighted by the Modelshift module and/or the vetters, indicating an eclipsing binary star.
(p)Vshape |  V-shape |  The shape of the signal is not U-shaped, as expected from a typical transit, but rather V-shaped. A transiting planet (usually with $R<<R_{star}$) is expected to produce a sharp ingress, a flat bottom, and a sharp egress. In contrast, an eclipsing star (usually with $R\sim R_{star}$) often produces gradual ingress and egress.
TD |  Too Deep |  For a given stellar radius, the deeper the eclipse, the larger the eclipser. As the deepest confirmed exoplanet eclipses range between 3-4%1111, e.g. (Noyes et al., 2008),(Triaud et al., 2013), we flag signals with depth greater than 2.5%. We note that we do not use TD as the only indicator for a (p)FP as the stellar radius provided in by ExoFOP-TESS might be under/overestimated.
FSCP |  Field Star in Central Pixel |  TESS pixels cover a sky-projected area of 21"x21" each. This means that other sources might fall within the same pixel of the target and be the true source of the signal.
LCMOD |  Light Curve MODulation |  Oscillations in the light curve due to intrinsic and/or rotational variability (potentially due to pulsations or spots) that are not synchronized with the orbital period. These can be produced by either the target itself of by a nearby field star that falls in the aperture used to extract the light curve. Such lightcurves are generally not indicative of a potential false positive.
sync |  Synchronous Modulations |  Lightcurve variations synchronized with the TOI orbital period (or half) are common features of variable binary stars. For example, stars in binary systems with small orbital separations can be deformed into ellipsoids due to tidal effects. The light curves of these systems can show sinusoidal oscillations on half of the eclipses period due to changes in the light-emitting area of the components along the orbital phase, (Shporer, 2017). Such lightcurves are generally indicative of a potential false positive.
EB |  Eclipsing Binary |  An eclipsing binary system.
HPMS |  High Proper Motion Star |  A high proper motion star as listed on SIMBAD333http://simbad.u-strasbg.fr/simbad/.
SSys |  Strong Systematics |  The long-cadence eleanor light curve displays strong artifacts.
Fla |  Flares |  TESS light curve shows potential flaring events. These are sharp rises in flux followed by an exponential decay.
NT |  No Transit |  The eleanor light curve does not show noticeable transit-like signals for QLP-detected TOIs. The Modelshift module flags the candidate due to the low statistical significance of the expected signal.
NS |  Nearby Source |  We identify a nearby source that is the true source of the signal. This has been achieved using the eleanor pixel_by_pixel() function, as explained in Section 4.2.
MD |  Momentum Dump |  TESS thruster-firing-induced artifact in the light curve.
LOWSNR |  Low Signal to Noise Ratio |  The SNR of the expected transits is too low for reliable vetting.
Table 1: List of abbreviations used during the vetting process. The first part
of the table shows the final disposition abbreviation. The second half
displays the general comments that have been used to support the disposition.
A (p) is used next to certain comments and means potential. The human vetters
would add a (p) in case DAVE did not automatically flag the feature due to
lack of statistical significance.

###  2.1 Photocenter Analysis

As a first step, the vetter would start with the critical problem of
pinpointing the true source of the transit-like events. Individual TESS pixels
cover a sky-projected area of ${\rm 21\times 21}$ arcseconds2 . This often
means that one or more field stars fall in the same pixel as the target or in
the aperture used to extract the light curve. First, the vetter would inspect
stellar charts and catalogs to evaluate whether any known sources fall within
the aperture. Next, they would check if these sources are bright enough to
cause the transit-like signal in the light curve based on the measured transit
depth and on the magnitude difference between the target and the source. For
every target, we considered a threshold $\Delta mag$ such that only field
stars with a magnitude within this limit could cause a signal with the same
depth of the one under consideration. In cases for which one or more such
sources are identified, the final disposition for the TOI is accompanied by
the comment FSCP (Field Star in Central Pixels). This procedure is
complementary to the photocenter inspection, during which the vetter would
examine the TESS difference images generated by DAVE and identify the true
source of the signal (see Fig.2).

If the difference images present a photocenter shift away from the target, the
candidate signal is considered a false positive since the signal is coming
from a different star and the disposition is accompanied by the comment ’CO’
(Centroid Offset). If the corresponding difference images show a well-defined
group of pixels centered on the target, the candidate signal would pass the
centroid analysis. There are cases for which the centroid difference images
are difficult to interpret due to artifacts, stray-light or low SNR. When this
was observed to be the case, vetters would comment with ’UC’ (Unreliable
Centroids).

We note that small exoplanets could produce low SNR signal in TESS data which
would also result in poor photocenter images. If the light curves did not not
show any other red flag, we would flag the poor quality of the measured
photocenters but still label the signal as planetary candidates (PC).

Figure 2: DAVE photocenter analysis for TIC 43647325. Upper right: the average
out-of-transit TESS image. The white contour represents the pixels used to
extract the light curve, the black star represents the catalog position of the
target, the purple triangle represents the measured average out-of-transit
photocenter, and the red circle represents the measured difference image
photocenter; lower left: the average in-transit image; upper left: difference
image highlighting the location of the source (black star symbol) varying in
brightness during the detected transits. The colorbars units are in electrons
per second ($e^{-}/s$). Taking into account DAVE results as a whole, this TOI
has been marked as a planet candidate (PC).

###  2.2 Light Curve Analysis

Once the measured photocenters have been analysed, we inspect the full light
curve for every available sector. It is worth noting that we use eleanor
“corrected flux” light curves (Feinstein et al., 2019) without any processing
such as custom detrending or removal of any data point. This visual inspection
is necessary to assess whether prominent systematics could make DAVE automated
dispositions unreliable. During our work, unreliable detections were mostly
observed when using TESS Full Frame Images-extracted light curves. Since DAVE
uses un-detrended data by design, this step is also critical in situations
where strong stellar variability is present. The variability can throw off the
evaluation process of some features such as the depth difference between odd
and even signals (see Fig.3). The light curve variability is also quantified
with a Lomb-Scargle periodogram and a phase-folded light curve is provided as
part of a summary PDF for every TIC (as in Fig.3).

Thus, the full light curve and the LS PDFs aid the vetters in evaluating the
Modelshift result while taking into account potential irregular behavior in
the light curve.

This step of the analysis results in comments about the light curve and the
signals added to the final disposition. Typical comments used in this phase
are LCMOD, Fla, MD, WE and SSys (see Tab.1 for the details).

Figure 3: Prominent stellar variability seen in the light curve of TIC
33911302 (upper left panel). The variability is quantified by DAVE through a
Lomb-Scargle analysis and provided to the vetter as a phase-folded light curve
(lower left panels). Such light curve variability can result in nominal false
positive disposition by the Modelshift module (right panel) due to measured
odd-even difference – but can be overruled by the vetter as an incorrect
automated disposition. The first panel of the Modelshift shows the phase-
folded lightcurve along with the best-fitting transit model (black line); the
second panel of the Modelshift plot shows the same as the upper panel but the
light curve is convolved with the transit model; the lower panels shows zoom-
ins on the primary and secondary events (as labeled), the odd and even primary
events, as well as any tertiary or positive events. The uppermost table
displays the significance of the aforementioned features, see Kostov et al.
(2019a) for the details. This TOI has been labelled as a PC because the
variability is not reminiscent of beaming, reflection and/or ellipsoidal
effects. The observed oscillations could be due to stellar variability of the
target itself or of another source that falls within the group of pixels used
to extract the light curve. There is no hint of a secondary eclipse standing
out of the light curve noise. We note that since the stellar variability for
this target is not coherent with the orbital period, the “Vshape” flag alone
is not sufficient to indicate a TOI as a potential false positive.

The next step is to inspect the Modelshift PDF, in which the folded light
curve and the primary and second-order features are highlighted.

The vetter would inspect the DAVE evaluations on top of the PDF (see Fig.9),
listed in a table where numbers in red indicate potential issues for the
respective candidate signal (see Kostov et al. (2019a) for details.). This
table could highlight significant additional eclipses, odd-even depth
difference or even positive spikes. However, vetters have been trained to
evaluate the issues raised by Modelshift based on what they have seen in the
previous step, i.e. the full light curve inspection. For example, when the
light curves present a modulation, the vetters do not automatically interpret
an odd-even difference flagged by DAVE as a strong indicator of a false
positive. Instead, the vetters would point out the necessity to detrend the
light curve to obtain a final disposition based on the consistency of the
depth of consecutive signals. Again, during the Modelshift inspection, vetters
could identify features with the comments listed in Table 1.

###  2.3 Final Dispositions and Comments

For each target, at least three vetters, plus one science team member, have
performed a complete and thorough vetting as described, in an effort to
minimize the subjectivity of the process by mediating among different
dispositions. The results of this effort are described in Section 4. The final
dispositions presented in our catalog are as follows:

  * •

a planet candidate (PC) is a TOI that passes all of the described vetting
tests.

  * •

a false positive (FP) is a TOI that does not pass one or more tests. It is one
with unambiguous non-planetary nature.

  * •

a potential false positive (pFP) is a TOI for which multiple issues have been
identified by DAVE , by the vetters, or by both – yet we cannot securely rule
it out as a a clear false positive because, for example, the signal-to-noise
ratio is low, or the signal is coming from a pixel containing comparably-
bright multiple sources.

To clarify, an example of a pFP might be a TOI presenting a potential
secondary eclipse in an otherwise flat light curve (pSS, defined as an
additional eclipse that does not stands out beyond the blue $1\sigma$ noise
level lines in the Modelshift PDF, see Fig. 10. A second example is a
candidate signal that exhibits prominent V-shaped eclipses and light curve
modulations coherent with the orbital period of the TOI (see Fig. 6) yet no
significant secondary eclipses or centroid offset were found. Furthermore, a
candidate for which there are no discernible transit-like features in the
lightcurve is also flagged as a pFP. In contrast to pFPs, an FP disposition is
representative of a clear secondary eclipse on the same period of the primary,
odd-even transit depth differences, or a photo-center shift during the
detected transits. Along with these dispositions, we also provide comments on
any other noticeable features using the abbreviations listed in Tab. 1.

##  3 Citizen Scientists-led Development of Additional Vetting Tools and
Resources

The Planet Patrol citizens scientists have not only been fundamental to the
vetting process, but have also been responsible for major contributions to the
presentation and visualization of the results.

LC led a team of volunteers to produce an introductory Video Tutorial hosted
on YouTube, in order to help lower the bar for further involvement in the
vetting process by the wider community. The video introduces the key concepts
about the search for transiting exoplanets and describes the vetting process
through examples.

HADL has produced video recordings of all of our meetings, which can serve as
further introduction to vetting as well as informal tutorials,
recommendations, guidance, resources, etc. for newcomers. The recordings are
posted on Youtube. If there is a sufficient interest in the community, we will
migrate these recordings to a dedicated repository.

Yet another project that emerged from the superuser group (led by RS, a
student at Staples High School) is the development of a custom web-based
interface designed to streamline the vetting process – in essence, a dedicated
vetting portal. Currently, we are using Google Sheets to collect the
dispositions from each vetter for each target, which involves manual input of
thousands of entries (e.g. using the abbreviations listed in Table 1 plus free
text). This is a slow and sometimes cumbersome process prone to errors. To
address this issue, we are transitioning to a custom Graphical User Interface
which allows user-friendly drop-down menus, multiple-choice answers, free
text, automated uploading of targets that still need to be vetted, etc. For
completeness, we briefly describe the development of the vetting portal below.

The website was written in Typescript React, is hosted by Heroku at Planet
Patrol Website, and stores data in an IBM Cloud; the source code can be found
at GitHub. While designing the website, it was important to be mindful of the
workflow that vetters were accustomed to. For example, the vetting portal has
to A) present information in a condensed and easy-to-navigate layout; B) allow
users to find relevant information regarding each target, including the
corresponding ExoFOP-TESS link and the PDF files produced by DAVE in PDF
format; C) let users write dispositions on each target, giving them as much
freedom in their comments while requiring a machine-readable format; D)
facilitate effective analysis of user dispositions. The portal expanded the
user experience by including options to easily sort the table by parameters,
find targets by number of dispositions, and by providing direct links to the
vetting PDFs produced by DAVE (using the Google Drive API to automatically
search for the relevant document). Also, the portal offers drop-down menus to
input dispositions, buttons with the pre-defined comments of Table 1 and free-
text input – all of which can be accesses in machine-readable format. Lastly,
the new website includes features that automatically highlight targets that
still need to be vetted by the user, the dispositions and comments of other
users, as well as the final group dispositions.

TIC ID | Sectors | Epoch [BTJD] | Period [d] | Duration [hrs] | Depth [ppm] |  $R_{p}$ [$R_{J}$] |  $R_{s}$ [$R_{\odot}$] | $TESS_{mag}$ | Disposition | Comment
---|---|---|---|---|---|---|---|---|---|---
1003831 | 8 | 2458518.203 | 1.651142 | 0.76 | 3007 | 0.49 | 0.98 | 10.6701 | PC | TOI 564 b
1103432 | 8 | 2458519.87 | 3.727891 | 3.87 | 17864 | 1.66 | 1.34 | 12.8283 | pFP | pTD, pVshape, FSCP, pSS
1129033 | 4 | 2458410.985 | 1.360025 | 2.16 | 16381 | 1.09 | 0.95 | 9.62713 | PC | WASP-77 A b
1133072 | 8 | 2458517.532 | 0.846542 | 2.41 | 2525 | 0.15 | 0.33 | 12.625 | pFP | FSCP , LCMOD, short-P
1449640 | 5 | 2458440.434 | 3.50175 | 3.80 | 14123 | 1.98 | 1.70 | 11.921 | FP | SS
1528696 | 5 | 2458438.406 | 0.88202 | 0.58 | 9750 | 1.50 | 0.78 | 13.1686 | PC | NGTS-6 b
2758565 | 2 | 2458356.031 | 3.78192 | 2.04 | 15390 | 2.35 | 1.17 | 12.3611 | pFP | pSS,TD
4616072 | 6 | 2458469.062 | 4.18599 | 2.85 | 12942 | 1.70 | 1.56 | 12.8893 | PC | HATS-45 b
4646810 | 4 | 2458416.346 | 14.490034 | 1.59 | 905 | 0.23 | 0.78 | 8.8723 | PC |
4897275 | 21 | 2458872.341 | 16.710042 | 5.40 | 615 | 0.24 | 1.09 | 7.6474 | PC | HPMS
5109298 | 7 | 2458491.797 | 1.6221 | 2.43 | 2310 | 0.94 | 2.22 | 10.6698 | FP | Vshape, CO
5772442 | 7 | 2458492.502 | 1.10578 | 1.87 | 673 | 0.67 | 2.62 | 10.351 | FP | NS

Table 2: The final result of our work is a table of dispositions for 999 TESS
Objects of Interest (TOIs). Here, we report an extract of it. For each TOI, we
display the TESS Input Catalog (TIC) identifier of the star, the signal
parameters input in DAVE , the radius of the potential transiter, the radius
of the star, the TESS magnitude of the star and the final disposition with
related comments. The full table is available as an electronic supplement.

###  3.1 Lessons Learned

As part of this work, we kept record of aspects that could be useful for
similar citizen science efforts. For the benefit of the community, here we
share our experience.

1) Zooniverse talk board is a convenient method for engaging with the
volunteers. We tried to respond to every question as quickly as possible and
also provided a Google Form for them to exploit their interest (if any) and
ask to join the group of SuperUsers.

2) Only about a third of the volunteers who signed up for further involvement
in the project were active and participated in the weekly meetings. We suspect
part of the reason is the time difference, as outlined below.

3) Regular interactions between the science team and the SuperUsers were vital
for the success of the effort. For offline discussions, we used a dedicated
Google Group and a Slack channel; both worked well. For live discussions, we
used Google Meet. However, scheduling a weekly meeting that worked for people
spread across the world was difficult and we believe this would be an issue
for any citizen science project that attract the interest of people worldwide.
While the North/South American and European/African time zones could be
covered simultaneously from a single location, volunteers from the
Asian/Australian time zones were practically left out due to the time
difference. One solution to this issue could be to have two separate weekly
meetings, led from two locations separated by 12 hours.

4) Keeping a video record of the meetings provided a convenient catch-up
option for SuperUsers who could not participate in the meetings.

5) An easy to use, intuitive, and comprehensive tutorial is crucial to get
volunteers who join further along during the project up to speed in a
reasonable time frame. The resources and tools developed as part of the Planet
Patrol project, and the information gathered and synthesized, could be used in
the future as such tutorials.

6) The information provided by ExoFOP-TESS, Gaia, MAST, Simbad, Vizier, and
other publicly-available databases was an important contribution for the
success of the project. It allowed volunteers to not only gain deeper
understanding of the particular target they were vetting but, as a natural
consequence, also of general astronomical concepts such as spectrosopic binary
stars, parallax, proper motion, SED, etc.

##  4 The TESS Triple-9 Catalog

The TT9 catalog contains half the TOIs listed on ExoFOP TESS as of October
2020; we plan to present the rest of our vetting analysis in future work. The
TOIs have been detected by the spoc and QLP pipelines using the 2-minute and
30-minute cadence data. The specifics of these pipelines are presented in
Jenkins et al. (2016) and Huang et al. (2020). Our analysis led to the vetting
of 709 signals as true planetary candidates (PC), i.e. TOIs that have passed
all our vetting tests. We identified 144 TOIs as FPs and 146 as potential
false positives (pFP). The dispositions have been collected in a table with
period of the eclipses, epoch, depth, duration, stellar and TOI estimated
radii, the stellar TESS magnitude and general comments (see Table 2).

Figure 4: Comparison between the 709 planetary candidates in the TT9 catalog
(orange circles) with a subset of the entirety of confirmed exoplanets (purple
squares). We report confirmed exoplanets discovered via the precise radial
velocities and transits techniques with a period lower than 200 days for
clarity.

###  4.1 Planetary Candidates

Of the 709 TOIs marked as PCs in TT9, 146 have already been confirmed
according to ExoFOP TESS. The largest among these are hot Jupiters discovered
by ground-based projects like the Wide Angle Search for Planets (WASP)
(Pollacco et al., 2006), the Hungarian-made Automated Telescope Network
project (HAT-N) (Bakos et al., 2013), the XO effort (McCullough et al., 2005),
the Qatar Exoplanet Survey (QAS) (Alsubai et al., 2013), the Kilodegree
Extremely Little Telescope survey (Pepper et al., 2004) and the Next
Generation Transit Survey (Wheatley et al., 2018). The smallest planets are
instead discoveries of space-based missions: CoRoT (Catala et al., 1995),
Kepler (Borucki et al., 2008), K2 (Howard, 2015) and TESS.

The most common comments for our PCs are “Field Star in Central Pixel” (FSCP,
81 times), “Light Curve MODulation” (LCMOD, 77 times) and “V-shaped signal”
(Vshape, 55 times). The first is expected since TESS cameras pixels cover a
${\rm 21\times 21}$ arcseconds area on the sky, resulting in a frequent blend
of multiple sources. The second can either be caused by modulation of the
target star itself or from sources that fall within the aperture used to
extract the light curve. Finally, a V-shaped transit is not a conclusive proof
against the planetary nature of a transiter, but it suggests that the
dimensions of the two celestial objects at play are comparable. This is
expected to happen for binary star systems more than a planet-star pair due to
geometrical reasons even though large hot Jupiters with grazing transits have
also been shown to produce this type of signal, e.g. (Smalley et al., 2011),
(Mancini et al., 2014), (Bento et al., 2017). We note that we label as PC also
4 TOIs for which we see no obvious signals in the light curves displaying
strong modulations (commented with “NT, strong LCMOD”). In these cases the
modulations are strong enough to hide the weak transit signal without
detrending, which we do not perform ad hoc. Hence, we keep these as PCs since
a signal might have been actually detected if the data were properly
detrended.

The final PC yield of the TT9 catalog is shown in Fig.4, in terms of orbital
period and planet radius, and also compared to a subset of confirmed
exoplanets. We only display confirmed exoplanets that have been discovered via
the radial velocity and transit photometry techniques, with periods shorter
than 200 days for clarity (3,884 out of the total 4,935 confirmed exoplanets
listed on the NASA Exoplanet Archive, as of mid-January 2022). A depression in
the number of planetary candidates is observed for the TT9 PCs as well as for
the confirmed exoplanets when $0.3R_{J}<R<1.0R_{J}$.

Figure 5: The left panel shows the 10x10 TESS pixels centered on the target
(yellow star), the field stars brighter than $T_{mag}$ = 21 mag (dots) within
a 4-pixel radius, the eleanor aperture (red solid outline), and the source of
the signals detected in QLP (blue dashed arrow). The colorbar show the TESS
magnitude of the field sources. The image has been created using the code
triceratops (Giacalone et al., 2021). Upper right panel: TESS sector 7 QLP
(blu) and eleanor (red) light curves for TIC 5772442. While the transits of
the TOI are clear with a period of approximately 2.5 days in the QLP light
curve, no sign of them is present in the eleanor data. In this case, we use
the eleanor pixel-by-pixel light curves (lower right panel) to find that the
signal is coming from a nearby source (blue outline in upper right panel)
while the light curve extracted for the target using the eleanor aperture (red
solid outline in middle panel) is flat. This TOI is thus a false positive
(FP). We note that (i) the eleanor aperture does not include the target star;
and (ii) the lower panel is flipped along the horizontal axis compared to the
lower right panel.

###  4.2 False Positives

A total of 144 TOIs in our catalog have been flagged as false positives (FPs).
The most common reason is a significant photocenter shift during transit – 76
targets – while secondary eclipses have been registered for 48 candidate
signals. In 6 cases the poor quality of the light curve might have deceived
the detection pipeline that locked onto features like gaps in the sectors,
momentum dump spikes or other systematics (TICs 24364065, 47384844, 101929303,
150247134, 169177766, 198384408). Only 2 TOIs show significant odd-even
differences between consecutive transits (TIC 9033144 and TIC 230086768).
Light curve modulations are present in 32 cases. Of these, 8 display
modulations coherent with the orbital period (TICs 2758565, 93963408,
96246348, 97158538, 141663460, 141663464, 198457103, 233720539). We kept track
of such modulations for the following reasons. First, in some cases they could
be triggering the detection pipelines and be mistaken for candidate planets.
Secondly, light curve modulations coherent with the orbital period of the
transiting exoplanet candidate can be indicative of a short-period eclipsing
binary. Finally, even though the detected modulation is not the deciding
factor for the FP disposition, we include it for the sake of completeness.

We note that in some cases more than one FP red flags were raised.

Another frequent source of false positives we identified is non-detections of
QLP-based planet candidates in eleanor light curves. As mentioned above, our
analysis is based on the latter because vetting transiting planet candidates
detected by one pipeline but using light curves produced by another is highly
valuable as it provides an independent set of tests and and confirmations
(e.g. Kostov et al. 2019)444We note that the QLP light curves were not
publicly-available at the start of this project. As we aim for consistency and
uniformity instead of completeness, we continued our analysis with eleanor
data even after the QLP light curves were uploaded to MAST. Specifically, DAVE
uses eleanor -generated “Corrected Flux” light curves to perform its tests on
the TOIs for which only FFI data is available. For some targets in our
catalog, the eleanor light curves do not show obvious transit signals at the
ephemeris provided by ExoFOP-TESS. In these cases, the vetters would label
them as being potential false positives rather than certain false positives.
This is mainly because the apparent lack of transit signals in eleanor data
can be caused by differences between the QLP pipeline (through which the
signals where discovered) and the eleanor pipeline (through which we vet
them). Altogether, there are 53 such “No Transit” (NT) cases. We have already
mentioned why 4 of them have been labeled as PC in the previous subsection.

We note that because there is no apparent transit-like signal for these NT
TOIs in eleanor data, we cannot perform a photocenter analysis. Thus to
analyse them and produce a final disposition, we visually inspected the
corresponding light curves using the eleanor pixel_by_pixel() routine. This
function allows the user to inspect the light curves of single pixels within
the downloaded TESS pixel cutout. Specifically, we would compare the eleanor
aperture used to extract the light curve to the single pixels light curves.
The fact that eleanor uses a different aperture optimization algorithm implies
that the final light curve is extracted from a different set of pixels and
might be different with respect to the QLP one.

Indeed, we found 14 (out of the 53 NT cases) for which the QLP-detected signal
is coming from a nearby source (NS) instead of the target under investigation
and we finally label these as false positives. We report an example of this
case in Fig. 5.

Finally, we note that 6 confirmed planets have been labelled in TT9 as FPs due
to the presence of significant secondary eclipses – despite the fact that
these are confirmed hot Jupiters. These dispositions, however, are based on
our definition of an FP, i.e. a candidate exhibiting clear secondary eclipses
(see Fig.11). Thus even though we are aware that these ‘secondary eclipses’
are in fact planetary occultations, for consistency we flag the 6 TOIs as FPs
(see also Table 3 and Section 4.4).

###  4.3 Potential False Positives

As described in Section 2, we have labelled as potential false positives (pFP)
146 TOIs for which we could not confirm the suspected non-planetary nature yet
identified a number of potential issues. For these TOIs, we identified various
combinations of: V-shape morphology, ultra-short periods, potential secondary
eclipses, potential photocenter offset, field stars in the same pixel of the
target that are bight enough to produce the detected signals as contamination,
and/or coherent light curve modulations as commonly seen in short-period
eclipsing binaries. We reported examples of potential false positives that
could be due to unresolved background sources in Fig.6 and Fig.10. Programs
such as TFOP555TESS Follow-Up Observing Programs could clear these cases
thanks to higher-resolution photometric measurements, spectroscopic
observations, high-resolution imaging or precise radial velocity measurements.
Specifically, we identified 49 cases showing potential photocenter offset and
32 cases for which a potential secondary eclipse has been identified. A total
of 69 pFPs display light curve modulations (LCMOD), of which 33 are
synchronous with the eclipses (sync). Finally, 35 TOIs show no transit event
but, as opposed to the FP cases, we could not identify nearby sources showing
the expected eclipses.

In some cases, more than one of these red flags have been raised by the
vetters.

TIC | Name |  $\sim$Depth (ppm) | Comments
---|---|---|---
16740101 | KELT9-b | 600 | Occ
22529346 | WASP-121 b | 200 | Occ
86396382 | WASP-12 b | 400 | FSCP, Occ
100100827 | WASP-18 b | 300 | Occ
129979528 | WASP-33 b | 500 | LCMOD, Occ
158324245 | KOI-13 b | 400 | Occ
Table 3: Confirmed planets in the TT9 catalog for which our vetting efforts
identify prominent secondary eclipses. According to our vetting workflow,
these planets had to be flagged as false positives in TT9 because we do not
have sufficient information to distinguish between an occultation and a
secondary eclipse.

###  4.4 Confirmed Planets

For consistency, we vet targets independent of any prior knowledge of the
system, and regardless of their current disposition on ExoFOP. Some of them
are listed as confirmed planets and were added to our catalog for
completeness. Our analysis of confirmed planets supports the planetary nature
interpretation of 140 of them and flagged 6 as false positives. The reason for
the latter dispositions is as follows.

When large, close-in planets pass behind their host star along the line of
sight with the observer, a secondary dip in the light curve can be observed
due to the occultation of the light reflected and/or emitted by the planet.
Occultations allow direct measurement of the planet’s radiation (hence its
effective temperature) and helps to constrain the planet’s orbital
eccentricity. Our analysis highlighted this kind of signal as a false positive
indicator since the event is effectively identical to a secondary eclipse. We
find such occultations in the light curves of 6 of the confirmed planets in
the TT9 and we label these signals as false positives since we vetted every
TOI without prior knowledge of its current status. An example is shown in Fig.
11. We report the TICs of these candidates, along with the approximate depth
of the secondary eclipse as seen in TESS bandpass (600-1000 nm), in Table 3.
For completeness, we also point out the other disposition comments for these
targets where applicable.

Figure 6: An example of potential false positive (pFP): TIC 97158538. The
upper panel presents a portion of the Modelshift results. Here, one can see
that the TOI presents light curve modulations coherent with eclipses. The
lower panels displays the out-of-transit TESS image (lower right) and the
difference image (lower left) between in- and out-of-transit frames. The
colorbars show units of electrons per second ($e^{-}/s$). Given the large
scatter in the measured photocenters, likely caused by the light curve
variability, we note a potential, sub-pixel centroids offset.

###  4.5 Potential New Signals

During the analysis presented above, the vetters picked up new signals not
previously reported on ExoFOP TESS. We inspect these one by one and check
whether they could be identified with known systematic effects such as
Momentum Dumps (MD), i.e. artifacts caused by TESS thruster firings. This can
be done by comparing the signals to the registered MDs events in TESS Data
Releases666https://archive.stsci.edu/tess/tess_drn.html. These new signals are
as follows.

An additional eclipse has been flagged for TIC 23740089, located at BTJD
1945.5 in sector 23. This TIC hosts what we have labelled as a PC with a 0.59
$R_{J}$ candidate orbiting a 1.14 $R_{\odot}$ star in 9.8 days. The
additional, deeper ($>2\%$) eclipse is only observed once between transits two
and three (see Fig. 7). According to TESS Data Release Notes for sector 23,
this event does not correspond to a thruster firing event. The separation
between the TOI and this additional signal is such that we could be missing
the same deeper eclipse event after transit number one due to a data gap and
the one after transit number three due to the finite duration of the sector.
Thus, the available eleanor-extracted TESS data does not have sufficient
coverage to assess whether this event might indicate a binary star and we
labeled the target as a PC with an extra eclipse. TESS will observe again the
target between February 26th 2022 and April 22nd 2022 which might allow to
clear this specific case.

Figure 7: Long-cadence light curve of TIC 23740089. The highlighted signals
are the events detected by the QLP pipeline that we vet with DAVE . An
additional eclipse is observed at BTJD 1945.5. If this were the primary
eclipse of a circular binary system, such that the detected TOI is in fact
secondary eclipses, we would have to observe the same type of feature at
approximately 1935.6 and 1955.3. Neither time was observed due to a data gap
and the finite time range of the TESS sector so we cannot conform this
hypothesis. Figure 8: Short-cadence sector 7 light curve of TIC 143526444. The
vertical gray band highlights the transit of the candidate in the ExoFOP TOIs
catalog. The additional event at BTJD 1501 (black oval) is likely caused by a
momentum dump as reported in TESS data release notes. We don’t find any
similar event in the other available sectors.

We detected two additional events in the light curve of TIC 74534430 at
approximately 1539.6 and 1545.0 BTJD (in sector 8 and 9) that do not fall on
any reported MD events. These events are shallower and shorter than the TOI
transits, which is labelled in TT9 as a PC with an orbital period of 18 days
and radius of 0.19 $R_{J}$, orbiting a 0.61 $R_{\odot}$ star. The target has
been observed by TESS in sectors 8, 9, 35 and 36.

We also observed additional, non-periodic events in the light curve of TIC
128501004 in sector 16, the only one available for the target. No additional
TESS observations are scheduled for this star. Two different deep events have
been noticed in the light curve at approximately 1747.9 and 1748.2 (BTJD),
neither of which corresponds to a MD event. TIC 128501004 is a 1.1 $R_{\odot}$
star hosting a 0.18 $R_{J}$ object (labelled as a PC in TT9) with an orbital
period of 0.8 days. However, taking into account the additional events, this
candidate could be a triple star system which will require further
observations to be confirmed.

An additional event is present in the light curve of TIC 143526444 in sector
7. The target is a 3.5 $R_{\odot}$ star hosting a 0.9 $R_{J}$ PC with an
orbital period of 15.3 days. The event is at time 1501 (BTJD) where a MD event
has been registered and reported in TESS Data Release Notes $\\#9$. Thus we
reject this extra event as an artifact.

Finally, a potential new candidate has been observed in the TESS sector 4
light curve of TIC 100990000 at an epoch of approximately 1425 (BTJD). This
transit-like feature is a single event within the four available TESS sectors
(3, 4, 30 and 31) and has a duration of roughly six hours which is consistent
with a long period. If confirmed, this candidate would be the outermost planet
in a system with other two planets (Earth- and Super-Earth-sized) orbiting a
G-type star in 4.04 and 9.57 days, respectively. The estimated radius of the
new candidate is 2.7 $R_{\oplus}$. The characterization of this system is
being carried out in Cacciapuoti et al, submitted. We note that we have
recently learned that this candidate has also been uploaded on ExoFOP-TESS as
TIC 100990000.03 by a citizen scientist of the Planet Hunters TESS project
(Eisner et al., 2021).

##  5 Conclusions

We presented the TT9 catalog, containing 999 uniformly-vetted transiting
exoplanet candidates from TESS. We marked 709 TOIs as bona fide planet
candidates, of which 146 are confirmed exoplanets. Additionally, 144 TOIs were
flagged as false positives due to photocenter motion during transit,
significant secondary eclipses, and/or light curve modulations indicative of
eclipsing binary stars. Finally, 146 TOIs were labelled as potential false
positives due to too many potential issues to be passed as PCs yet no
definitive evidence to be ruled out as clear false positives.

The TT9 catalog is provided to the community in a table format (see Table 2)
which provides our final dispositions along with additional comments for each
TOI. For completeness, the table also includes the transit ephemeris, depth,
and duration, and the TESS magnitude. The DAVE generated modelshifts and PDF
summaries, as well as the photocentre images, are made publicly available on
ExoFOP TESS at the dedicated webpage of each TOI. This work presents the first
stage of our effort to vet all TOIs listed on ExoFOP-TESS using DAVE and
citizen science. Our results provide an independent analysis of known TOIs and
could be used to prioritize follow-up observations. The dispositions provided
in the TT9 catalog can be also utilized as input for demographic studies of
transiting exoplanets from TESS or as a training set for machine learning
algorithms aimed at automating vetting efforts.

##  6 Acknowledgements

This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.

We acknowledge the use of public TESS Alert data from the pipelines at the
TESS Science Office and at the TESS Science Processing Operations Center
(SPOC) and from the Massachusetts Institute of Technology Quick Look Pipeline
(QLP).

This research has made use of the Exoplanet Follow-up Observation Program
website, which is operated by the California Institute of Technology, under
contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program.

This publication uses data generated via the Zooniverse.org platform,
development of which is funded by generous support, including a Global Impact
Award from Google, and by a grant from the Alfred P. Sloan Foundation.

We thank the referee for the insightful comments which helped improve the
manuscript.

Software: DAVE (Kostov et al., 2019a), eleanor (Feinstein et al., 2019), QLP
(Huang et al., 2020)

##  7 Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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  * Triaud et al. (2013) Triaud A. H. M. J., et al., 2013, A&A, 551, A80
  * Wenger et al. (2000) Wenger M., et al., 2000, A&AS, 143, 9
  * Wheatley et al. (2018) Wheatley P. J., et al., 2018, MNRAS, 475, 4476

##  Appendix A Modelshift figures

In this Appendix, we display the examples of DAVE Modelshift PDF products that
have served as examples during the explanation of the general vetting workflow
(Section 2) and of the reasons behind the TT9 dispositions (Section 4).

Figure 9: An example of DAVE Modelshift PDF for TIC 106402532 (sector 9). No
issues of potential concern have been raised by either DAVE or the vetters for
this TOI. Considering the photocenter analysis as well, this TOI is labelled
as a bona-fide planetary candidate (PC) in the TESS Triple-9 catalog. Figure
10: Modelshift PDF for TIC 1133072 (sector 8). The second panel from top (the
phase-folded light curve convolved with the transit model) shows a light curve
reminiscent of the modulations typical for $\beta$ Lyrae-type binary stars,
where the primary and secondary eclipses do not have sharp ingress and egress.
Given the low SNR, the short orbital period, the V-shaped eclipses and the
potential secondary eclipse near phase 0.5 (note that it is not flagged as
significant by DAVE ), we flag the target as a pFP. Figure 11: The phase-
folded light curve for TIC 16740101, also known as KELT-9 b. A secondary
eclipse on the same period of the input signal is clear in DAVE Modelshift
result. The secondary is due to the occultation of the bright side of the
planet when it orbits behind the star along TESS line of sight. Based on DAVE
results only, for consistency we label this TOI as FP due to the significant
secondary eclipses.


# Trineutron resonances in the SS-HORSE–NCSM approach

I. A. Mazur  Center for Exotic Nuclear Studies, Institute for Basic Science,
Daejeon 34126, Republic of Korea  M. K. Efimenko  Laboratory for Modeling of
Quantum Processes, Pacific National University, Khabarovsk 680035, Russia  A.
I. Mazur  Laboratory for Modeling of Quantum Processes, Pacific National
University, Khabarovsk 680035, Russia  I. J. Shin  Institute for Rare Isotope
Science, Institute for Basic Science, Daejeon 34000, Republic of Korea  V. A.
Kulikov  Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, Moscow 119991, Russia  A. M. Shirokov  Skobeltsyn Institute of
Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia  J.
P. Vary  Department of Physics and Astronomy, Iowa State University, Ames, IA
50011-3160, USA

###### Abstract

The SS-HORSE–NCSM method is generalized to the case of democratic decay into
an odd number of fragments. This method is applied to the search for
resonances in three-neutron system (trineutron) using ab initio No-Core Shell
Model calculations with realistic nucleon-nucleon potentials. The $3/2^{-}$
and $1/2^{-}$ strongly overlapping resonances are predicted when softened $NN$
interactions are used and are preferred over the case where bare $NN$
interactions of the chiral effective field theory are used with no resonance
obtained.

##  I Introduction

In this paper, we develop and apply an ab initio method of calculating the
democratic decay of light nuclei into an odd number of fragments within the
No-Core Shell Model (NCSM) Barrett_ProgrPartNuclPhys_2013 . Such an approach
is of a current interest for the studies of neutron-excess light nuclei and,
in particular, Borromean neutron-excess nuclei near and beyond the neutron
drip line.

We apply this method to the search for resonant states in the three-neutron
system (trineutron). There is an increasing interest in theoretical and
experimental investigations of multi-neutron systems following the
experimental observation of the tetraneutron resonance Kisamori ;
Duer_Nature_2022 . By studying the multi-neutron systems we can sensitively
probe the interaction between neutrons for details that are not available from
neutron-neutron scattering experiments.

The first experimental investigations of the three-neutron system were
published in 1960s. In particular, the bound trineutron search failed in the
studies of the 3H($n$,$p$)${}^{3}n$ reaction in Ref. Ajdacic_PhysRevLett_1965
. A comprehensive description of the history of trineutron experimental
searches can be found in reviews of Refs. Kezerashvili_arXiv_2016 ;
Marques_EurPhysJA_2021 . The main conclusion of all experiments is the
exclusion of the bound trineutron. At the same time, the existence of a
resonant trineutron state is not ruled out.

References Kezerashvili_arXiv_2016 ; Marques_EurPhysJA_2021  present also the
history of theoretical investigations of the three-neutron system. Among those
we note the recent studies based on realistic $NN$ interactions
Hiyama_PhysRevC_2016 ; Gandolfi ; Deltuva_PhysRevC_2018 ; Li_PhysRevC_2019 .
The resonant trineutron has not been found in Refs. Hiyama_PhysRevC_2016 ;
Deltuva_PhysRevC_2018 . The binding energy of three neutrons confined by an
external potential (trap) has been extrapolated in Ref. Gandolfi  to the case
of the vanishing trap to estimate the trineutron resonance energy (without any
estimation for the resonance width). The obtained resonance energy of
$E_{r}=1.11(21)$ MeV is close to the result of Ref. Li_PhysRevC_2019  where
the trineutron resonance is predicted by the calculations in the ab initio No-
Core Gamow Shell Model at the energy of $E_{r}=1.29$ MeV with the width of
$\Gamma=0.91$ MeV.

In this work we will extend our SS-HORSE–NCSM approach Shirokov_PhysRevC_2016
; Mazur_PhysPartNucl_2017 ; Blokhintsev_YaF_2017_1 ; Blokhintsev_YaF_2017_2 ;
Shirokov_PhysRevC_2018 ; Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018
that generalizes the NCSM to the description of continuum spectrum states. The
advantage of the SS-HORSE–NCSM is that the scattering phase shifts are
computed by simple analytical expressions at the NCSM eigenenergies and there
is no need in additional numerical challenges for no-core systems as compared
to other continuum generalizations of NCSM like the NCSM with continuum
(NCSMC) NCSMC  or the No-Core Gamow Shell Model GSM ; Li_PhysRevC_2019 . Next
the $S$ matrix is parametrized and the resonant energies and widths are
obtained by a numerical location of the $S$-matrix poles. Recently this method
has been successfully applied to the description of resonant states in 5He
Shirokov_PhysRevC_2016 , 5Li Shirokov_PhysRevC_2018 , 7He Mazur_PhysRevC_2022
and 9Li Mazur_PhysAtNucl_2022  with JISP16 Shirokov_PhysLettB_2007  and
Daejeon16 Shirokov_PhysLettB_2016  realistic $NN$ interactions in the channels
of elastic scattering of protons in the case of 5Li or neutrons in all other
nuclei by the remaining nuclear fragment in the ground and sometimes in
excited states. This method has been also generalized to the case of four-body
democratic decays and applied to the description of resonances in the
tetraneutron Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018  and in the
7He nucleus in the channel of four-body decay into 4He and three neutrons
Mazur_PhysRevC_2022 . In short, we have previously applied the SS-HORSE–NCSM
approach up to now only to the decay channels with an even number of
fragments.

On the other hand, the SS-HORSE–NCSM has been applied to the hypernuclear
system $\Lambda nn$ in Ref. Htun_PhysRevC_2022 , a three body decay, for the
first time. The distinction from Refs. Shirokov_PhysRevC_2016 ;
Mazur_PhysPartNucl_2017 ; Blokhintsev_YaF_2017_1 ; Blokhintsev_YaF_2017_2 ;
Shirokov_PhysRevC_2018 ; Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018
is that Ref. Htun_PhysRevC_2022  did not search for the $S$-matrix poles but
extracted the resonance parameters from the slope of the phase shifts of the
true three-body ($3\to 3$) scattering.

In this paper we generalize the technique of locating the $S$-matrix poles
proposed in Refs. Shirokov_PhysRevC_2016 ; Mazur_PhysPartNucl_2017 ;
Blokhintsev_YaF_2017_1 ; Blokhintsev_YaF_2017_2 ; Shirokov_PhysRevC_2018 ;
Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018  to the case of
democratic decay into an odd number of fragments. We construct a family of
parameterizations of the ${3\to 3}$ scattering $S$ matrix in a minimal
approximation to enable the possibility of the $S$-matrix pole search.

The structure of the paper is the following. We discuss the $3\to 3$
scattering, the structure of the respective $S$ matrix and the generalization
of the SS-HORSE–NCSM approach to the case of the democratic decay into an odd
number of fragments using minimal approximations in Section II. We apply the
developed method to the search of resonances in the three-neutron system based
on the NCSM calculations with various realistic $NN$ interactions in Section
III. The conculsions are presented in Section IV.

##  II SS-HORSE–NCSM method for democratic decay into odd number of fragments

We make use of the version of the $J$-matrix formalism Heller_PhysRevA_1974 ;
Yamani_JMathPhys_1975  in scattering theory utilizing the harmonic oscillator
basis which is also known as HORSE Bang_AnnPhys_2000  for the generalization
of the NCSM to the case of the continuum spectrum. The essence of the HORSE
formalism is the division of the many-body Hilbert space into a finite-
dimensional oscillator subspace where both the potential energy of the
interactions between particles and their kinetic energy are taken into account
($P$ space) and the remaining infinite-dimensional subspace where only the
kinetic energy is retained and the interaction is neglected ($Q$ space). The
$P$ space conventionally includes all many-body states with oscillator
excitation quanta which do not exceed some certain number $N_{\max}$. This
definition is well matched with the NCSM where $N_{\max}$ is used to restrict
the model space.

We use a generalization of the HORSE formalism to the case of the true many-
body ($A\to A$) scattering developed in Ref. Zaytsev_TeorMatFiz_1998  to
describe states in the many-body continuum. The version of HORSE for $A\to A$
scattering utilizes the ideas of the method of hyperspherical harmonics (HH)
(see, e. g., Refs. Jibuti ; HHbook ) which was widely used in studies of
various atomic and nuclear systems, in particular, of the trineutron
Dzhibuti_SovJNuclPhys264_1984 ; Dzhibuti_SovJNuclPhys700_1984 ;
Jibuti_NuclPhysA_1985 ; Dzhibuti_SovJNuclPhys_1985 ;
Kezerashvili_Preprint_1993 .

In the case of continuum states, the HH method is an adequate tool for the
description of the so-called democratic decays of an $A$-body system when no
subgroup of the $A$-particles has a bound state. This condition appears to be
satisfied for the trineutron or tetraneutron. The wave function dependence on
the “democratic” hyperradius,

$\rho=\sqrt{\sum_{i=1}^{A}(\mathbf{r}_{i}-\mathbf{R})^{2}},$ (1)

is of a primary importance within the HH approach. Here $\mathbf{r}_{i}$ are
the individual neutron coordinates and $\mathbf{R}$ is the center-of-mass
coordinate. The remaining degrees of freedom are described by hyperspherical
functions depending on some set of $3A-4$ angles $\Omega_{i}$ on the
${(3A-3)}$-dimensional sphere coupled with neutron spins and a function
describing the center-of-mass motion. Both the hyperspherical and the
hyperradial functions are characterized by the hypermomentum $K$ and some
other quantum numbers $\alpha$ distinguishing different states with the same
hypermomentum which are of no interest for us in this research. For the states
of a definite total angular momentum $J$ and parity,
$K=K_{\min},K_{\min}+2,\ldots$, where generally $K_{\min}\geq 0$ is integer,
and $K_{\min}=1$ in the case of trineutron natural (negative) parity states
with $J=3/2$ or 1/2.

In the HH approach, the Schrödinger equation takes the form of a set of
coupled equations which is equivalent to a set of equations describing a
multichannel scattering with the same threshold in all channels. Each of the
equations includes a centrifugal term ${\cal L}({\cal L}+1)/\rho^{2}$, where
the effective orbital momentum Zaytsev_TeorMatFiz_1998

${\cal L}=K+\frac{3A-6}{2}.$ (2)

We utilize the democratic decay minimal approximation that implies retaining
only one HH with $K=K_{\min}$ in the $Q$-space which is associated with the
long-range behavior of the wave functions since the HH with $K>K_{\min}$ are
suppressed at large distances by the high centrifugal barrier. So, the wave
function is characterized by a single phase shift $\delta$ of $A\to A$
scattering. This phase shift can be calculated using NCSM eigenenergies
$E_{d}$ obtained with given values of $N_{\max}$ and the NCSM oscillator basis
parameter $\hbar\omega_{d}$ within the SS-HORSE–NCSM approach as
Shirokov_PhysRevLett_2016

$\tan\delta(E_{d})=-\frac{S_{N_{\max}+N_{\min}+2,{\cal
L}}(E_{d})}{C_{N_{\max}+N_{\min}+2,{\cal L}}(E_{d})},$ (3)

where $N_{\min}$ is the minimal number of oscillator quanta allowed by the
Pauli principle, $S_{n{\cal L}}(E)$ and $C_{n{\cal L}}(E)$ are regular and
irregular solutions for a free motion in the HORSE formalism which explicit
analitycal expressions can be found in Ref. Zaytsev_TeorMatFiz_1998 . Note,
$S_{n{\cal L}}(E)$ and $C_{n{\cal L}}(E)$ depend on the oscillator parameter
$\hbar\omega$.

The accuracy of the approximation retaining the single lowest HH in the
$Q$-space was confirmed in the studies of three-body democratic decays in
Refs. Lurie_IzvRossAkadNaukSerFiz_1993 ; Lurie_IzvRossAkadNaukSerFiz_1997 ;
Lurie_AnnPhys_2004 ; Lurie_Jmatrix_book . It is important to emphasize here
that the NCSM calculations performed in the $P$ space utilize a complete set
of HH with $K\leq N_{\max}+N_{\min}$.

The $A\to A$ $S$ matrix is related to the phase shift $\delta$,

$S(k)=e^{2i\delta(E)}.$ (4)

To study the $S$ matrix analytical properties, it is more convenient to
analyze it as a function of the momentum $k$ instead of the energy $E$,

$E=\frac{\hbar^{2}k^{2}}{2M},$ (5)

where $M$ is total mass of the system.

In the case of even $A$, ${\cal L}$ is integer, and the ${A\to A}$ $S$ matrix
analytical properties are similar to those of two-body scattering. In
particular Baz ; Newton ,

$S(-k)=S^{-1}(k)$ (6)

and

$S^{*}(k)=\frac{1}{S(k^{*})},$ (7)

which are crucial for the $S$-matrix parameterization. The parameterized $S$
matrix can be analytically continued to the complex $k$ plane for the search
of its poles associated with resonant and bound states. This technique has
been used to estimate the energy and width of the resonant state in the
tetraneutron Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018 .

Analytical properties of the ${A\to A}$ $S$ matrix become more complicated in
case of an odd $A$ due to a half-integer value of the effective angular
momentum ${\cal L}$ as follows from Eq. (2). The $S$ matrix properties in the
case of arbitrary non-integer angular momentum are discussed in Ref. Newton .
In this case Eq. (6) is generalized to

$S(ke^{i\pi})=e^{2\pi i{\cal L}}S^{-1}(k)+1-e^{2\pi i{\cal L}},$ (8)

which holds for any complex value of $k$. As a result, for a half-integer
${\cal L}$ we have

$S(ke^{i\pi})=-S^{-1}(k)+2.$ (9)

Note that Eq. (7) is valid for any real value of angular momentum. We
attribute properties (7) and (9) to the ${A\to A}$ $S$ matrix in the case of
an odd $A$.

The $S$ matrix has multiple sheets and its properties are complicated in the
case of a non-integer angular momentum. The $S$ matrix can be expressed as
Alfaro

$S(k)=\frac{Z(k)-ik^{2{\cal L}+1}e^{i\pi(2{\cal L}+1)}}{Z(k)-ik^{2{\cal
L}+1}},$ (10)

where $Z(k)$ has the following property:

$Z(ke^{i\pi})=Z(k).$ (11)

Equation (10) cannot be used directly in the case of a half-integer ${\cal
L}$: we have an uncertainty of the $0/0$ type that should be resolved using
the L’Hôpital’s theorem considering ${\cal L}$ as a continuous variable and
investigate the limit ${\cal L}\to K+(3A-6)/2$ Alfaro  to obtain

$S(k)=1+\frac{2\pi k^{2{\cal L}+1}}{Y(k)-2ik^{2{\cal L}+1}\ln(k/q_{0})},$ (12)

where $Y(k)=\left.\frac{\partial Z(k)}{\partial{\cal L}}\right|_{{\cal
L}=K+(3A-6)/2}$ and $q_{0}$ is a real-valued momentum needed to make
dimensionless the argument of $\ln$ in the denominator. We note that our final
results for the $S$-matrix poles are independent of $q_{0}$. Using Eqs. (4)
and (12), it is easy to deduce

$\tan\delta=\frac{\pi k^{2{\cal L}+1}}{2k^{2{\cal
L}+1}\ln(k/q_{0})+i\left(Y(k)+\pi k^{2{\cal L}+1}\right)}.$ (13)

The phase shift is a real-valued function for real $k>0$. Therefore it is
convenient to introduce a real-valued at real $k$ function

$X(k)=i\left(Y(k)+\pi k^{2{\cal L}+1}\right)\\!.$ (14)

It is easy to show that $Y(ke^{i\pi})=Y(k)$, that leads to the following
symmetry property of the function $X(k)$:

$X(ke^{i\pi})=X(k).$ (15)

The $A\to A$ $S$ matrix and phase shift are expressed in terms of $X(k)$ as

$S(k)=\frac{X(k)+2k^{2{\cal L}+1}\ln(k/q_{0})+i\pi k^{2{\cal
L}+1}}{X(k)+2k^{2{\cal L}+1}\ln(k/q_{0})-i\pi k^{2{\cal L}+1}},$ (16)
$\tan\delta=\frac{\pi k^{2{\cal L}+1}}{2k^{2{\cal
L}+1}\ln(k/q_{0})+X(k)}{\color[rgb]{1,0,1}.}$ (17)

Due to Eq. (15), the function $X(k)$ can be parameterized as a series
expansion in even powers of $k$,

$X(k)=\sum_{i=0}^{W}w_{i}k^{2i}.$ (18)

The parameterization (18) provides for an estimation of the phase-shift
behavior in the limit $k\to 0$. For example, for the three-body problem
($A=3$), supposing that $X(k)\xrightarrow{k\to 0}w_{0}$, we obtain from Eq.
(17):

$\tan\delta\sim\delta\sim k^{2K+4}\sim k^{2{\cal L}+1}\sim E^{K+2\\!}.$ (19)

This behavior is in line with the analysis presented in Ref. Jibuti
justifying the parameterization (18).

Following the ideas of the SS-HORSE–NCSM approach Shirokov_PhysRevC_2016 ;
Mazur_PhysPartNucl_2017 ; Blokhintsev_YaF_2017_1 ; Blokhintsev_YaF_2017_2 ;
Shirokov_PhysRevC_2018 ; Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018
, we can obtain the parameters $w_{i}$ of the expansion (18) by calculating a
set of the $A\to A$ phase shifts $\delta(E_{d})$ using Eq. (3) at the NCSM
eigenenergies $E_{d}$ obtained with a chosen $N_{\max}$ and a set of the
$\hbar\omega_{d}$ values, and next parameterize this set of $\delta(E_{d})$ by
means of Eqs. (17) and (18) (see the next Section for more detals). To
calculate energies and widths of resonances, we locate the $S$-matrix poles by
searching for zeros of the denominator in the r.h.s. of Eq. (16) which is
equivalent to solving numerically in the complex $k$ plane
($-\pi<\arg(k)<\pi$) equation

$X(k)+2k^{2{\cal L}+1}\ln(k/q_{0})-i\pi k^{2{\cal L}+1}=0$ (20)

using the technique suggested in Ref. Shirokov_PhysRevC_2018  or the
Newton–Raphson method (see, e. g., Ref. Rakityansky ).

##  III Trineutron

The above method is applied to the search of resonances in the three-neutron
system. We use various realistic $NN$ interactions, the same as employed in
our analysis of the tetraneutron Shirokov_PhysRevLett_2016 ;
Shirokov_AIPConfProc_2018 . We utilize the MFDn code Maris_ProcComputSci_2010
; Aktulga_ConcurComputPractExper_2014  to perform the NCSM calculations with
$N_{\max}$ ranging from 4 to 20 and $\hbar\omega$ spanning from 2 to 50 MeV.

The results for the $3/2^{-}$ ground state energy obtained with the Daejeon16
Shirokov_PhysLettB_2016  interaction, are shown in the upper panel of Fig. 1.
The $3\to 3$ phase shifts at the NCSM eigenenergies obtained using Eq. (3) are
presented in the lower panel. It is seen that the phase shifts tend to the
same smooth resonance-like curve as $N_{\max}$ is increasing demonstrating a
convergence of the $3\to 3$ phase shift calculations.

Figure 1: Upper panel: NCSM results for the trineutron $3/2^{-}$ ground state
energy obtained with Daejeon16 $NN$ interaction with various $N_{\max}$
plotted as functions of $\hbar\omega$. Lower panel: $3\to 3$ phase shifts at
the NCSM eigenenergies obtained using Eq. (3).

We parameterize the function $X(k)$ for each individual value of
${N_{\max}\geq 16}$ used in the NCSM calculations of the trineutron ground
state energies. For a given $N_{\max}$, we use a set of parameters $w_{i}$
defining $X(k)$ to find the energies ${\cal E}_{d}$ by solving the equation

$-\frac{S_{N_{\max}+3,{\cal L}}({\cal E}_{d})}{C_{N_{\max}+3,{\cal L}}({\cal
E}_{d})}=\frac{\pi\kappa^{6}_{d}}{2\kappa^{6}_{d}\ln(\kappa_{d}/q_{0})+X(\kappa_{d})},$
(21)

for each value of $\hbar\omega_{d}$ used in the respective NCSM calculations.
Here $\kappa_{d}=\frac{\sqrt{2M{\cal E}_{d}}}{\hbar}$ and ${\cal
L}=K_{\min}+\frac{3}{2}=\frac{5}{2}$. To find the optimal values of $w_{i}$,
we minimize the function

$\Xi=\sqrt{\frac{1}{D}\sum_{d=1}^{D}\left[({\cal
E}_{d}-E_{d})^{2}\left(\frac{\hbar\omega_{M}}{\hbar\omega_{d}}\right)^{2}\right]},$
(22)

where $D$ is the number of the NCSM energies $E_{d}$ obtained with the same
$N_{\max}$ and the same $\hbar\omega_{d}$ as each of the respective energies
${\cal E}_{d}$,
$\displaystyle\hbar\omega_{M}=\max_{d=1,\ldots,D}\hbar\omega_{d}$, and
$\left(\hbar\omega_{M}/\hbar\omega_{d}\right)$ is the weight increasing the
importance of states with smaller $\hbar\omega_{d}$ corresponding to energies
closer to the resonance region.

The quality of the fits can be estimated by the r.m.s. deviation

$\xi=\sqrt{\frac{1}{D}\sum_{d=1}^{D}({\cal E}_{d}-E_{d})^{2}}.$ (23)

In our case we get approximately the same r.m.s. deviations $\xi$ obtained
with 5 ($W=4$) or 6 ($W=5$) parameters $w_{i}$ in Eq. (18), which are,
however, significantly smaller than the r.m.s. deviations obtained with 4
parameters ($W=3$). Energies and widths obtained by locating the $S$-matrix
poles using Eq. (20) and the NCSM results from calculations with
$N_{\max}=16$, 18, 20 and parameterizations with $W=4$ and 5 together with the
respective $\xi$ values are presented in Table 1.

Table 1: Convergence of energy $E_{r}$ and width $\Gamma$ of the trineutron $3/2^{-}$ resonant state obtained with $NN$ interaction Daejeon16 with increasing $N_{\max}$. $\xi$ is the r.m.s. deviation defined by Eq. (23). $N_{\max}$ | 16 | 18 | 20 | 16 | 18 | 20
---|---|---|---|---|---|---
$W$ | 4 | 4 | 4 | 5 | 5 | 5
$E_{r}$, MeV | 0.560 | 0.508 | 0.483 | 0.607 | 0.537 | 0.481
$\Gamma$, MeV | 1.458 | 1.152 | 0.924 | 1.524 | 1.176 | 0.963
$\xi$, keV | 3.3 | 3.9 | 2.5 | 2.7 | 2.0 | 1.8

Fits of the $3\to 3$ phase shifts in the $3/2^{-}$ state with 6 parameters
$w_{i}$ in Eq. (18) ($W=5$) are presented by solid curves in Fig. 2. The $3\to
3$ phase shifts at the NCSM eigenenergies obtained by Eq. (3) and used for the
fitting are shown by symbols in Fig. 2. The trineutron resonance energy and
width obtained by locating the $S$-matrix pole based on the NCSM calculation
with $N_{\max}=20$ and fit with ${W=5}$ are adopted as the final result
presented in the Table 2 together with their uncertainties estimated as
deviations of results obtained with $N_{\max}=18$, 20 and $W=4$, 5 from the
final result.

Figure 2: Fits of $3\to 3$ phase shifts in the $3/2^{-}$ trineutron state obtained with $NN$ interaction Daejeon16 and ${W=5}$ in the $X(k)$ expansion (18). Table 2: Energies $E_{r}$ and widths $\Gamma$ of trineutron resonant states obtained with soft $NN$ interactions Daejeon16 Shirokov_PhysLettB_2016 , JISP16 Shirokov_PhysLettB_2007  and SRG-evolved Idaho N3LO Entem_PhysRevC_2003 . Uncertainties are presented in parentheses. All values are in MeV. | $3/2^{-}$ | $1/2^{-}$
---|---|---
Interaction | $E_{r}$ | $\Gamma$ | $E_{r}$ | $\Gamma$
Daejeon16 | 0.48(6) | 0.96(21) | 0.48(8) | 0.96(17)
JISP16 | 0.35(8) | 0.70(9) | 0.35(11) | 0.67(22)
N3LO, SRG, $\Lambda=2$ fm-1 | 0.34(8) | 0.70(19) | 0.35(9) | 0.68(16)

It is interesting that we obtain in the trineutron NCSM calculations the
$1/2^{-}$ state very close to the lowest $3/2^{-}$ state. We perform the same
analysis of the $1/2^{-}$ trineutron resonance. The $3\to 3$ $1/2^{-}$ phase
shifts are very close to those in the $3/2^{-}$ state and the obtained
$1/2^{-}$ resonance energy and width are presented in Table 2. It is seen that
the $3/2^{-}$ and $1/2^{-}$ resonance energies and widths are the same within
the uncertainty estimations and these resonances completely overlap.

We employ the same technique to search for resonances with other soft $NN$
interactions, in particular, with Idaho N3LO Entem_PhysRevC_2003  softened by
the similarity renormalization group (SRG) transformation Glazek_PhysRevD_1993
; Wegner_AnnPhys_1994  with the flow parameter $\Lambda=2$ fm-1 and JISP16
Shirokov_PhysLettB_2007 . The respective results are also presented in the
Table 2. Note, in these cases the $3/2^{-}$ and $1/2^{-}$ resonance also
degenerate and strongly overlap.

We also have analyzed the trineutron resonance with bare realistic $NN$
interactions derived in chiral effective field theory: Idaho N3LO and LENPIC
N4LO Semi-local Coordinate Space interaction Epelbaum_PhysRevLett_2015  with
regulator $R=0.9$ fm. In these cases, the $3\to 3$ phase shifts do not
demonstrate a resonant behavior. Due to the almost complete degeneracy of the
$3/2^{-}$ and $1/2^{-}$ states, these interactions also do not support the
$1/2^{-}$ resonance.

We obtained Shirokov_AIPConfProc_2018  a $4\to 4$ $S$-matrix pole at negative
imaginary momentum in the tetraneutron calculations with the Idaho N3LO
interaction which corresponds to a virtual state at the energy of
$E_{v}=-15.2$ keV. It is easy to prove, that the $A\to A$ $S$-matrix for an
odd number of fragments $A$ does not allow $S$-matrix poles at the negative
imaginary half-axis of momentum. Therefore, a virtual state is prohibited in
the trineutron treated as a democratic decaying system.

##  IV Summary and conclusions

We suggest an extension of the SS-HORSE–NCSM method to a democratic decay into
odd number of fragments. The first application of this method is the analysis
of the resonant trineutron state.

We conclude that the soft $NN$ interactions which we investigated predict two
low-lying nearly degenerate overlapping trineutron resonances with spin-
parities $3/2^{-}$ and $1/2^{-}$. On the other hand, these resonances are not
supported by bare $NN$ interactions of chiral effective field theory. We do
not include $NNN$ interaction in our calculations which has yet to be designed
for three-nucleon systems with isospin $T=3/2$.

We argue that the Daejeon16 $NN$ interaction is preferable for the trineutron
studies since it originates from the chiral effective field theory and is
fitted to stable light nuclei up to 16O by phase-equivalent off-shell
variations which effectively mimic effects of $NNN$ forces. The JISP16 $NN$
interaction, which was also fitted to light nuclei by off-shell variations,
leads to similar trineutron results as well as the SRG-evolved Idaho N3LO $NN$
interaction.

We predict two overlapping trineutron resonances with spin-parities $3/2^{-}$
and $1/2^{-}$ with nearly exactly the same energies $E_{r}$ and widths
$\Gamma$: $E_{r}\simeq 0.5$ MeV and $\Gamma\simeq 1$ MeV obtained in
calculations with the Daejeon16 and ${E_{r}\simeq 0.35}$ MeV and $\Gamma\simeq
0.7$ MeV obtained in calculations with the JISP16 and SRG-evolved N3LO $NN$
interactions.

Our results are in line with the conclusions of Refs. Gandolfi ;
Li_PhysRevC_2019  predicting the trineutron resonance at lower energy than the
tetraneutron resonance Shirokov_PhysRevLett_2016 ; Shirokov_AIPConfProc_2018 .
However, in our calculations we obtain the trineutron resonance at lower
energies as compared to $E_{r}=1.29$ MeV in Ref. Li_PhysRevC_2019  and
$E_{r}=1.11(21)$ MeV in Ref. Gandolfi . Meanwhile, its width obtained with the
Daejeon16 $NN$ interaction is in agreement with $\Gamma=0.91$ MeV proposed in
Ref. Li_PhysRevC_2019 . We note that Refs. Gandolfi ; Li_PhysRevC_2019  do not
specify the spin-parity of the predicted trineutron resonance.

## Acknowledgments

The work of I. A. Mazur is supported by the Institute for Basic Science
(IBS-R031-D1). The work of E. K. Efimenko and A. I. Mazur is supported by the
Ministry of Science and Higher Education of Russian Federation (project No.
FEME-2024-0005). The work of I. J. Shin is supported by the National Research
Foundation of Korea (NRF) funded by Ministry of Science and ICT
(2013M7A1A1075764). A. M. Shirokov is thankful to the Chinese Academy of
Sciences President’s International Fellowship Initiative Program (Grant No.
2023VMA0013) which supported his visits to the Institute of Modern Physics,
Chinese Academy of Sciences in Lanzhou, China, where a part of this work was
performed and acknowledges the hospitality of Chinese colleagues during these
visits. The work of J. P. Vary is supported by the U.S. Department of Energy,
Division of Nuclear Physics, Grant No. DE-SC0023692.

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