# Quasi-Normal Modes of Loop Quantum Black Holes Formed from Gravitational
Collapse

Chao Zhanga,b<EMAIL_ADDRESS>Anzhong Wangc<EMAIL_ADDRESS>Corresponding author aBasic Research Center for Energy Interdisciplinary,
College of Science, China University of Petroleum-Beijing, Beijing 102249,
China
bBeijing Key Laboratory of Optical Detection Technology for Oil and Gas, China
University of Petroleum-Beijing, Beijing, 102249, China
c GCAP-CASPER, Physics Department, Baylor University, Waco, Texas, 76798-7316,
USA

###### Abstract

In this paper, we study the quasi-normal modes (QNMs) of a scalar field in the
background of a large class of quantum black holes that can be formed from
gravitational collapse of a dust fluid in the framework of effective loop
quantum gravity. The loop quantum black holes (LQBHs) are characterized by
three free parameters, one of which is the mass parameter, while the other two
are purely due to quantum geometric effects. Among these two quantum
parameters, one is completely fixed by black hole thermodynamics and its
effects are negligible for macroscopic black holes, while the second parameter
is completely free (in principle). In the studies of the QNMs of such LQBHs,
we pay particular attention to the difference of the QNMs between LQBHs and
classical ones, so that they can be observed for the current and forthcoming
gravitational wave observations, whereby place the LQBH theory directly under
the test of observations.

##  I Introduction

Black holes (BHs) are one of the most mysterious phenomena in the universe.
The existence of BHs provides us a perfect way to test gravitational effects
under extremely strong gravitational fields, such as the formation of gigantic
jets and disruption of neighboring stars. On the other hand, from the
theoretical point of view, BHs are also excellent labs to test theories of
gravity that are different from general relativity (GR) (see, e.g., test_GR1 ;
test_GR2 ; test_GR3 ; Xiang2019 ; Chao2020 ; Chao2020b ; Chao2023a ; Zack2020
). It is true that BHs were initially only discovered as mathematical
solutions of GR Schwarzschild1916 . Nonetheless, observations have already
confirmed their existence, including those from gravitational wave (GW)
observations Ref1 , which marks the beginning of a new era — the GW astronomy.

Following the first observation of GW, more than 90 GW events, which are all
from coalescences of compact bodies, black holes and/or neutron stars, have
been identified by the LIGO/Virgo/KAGRA (LVK) scientific collaborations GWs ;
GWs19a ; GWs19b ; GWsO3b . In the future, more advanced ground- and space-
based GW detectors will be constructed Moore2015 ; Gong:2021gvw , such as
Cosmic Explorer CE , Einstein Telescope ET , LISA LISA , TianQin Liu2020 ;
Shi2019 , Taiji Taiji2 , and DECIGO DECIGO . These detectors will enable us to
probe signals with a much wider frequency band and long distances. This has
triggered the interest in the observation of quasi-normal mode (QNM) of black
holes Berti2009 ; Berti18 , extreme mass ratio inspirals (EMRIs) Gong2023 ;
Tu2023 , the effect of dark matter on GWs Chao2022 ; Shen2024 , to name only a
few of them.

GWs emitted during the ringdown stage of a coalescence, where a massive black
hole is just formed, can be considered as the linear combination of the QNMs
of the just formed BH 111The BH just formed from the merge stage of two
compact objects is highly nonlinear, and the linear perturbation theory cannot
be applied to the initial stage of such just formed BH. Thus, how to connect
the ringdown waveform determined by QNMs is still an open question Cheung2023
; Mitman2023 . In this paper, we shall not consider such a matching, and
simply consider only the times during which the GWs can be well expressed as
linear combinations of QNMs., and are in general studied with the perturbation
theory Regge1957 ; Teukolsky1973 ; Will1985 ; Leaver1985 ; Iyer1987 ;
Chandra92 . QNMs of such perturbations, including scalar, vector
(electromagnetic), and tensor (gravitational), have been extensively studied
in GR as well as in various modified theories of gravity. From the theoretical
point of view, QNMs are eigenmodes of dissipative systems and usually contain
two parts, the real part and the imaginary part. Its real part gives the
frequency of vibration while its imaginary part provides the damping time. The
information contained in QNMs provide the keys in revealing whether BHs are
ubiquitous in our universe, and more important whether GR is the correct
theory to describe gravity even in the strong field regime. As a matter of
fact, according to the no-hair theorem of GR, BHs are described uniquely by
only three parameters, mass, charge and angular momentum. Thus, in GR QNMs are
functions of only these three parameters. Hence, observing such modes can
directly test such relations. For more details, we refer readers to Berti2009
; Berti18 ; Kokkotas1999 ; Konoplya2011 .

On the other hand, from the observational point of view, the sensitivities of
the current detectors limit the detection of QNMs mainly to the dominant
modes, and it is still an ongoing debate on whether other modes can be
observed by LIGO, Virgo and KAGRA Carullo2018 ; Isi2019 ; Bustillo2021 ;
Finch2022 ; Ma2023 ; Isi2023 ; Carullo2023 ; Capano2023 . However, with the
technological advances, it is strongly believed that the upcoming third-
generation detectors, either the ground-based or the space-based ones, will be
able to detect various sub-leading modes, see for example, Shi2024  and
references therein.

In this paper, we shall focus on the QNMs of a large class of quantum BHs in
the context of effective loop quantum gravity (LQG) Lewandowski2023 ; Luca2024
. It should be noted that QNMs of loop quantum black holes (LQBHs) 222Loop
quantum black holes have been studied extensively in the past decades or so.
For details, see Gambini2023 ; Ashtekar2023 ; OG2024 ; GZO2024  and references
therein. have been already studied extensively Tu2023 ; Yan2023 ; Yan2022 ;
Liu2021 ; Ramin2021 ; Arbey2021 ; Livine2024 ; Xia2024 ; Fu2023 ; Shao2024 ;
Yang2023 ; Okounkova2020 ; LiuC2020 . However, in most of these models, the
observational effects are negligible for macroscopic BHs. The LQBHs to be
studied in this paper are distinguishable from the previous ones in the
senses: (a) They can be directly formed from the gravitational collapse of a
spherically symmetric dust cloud; and (b) the solutions usually contain three
free parameters, one is the mass of the BH, and the other two characterize the
quantum geometric effects. Among the two quantum parameters, one is completely
fixed by the black hole thermodynamics, whose observational effects are always
negligible for macroscopic BHs. On the other hand, the second quantum
parameter is free Luca2024 . Our major goal here is to study its effects to
QNMs, in order to obtain observational constraints on this second quantum
parameter.

The rest of the paper is organized as follows: Sec.II describes briefly the
LQBH solutions we are going to study in this paper and the equations for a
scalar field moving in such backgrounds. The latter leads to the master
equation for the study of QNMs. Knowing this, in Sec.III we calculate the QNMs
for both the time- and frequency-domains since the former reflects a
comprehensive effect of various different modes (to be introduced below),
while the latter provides a more quantitative evidence for the confinement to
the coupling-parameter phase space with future GW missions. Some of our
concluding remarks and future outlooks are given in Sec.IV.

Before proceeding further, we would like to note that linear perturbations of
spherically symmetric LQBH spacetimes have not been worked out yet in the
framework of LQG, although some important steps were already taken Mena2024 .
In particular, the QNMs of the scalar perturbations studied in this paper
cannot be compared directly with the QNMs of spin-2 gravitons of black holes
studied in GR, which are directly related to the gravitational waveforms
emitted in the ringdown phase Berti2009 ; Berti18 ; Kokkotas1999 ;
Konoplya2011 . So, in this paper, what we are comparing with are the QNMs of a
scalar field moving in the backgrounds of LQBHs and the QNMs of the same
scalar field moving in the Schwarzschild BH background of GR, as we believe
that the deviations of the scalar perturbations between these two theories are
of the same order as that of the deviations of the spin-2 gravitons between
the two same theories.

In addition, in this paper we are adopting the unit system so that
$c=G=\hbar=1$, where $c$ denotes the speed of light, $G$ is the gravitational
constant, and $\hbar$ the Planck constant divided by $2\pi$. All the Greek
letter in indices run from 0 to 3. The other usages of indices will be
described at suitable places.

##  II LQBH Background and scalar perturbation

In this section, we shall first introduce briefly the background of a
spherically symmetric loop quantum black hole formed from the gravitational
collapse of a homogeneous and isotropic dust fluid in the framework of
effective theory of loop quantum gravityLewandowski2023 ; Luca2024 , and then
consider the linear perturbations of a scalar field in such backgrounds. In
doing so, we ignore the back-reactions of the scalar field.

###  II.1 Loop Quantum Black Holes formed from Gravitational Collapse

The loop quantum black hole (LQBH) solution we are studying is given by
Luca2024

$\displaystyle ds_{+}^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d{\Omega}^{2},~{}~{}$
(2.1)

where

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{\alpha}{r^{2}}\left(\frac{M}{r}+\frac{B}{2}\right)^{2},$
(2.2)

and $d\Omega^{2}=d\theta^{2}+\sin^{2}{\theta}d\varphi^{2}$. Here $B$ is a
dimensionless coupling parameter. Considering the above solution as describing
the external spacetime of a collapsing ball, while inside the ball the
spacetime is described by the FLRW universe

$\displaystyle ds_{-}^{2}$ $\displaystyle=$ $\displaystyle-
dT^{2}+a^{2}\left(dR^{2}+\chi_{k}^{2}(R)d{\Omega}^{2}\right),~{}~{}$ (2.3)

it was found that the junction conditions lead to Luca2024

$\displaystyle B=-k\chi_{k}^{2}(R_{0})=\begin{cases}-\sin^{2}(R_{0}),&k=1,\cr
0,&k=0,\cr\sinh^{2}(R_{0}),&k=-1,\cr\end{cases}$ (2.4)

where $R_{0}$ denotes the junction surface. The factor $\alpha$, which has the
dimension of $r^{2}$, is given by Chiou2008 ; Luca2024

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
16\sqrt{3}\pi\gamma^{3}\ell_{\text{pl}}^{2},$ (2.5)

where $\ell_{\text{pl}}$ denotes the Planck length, and $\gamma$ is known as
the Barbero-Immirzi parameter whose value is set to $\gamma\approx 0.2375$
using black hole thermodynamics in LQG Meissner2004 .

To calculate QNMs of the above BHs, let us first introduce the dimensionless
coordinates $\tilde{t}$ and $\tilde{r}$

$\displaystyle t=r_{s}\tilde{t},\quad r=r_{s}\tilde{r},$ (2.6)

where $r_{s}\equiv 2M$. Then, in terms of $\tilde{t}$ and $\tilde{r}$, the
metric reduces to

$\displaystyle ds_{+}^{2}$ $\displaystyle=$ $\displaystyle
r_{s}^{2}\left(-f(\tilde{r})d\tilde{t}^{2}+\frac{d^{2}\tilde{r}}{f(\tilde{r})}+\tilde{r}^{2}d^{2}\Omega\right)$
(2.7) $\displaystyle\equiv$ $\displaystyle r_{s}^{2}d\tilde{s}^{2},$

where

$\displaystyle f(\tilde{r})$ $\displaystyle=$ $\displaystyle
1-\frac{1}{\tilde{r}}+\frac{\tilde{\alpha}}{\tilde{r}^{2}}\left(\frac{1}{\tilde{r}}+B\right)^{2},$
$\displaystyle\tilde{\alpha}$ $\displaystyle\equiv$
$\displaystyle\frac{\alpha}{4r_{s}^{2}}=\sqrt{3}\pi\gamma^{3}\left(\frac{\ell_{\text{pl}}}{M}\right)^{2}.$
(2.8)

Note that for solar-mass BHs, we have $M\simeq 10^{38}\;\ell_{\text{pl}}$, so
that (the dimensionless) $\tilde{\alpha}\simeq{\cal{O}}\left(10^{-77}\right)$.
That is, for macroscopic BHs, the quantum gravitational effects from the
$\alpha$ terms are negligible, unless the combination of $\tilde{\alpha}B$ is
large, a condition that we will assume in order to compare the resultant QNMs
of LQBHs with GW observations. Considering Eq.(2.4), we find that this is
possible only when $k=-1$. As a result, the values of $B$ to be considered
will be a non-negative number.

On the other hand, since $ds_{+}^{2}$ and $d\tilde{s}^{2}$ are related
conformally with a constant conformal factor [cf., (2.7)], the physics of the
spacetimes of $ds_{+}^{2}$ can be obtained directly from that of the
spacetimes of $d\tilde{s}^{2}$. In particular, by following the same manner of
definition, we have $R=\tilde{R}/r_{s}^{2}$ and

$\displaystyle\tilde{\omega}$ $\displaystyle\equiv$ $\displaystyle
r_{s}\omega,$ (2.9)

where $\omega$ and the dimensionless $\tilde{\omega}$ are the QNMs of
$ds_{+}^{2}$ and $d\tilde{s}^{2}$, respectively, and $R$ and $\tilde{R}$ are
their corresponding curvatures333One should avoid confusing the curvature $R$
in here with the $R$ coordinate appearing in (2.3).. Therefore, in the rest of
the paper we shall work with the spacetime described by $d\tilde{s}^{2}$, i.e.

$\displaystyle d\tilde{s}^{2}$ $\displaystyle=$
$\displaystyle-f(\tilde{r})d\tilde{t}^{2}+\frac{d^{2}\tilde{r}}{f(\tilde{r})}+\tilde{r}^{2}d^{2}\Omega,$
(2.10)

when we calculate the QNMs of the quantum BHs. Then, using Eq.(2.9) we can
easily read off $\omega$ for a given BH with mass $M$.

Moreover, by looking at Eq.(2.2), we immediately notice that a positive radius
of the metric horizon $r_{MH}$, which is defined through $f(r=r_{MH})=0$, is
absent at the $M\to 0$ limit. Therefore, by requiring $r_{MH}>0$, we have to
look at the cases where $M$ is large enough. Keeping this in mind, in this
paper we shall consider the cases of $\lambda\equiv
M/M_{\odot}\in(3,100)$444In order to study their observational results, in
this paper we mainly consider BHs with masses in the order of the solar mass
or larger. In particular, the current GW observations set the lower limit to
about $3M_{\odot}$ (See., e.g., LVK2024 ), which makes it reasonable for us to
consider the range $M/M_{\odot}\in(3,100)$. As will be seen later, the
difference on QNMs between a LQBH and a Schwarzschild BH sensitively depends
on the ratio $M/M_{\odot}$ when that is relatively small. After it has reached
to about 100 (and even before that), the difference is observationallly
negligible. That is why we set the upper limit to $100$ in here. in this
current paper, where $M_{\odot}$ denotes the solar mass.

Figure 1: The behavior of the dimensionless quantity $r_{MH}/(2M)$ as a
function of $B$ for different values of $M$.

Knowing this, we are on the position of determining an approximate upper limit
for $B$ by requiring $r_{MH}>0$ for $\lambda\in(3,100)$. By plotting out the
behavior of $r_{MH}$ as a function of $B$ for the range of $M$ under
consideration in Fig. 1, we can notice an approximate upper limit of $B$. A
more careful study to the equation $f(r=r_{MH})=0$ shows that, for a positive
$r_{MH}$ to exist for any $\lambda\in(3,100)$, the upper limit of $B$ is about
$B_{max}\approx 7.176\times 10^{38}$. It is worth mentioning here that, to
avoid confusions arising by using different unit systems, in Fig. 1 we are
focusing on the dimensionless quantity $r_{MH}/(2M)$ (instead of the quantity
$r_{MH}$ itself). Therefore, the curves of Fig. 1 will not be changed when
considering different unit systems.

In addition, by observing Fig.1, we also notice that the value of
$r_{MH}/(2M)$ will soon deviate from that of GR when $B$ reaches a critical
point (For the $M/M_{\odot}=3$ case, that critical point is located around
$B=10^{38}$, which is very close to the upper limit of $B$). Before getting
close to such a critical point, it is very hard to distinguish the value of
$r_{MH}/(2M)$ from its GR counterpart. This implies that one may find the
observable difference between a LQBH and a Schwarzschild one only for
sufficiently large $B$’s (which should be quite close to $B_{max}$). Yet, such
a critical point is getting larger and larger as $M$ increases. Therefore,
given an overall $B_{max}$ (obtained from the $M/M_{\odot}=3$ case), we expect
to barely see deviations between a LQBH and the corresponding Schwarzschild
one for very large $M$’s. We shall come back to these points later.

###  II.2 Perturbations of a Scalar Field

Since in the rest of this paper we shall work only with the background
described by the metric (2.10) and the dimensionless coordinates $\tilde{t}$
as well as $\tilde{r}$, without causing any confusion, we shall drop the
tildes from the dimensionless coordinates $(\tilde{t},\tilde{r})$ and the
corresponding quantities. Then, following Ramin2021  we consider the scalar
perturbations that obey the Klein-Gordon equation

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)$
$\displaystyle=$ $\displaystyle 0,$ (2.11)

where $\Phi=\Phi(t,r,\theta,\varphi)$ denotes the scalar field. By decomposing
it through the spherical harmonics $Y_{lm}(\theta,\varphi)$ as
$\Phi=Y_{lm}\Psi(t,r)/r$, a master equation of $\Psi$ is obtained as

$\displaystyle\frac{\partial^{2}}{\partial
x^{2}}\Psi+\left(-\frac{\partial^{2}}{\partial t^{2}}+V_{eff}(r)\right)\Psi$
$\displaystyle=$ $\displaystyle 0,$ (2.12)

where the effective potential $V_{\text{eff}}$ is given by

$\displaystyle V_{\text{eff}}$ $\displaystyle\equiv$ $\displaystyle
f\frac{L}{r^{2}}+\frac{1}{2r}\frac{df^{2}}{dr},$ (2.13)

with $L\equiv l(l+1)$, and $x$ the tortoise coordinate defined through

$\displaystyle\frac{dr}{dx}$ $\displaystyle=$ $\displaystyle f(r).$ (2.14)

Notice that, since $r$ is dimensionless, $x$ is actually also dimensionless.
Also notice that, when the background has spherically symmetry, the
perturbations are independent of the chociue of the angular index $m$
Chandra92 . Thus, without loss of the generality, we shall set $m=0$. For a
given function $f(r)$ and choosing a suitable value of $l$, we can solve
(2.12) for $\Psi$ (which contains the information of QNMs) for a BH.

Figure 2: The behavior of $V_{\text{eff}}$ as a function of $r$ (which is in
its dimensionless form). Upper panel: For the $\lambda=M/M_{\odot}=3$ and
$l=2$ case by varying the parameter $B$. Lower panel: For the $B=7.176\times
10^{38}$ and $l=2$ case by varying the parameter $\lambda$. The GR case is
also exhibited in here as a comparison.

In fact, the behavior of $V_{\text{eff}}$ is directly related to the results
of QNMs. Therefore, to take a glance to the influence of $\\{B,\lambda\\}$ on
the consequence, we first check their influence on $V_{\text{eff}}(r)$. The
behavior of $V_{\text{eff}}$ is plotted in Fig.2 by setting $l=2$. In the
upper panel we fix $\lambda=M/M_{\odot}=3$ and plot out different curves with
a varying $B$ factor. In contrast, in the lower panel we fix
$B=B_{max}=7.176\times 10^{38}$ and plot out different curves with a varying
$\lambda\in(3,100)$. Notice that, the $V_{\text{eff}}$ from GR is added to
there as a comparison.

The upper panel of Fig.2 implies that an observable deviation between a LQBH
and a Schwarzschild one requires an extremely large $B$, which is consistent
with what we observed from Fig.1. On the other hand, as expected (again by
observing Fig.1), the lower panel reflects that the deviation on
$V_{\text{eff}}$ from its GR counterpart will soon fade away once $\lambda$
become relatively large (e.g., when it is larger than $20$), even with
$B=B_{max}$. From the qualitative point of view, Fig.2 provides us with a good
reference on choosing the parameters from the phase space of $\\{B,\lambda\\}$
when calculating for QNMs in the next section. Although we are not going to
follow Fig.2’s choices precisely, that gives some hints to the choice of the
parameters.

##  III Quasi-Normal Modes of LQBHs

First of all, we try to solve Eq.(2.12) in the time-domain with the finite
difference method (FDM) Chao2023a ; Chao2022 ; RichardB . One advantage of
such a method is that, for any chosen $l$, the final result from it can
reflect comprehensively the influence of a bunch of different modes
(represented by $\\{m,n\\}$, although $m$ can be always set to zero in the
spherical case without loss of the generality, as mentioned earlier. It must
not be confused this m with the mass parameter M of the LQBHs.). To apply the
FDM, we first introduce two new variables $\mu\equiv t-x$ and $\nu\equiv t+x$
[so that $t=(\nu+\mu)/2$ and $x=(\nu-\mu)/2$]555 An important feature of the
FDM is that in using it, we need the exact form of $x(r)$, in addition to its
derivative with respect to $r$. Therefore, according to the definition (2.14),
we have to assign $x$ an integral constant to absolutely fix it. In fact, such
a constant could be chosen arbitrarily and it’s independent of our results for
calculating QNMs. Thus, we made a simple choice by letting $x(r=2)=2$, which
is consistent with the general choice in GR (See e.g., Leaver1985 )..
Therefore, on a $(N+1)\times(N+1)$ lattice (where $N$ is a positive integer
that will be chosen properly according to our usage), we perform the
calculation of $\Psi(\mu,\nu)$ by using the recursion formula

$\displaystyle\Psi(\mu+\delta h,\nu+\delta h)$ $\displaystyle\cong$
$\displaystyle\Psi(\mu,\nu+\delta h)+\Psi(\mu+\delta
h,\nu)-\Psi(\mu,\nu)-\frac{1}{8}\delta h^{2}V_{\rm
eff}\left(r\right)\left[\Psi(\mu,\nu+\delta h)+\Psi(\mu+\delta h,\nu)\right],$
(3.1)

where $\delta h$ is the step size (which will be chosen properly in reality by
measuring our tolerance of accuracy). The boundary conditions are given by
$\Psi(\mu,\nu=0)=0\;(\mu\neq 0)$ and $\Psi(\mu=0,\nu)=\exp[-(\nu-1)^{2}/2]$.
Thus, after $N^{2}$ iterations, we find all the $\Psi(n\delta h,n\delta h)$’s
for $n\in[0,N]\cap\mathbb{Z}$. From that we can calculate $\Psi(t,x=0)$ by
using the relation $\Psi(t,x=0)=\Psi(\mu,\mu)$.

Figure 3: The behavior of $\ln|\Psi(t,x=0)|$ as a function of $t$ (which is in
its dimensionless form). Upper panel: For the $\lambda=M/M_{\odot}=3$ and
$l=2$ case with the variation of the parameter $B$. Lower panel: For the
$B=7.176\times 10^{38}$ and $l=2$ case with the variation of the parameter
$\lambda$. The GR case is also exhibited in here as a comparison.

Then, the results are shown in the upper panel of Fig. 3. For the case with
$\lambda=M/M_{\odot}=3$ and $l=2$, the behavior of $\ln|\Psi(t,x=0)|$ (which
encodes the information of various QNM frequencies) is plotted out as a
function of $t$ for different values of $B$ (with these values chosen by
referring Fig.2). It should be noted that the variable $t$ here is
dimensionless, which was previously written as ${\tilde{t}}=t/r_{s}$. The GR
case is also added as a comparison. Just like what we have observed from
Fig.2, the result is not very sensitive to the value of $B$, especially when
it is not large enough, so that the deviation between the curves of GR
(represented by the red dashed line) and that of the case $B=3\times 10^{38}$
(represented by the green solid line) is still moderate. In contrast, once $B$
reached certain critical values, the difference becomes obvious. That is why
we observe a relatively large deviation between the curves of the $B=4.5\times
10^{38}$ case (represented by the blue dashed line) and that of the $B=7\times
10^{38}$ case (represented by the purple solid line). On the other hand, by
observing the upper panel of Fig. 3 we also notice that the frequency of
vibration is getting bigger as the factor $B$ increases.

Similarly, we show the corresponding results when changing $\lambda$ in the
lower panel of Fig. 3 (with its values chosen by referring Fig.2). By setting
$B=7.176\times 10^{38}$ and $l=2$, the behavior of $\ln|\Psi(t,x=0)|$ is
plotted out as a function of $t$. We also add the GR case in order to make a
comparison. As we expect from Fig. 1, we can barely see the deviations between
the curve of a LQBH with large masses (e.g., the one represented by the green
solid line) and that of GR (represented by the red dashed line), even if the
parameter $B$ is chosen to be $B_{max}$. In contrast, given a relatively small
$\lambda$ (e.g., the one represented by the blue dashed line), we can observe
a significant deviation. On the other hand, by observing the lower panel of
Fig. 3 we also notice that the frequency of vibrations is getting bigger with
the decrease of the factor $\lambda$.

Table 1: Various modes of $\omega$ (in its dimensionless form, for which we dropped the “tilde” as mentioned earlier) for LQBHs with different masses that characterized by $\lambda$. The GR limit of $\omega$ for each mode is provided and the percent difference with the GR limit is shown. The factor $B$ is fixed to be $B=7.176\times 10^{38}$.  |  | GR | $\lambda=20$ |  |  | $\lambda=10$ |  |  | $\lambda=5$ |  |
---|---|---|---|---|---|---|---|---|---|---|---
$l$ | $n$ | $\omega_{GR}$ | $\omega$ | $\Re{(\delta\omega)}$ | $\Im{(\delta\omega)}$ | $\omega$ | $\Re{(\delta\omega)}$ | $\Im{(\delta\omega)}$ | $\omega$ | $\Re{(\delta\omega)}$ | $\Im{(\delta\omega)}$
2 | 0 | $0.96728-0.19353i$ | $0.97096-0.19377i$ | $0.4\%$ | $0.1\%$ | $0.98235-0.19445i$ | $1.6\%$ | $0.5\%$ | $1.03475-0.19666i$ | $7.0\%$ | $1.6\%$
2 | 1 | $0.92769-0.59125i$ | $0.93154-0.59191i$ | $0.4\%$ | $0.1\%$ | $0.94349-0.59380i$ | $1.7\%$ | $0.4\%$ | $0.99872-0.59958i$ | $7.6\%$ | $1.4\%$
2 | 2 | $0.86077-1.01740i$ | $0.86491-1.01833i$ | $0.5\%$ | $0.1\%$ | $0.87779-1.02096i$ | $2.0\%$ | $0.4\%$ | $0.93776-1.02794i$ | $8.9\%$ | $1.0\%$
3 | 0 | $1.35073-0.19300i$ | $1.35585-0.19324i$ | $0.4\%$ | $0.1\%$ | $1.37173-0.19393i$ | $1.6\%$ | $0.5\%$ | $1.44474-0.19618i$ | $7.0\%$ | $1.6\%$
3 | 1 | $1.32134-0.58458i$ | $1.32659-0.58526i$ | $0.4\%$ | $0.1\%$ | $1.34287-0.58725i$ | $1.6\%$ | $0.4\%$ | $1.41799-0.59356i$ | $7.3\%$ | $1.5\%$
3 | 2 | $1.26718-0.99202i$ | $1.27267-0.99308i$ | $0.4\%$ | $0.1\%$ | $1.28971-0.99612i$ | $1.8\%$ | $0.4\%$ | $1.36870-1.00519i$ | $8.0\%$ | $1.3\%$
6 | 0 | $2.50377-0.19261i$ | $2.51324-0.19285i$ | $0.4\%$ | $0.1\%$ | $2.54262-0.19354i$ | $1.6\%$ | $0.5\%$ | $2.67774-0.19582i$ | $6.9\%$ | $1.7\%$
6 | 1 | $2.48750-0.57947i$ | $2.49705-0.58018i$ | $0.4\%$ | $0.1\%$ | $2.52665-0.58224i$ | $1.6\%$ | $0.5\%$ | $2.66293-0.58895i$ | $7.0\%$ | $1.6\%$
6 | 2 | $2.45569-0.97120i$ | $2.46537-0.97236i$ | $0.4\%$ | $0.1\%$ | $2.49542-0.97572i$ | $1.6\%$ | $0.5\%$ | $2.63397-0.98646i$ | $7.3\%$ | $1.6\%$

On the other hand, Eq.(2.12) can be written in its frequency-domain as
Chao2022

$\displaystyle\frac{\partial^{2}}{\partial
x^{2}}\Psi+\left(\omega^{2}+V_{\text{eff}}(r)\right)\Psi$ $\displaystyle=$
$\displaystyle 0,$ (3.2)

where $\omega$, representing the dimensionless quantity $\tilde{\omega}$
introduced in Eq.(2.9), is the QNM frequency to be calculated. To solve for
$\omega$ from (3.2), we introduce the WKB method Chao2022 . The QNM frequency
can be calculated up to the 6th order by using

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\sqrt{-i\left[\left(n+\frac{1}{2}\right)+\sum_{k=2}^{6}\Lambda_{k}\right]\sqrt{-2V_{0}^{\prime\prime}}+V_{0}},~{}~{}~{}~{}$
(3.3)

where

$\displaystyle V_{0}\equiv\left.V_{\rm eff}\right|_{r=r_{\text{max}}},\quad
V_{0}^{\prime\prime}\equiv\left.\frac{dV_{\rm
eff}}{dr^{2}}\right|_{r=r_{\text{max}}},$ (3.4)

with $V_{\rm eff}(r=r_{\text{max}})$ gives the maximum of $V_{\rm eff}$ for
$r\in(r_{MH}/r_{s},~{}\infty)$. The expressions of $\Lambda_{k}$’s could be
found in Will1985 ; Iyer1987 ; Konoplya2003 . Note that $n=0,1,2,...$.

For the mode $\\{l,n\\}$ (recall that we we have set $m=0$ so that it is not
mentioned), some of the results of $\omega$ are shown in Table 1. In this
table we fix the factor $B$ as $B=7.176\times 10^{38}$. In addition, the GR
limit, which is denoted as $\omega_{GR}$, is also added as a comparison.
Notice that, to compare with the results given in Ramin2021 , the modes with
$l=2,6$ are included in Table 1. To see clearly the difference between a LQBH
and a classical Schwarzschild BH with the same mass, we have introduced the
percent difference defined by

$\delta\omega\equiv\frac{\omega-\omega_{GR}}{\omega_{GR}}\times 100\%,$ (3.5)

with $\Re{(\delta\omega)}$ and $\Im{(\delta\omega)}$ denote the real and
imaginary parts of $\delta\omega$, respectively.

From Table 1 we find that the value of $\delta\omega$ is sensitive to the
choice of $\lambda$. Just like what we have observed in the lower panel of
Fig.3, a huge $\lambda$ often results in negligible deviations on QNMs between
a LQBH and a classical Schwarzschild one. More importantly, given a
sufficiently large $B$ and a reasonable $\lambda$, the percent difference
$\delta\omega$ can well locate in the observational window once the LIGO-
Virgo-KAGRA detector network reaches its designed sensitivity Carullo2018 ,
whereby some constraints on these LQBHs can be found from GW observations.

In addition, it is worth mentioning here that the corresponding value of
$r_{MH}$ for each $\omega$ appearing in Table 1 effectively reflects its
deviation from that of GR (as implied by Fig.1). Let us take the $l=2$ case as
an example. Setting $\lambda=5$ leads to $r_{MH}\approx 0.90003$ while setting
$\lambda=20$ leads to $r_{MH}\approx 0.99434$. Clearly, the latter is quite
close to that of GR, for which we have $r_{MH}=1$ [cf., Eq.II.1], in comparing
to the former. It thus makes sense to us that the former tends to bring us an
potentially observable deviation from that of GR as mentioned earlier.
Actually, if we enlarge our scope, we shall find that setting $\lambda=3$ will
further putting $r_{MH}$ into approximately $0.50800$, which is obviously far
away from the GR case. Although we did not show it explicitly in here, one can
easily find out that the corresponding $\omega$’s for $\lambda=3$ are indeed
tremendously different from their counterparts of GR.

##  IV Conclusions

In this paper we have investigated QNMs of perturbations of a scalar field on
the backgrounds of a large class of LQBHs, which can be formed from
gravitational collapse of realistic matterLewandowski2023 ; Luca2024 . These
solutions are characterized by three parameters, $M$, $\alpha$ and $B$ [cf.
Eqs. (2.4) and (2.5), where $M$ is the mass parameter, and $\alpha$ and $B$
are the two quantum parameters, characterizing the quantum geometric effects
of the LQBHs. To see the effects of these two quantum parameters, it is
suggestive to introduce the dimensionless coordinates
$({\tilde{t}},{\tilde{r}})$, as defined by Eq.(2.6), for which the spacetime
can be cast in the form of Eq.(2.10) with

$\displaystyle f(\tilde{r})$ $\displaystyle=$ $\displaystyle
1-\frac{1}{\tilde{r}}+\frac{\tilde{\alpha}}{\tilde{r}^{2}}\left(\frac{1}{\tilde{r}}+B\right)^{2},$
(4.1)

where

$\displaystyle\tilde{\alpha}$ $\displaystyle\equiv$
$\displaystyle\frac{\alpha}{4r_{s}^{2}}=\sqrt{3}\pi\gamma^{3}\left(\frac{\ell_{\text{pl}}}{M}\right)^{2}\simeq{\cal{O}}\left(10^{-77}\right),$
(4.2)

for a solar massive LQBH, $M\simeq 10^{38}l_{pl}$. Therefore, when $B\simeq
0$, the corresponding LQBHs have negligible effects to the observations of
such BHs. From Eq. (2.4) it can be seen that this is indeed the case for
$k=0,+1$. On the other hand, for $k=-1$, we have $B=\sinh^{2}(R_{0})$, where
$R_{0}$ is the geometric radius of the collapsing dust ball. Therefore,
depending on $R_{0}$, the parameter $B$ can be arbitrarily large, so that the
leading term ${\tilde{\alpha}}B^{2}/{\tilde{r}}^{2}$ appearing in
$f({\tilde{r}})$ can have observational effects for macroscopic LQBHs.

Motivated by the above considerations, we have calculated the QNMs of
perturbations of a scalar field in the backgrounds of the above LQBHs. In
particular, we have found that they sensitively depend on the ratio
$\lambda=M/M_{\odot}$, when $B$ reaches a critical value, say, $B_{c}$, which
is of the order of $B_{c}\simeq M_{\odot}/l_{pl}\simeq 10^{38}$, as shown in
Fig.2. Therefore, with $B\simeq B_{c}$ one can find significant deviations
between a LQBH and a Schwarzschild one, for some values of $M$ (or
$\lambda\equiv M/M_{\odot}$) as one can see from Figs.1 and 2.

To be comparable with observations, in this paper we have mainly focused on
the cases $\lambda\in(3,100)$. For such an interval of the LQBH masses, there
exists an upper bound $B_{max}$ of $B$, only under which LQBHs exist. We have
found that this maximal value is $B_{max}\approx 7.176\times 10^{38}$. On the
other hand, we have also found that, working with the dimensionless
coordinates $({\tilde{t}},{\tilde{r}})$ can significantly simplify the
calculations of the QNMs of the LQBHs. With this in mind, we have first
studied the perturbations of the time-domain master equation (2.12). Using the
FDM method [cf., (3.1)], we have solved the equation and plotted the behavior
of $\ln|\Psi(t,x=0)|$ as a function of $\tilde{t}$ in Fig.3 (in which simply
written as $t$) for the dominant mode $l=2$. In the upper panel of this
figure, we have fixed $\lambda=3$, while varying the parameter $B$. On the
other hand, in the lower panel, we have fixed $B=7.176\times 10^{38}$ but now
varying $\lambda$. Then, we have shown that only in the neighborhood of
$(B,\lambda)\approx(B_{c},3)$ in the phase space, can the deviations of the
QNMs between LQBHs and the corresponding classical BH become significant.

To find out more details, we have further calculated explicitly the values of
$\omega$ by solving the master equation in the frequency-domain [cf., (3.2)].
The corresponding results are exhibited in Table 1. By examining the percent
difference between the classical and LQG BHs, defined by (3.5), we have shown
that the percent difference as large as $2\%$ (or larger) can be obtained for
various choices of $\\{B,\lambda\\}$. These differences could be well within
the detectability of the forthcoming third-generation detectors of GWs
Bustillo2021 ; Shi2024 , whereby one can either rule out some regions of the
phase space of the LQBHs or confirm them.

Finally, we note that our current work can be extended to several different
directions. For instance, one may study the QNMs of other LQBHs considered in
OG2024 . On the other hand, one may study the QNMs of metric perturbations of
LQBHs recently developed in Mena2024 .

## Acknowledgments

We thank Profs. J. Lewandowski and P. Singh for valuable discussions on loop
quantum black holes. We also thank Dr. Kai Lin for valuable discussions on the
calculations of QNMs. C.Z. is supported in part by the National Natural
Science Foundation of China under Grant No. 12205254, and the Science
Foundation of China University of Petroleum, Beijing under Grant No.
2462024BJRC005. A.W. is partially supported by the US National Science
Foundation (NSF) through the grant: PHY-2308845.

## References

  * (1) E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani, et. al., Testing General Relativity with Present and Future Astrophysical Observations, Class. Quantum Grav. 32, 243001 (2015).
  * (2) L. Barack, V. Cardoso, S. Nissanke, T. P. Sotiriou, et. al., Black Holes, Gravitational Waves and Fundamental Physics: A Roadmap, Class. Quantum Grav. 36, 143001 (2019).
  * (3) The LIGO Scientific Collaboration, the Virgo Collaboration, the KAGRA Collaboration,, Tests of General Relativity with GWTC-3, arXiv: 2112.06861 [astro-ph].
  * (4) X. Zhao, C. Zhang, K. Lin, T. Liu, R. Niu, B. Wang, S.-J. Zhang, X. Zhang, W. Zhao, T. Zhu, A. Wang, Gravitational waveforms and radiation powers of the triple system PSR J0337+1715 in modified theories of gravity, Phys. Rev. D100, 083012 (2019).
  * (5) C. Zhang, X. Zhao, A. Wang, B. Wang, K. Yagi, N. Yunes, W. Zhao and T. Zhu, Gravitational waves from the quasicircular inspiral of compact binaries in Einstein-aether theory, Phys. Rev. D101, 044002 (2020).
  * (6) C. Zhang, X. Zhao, K. Lin, S.-J. Zhang, W. Zhao and A.-Z. Wang, Spherically symmetric static black holes in Einstein-aether theory, Phys. Rev. D102, 064043 (2020).
  * (7) C. Zhang, A. Wang and T. Zhu, Odd-parity perturbations of the wormhole-like geometries and quasi-normal modes in Einstein-Æther theory, JCAP 05 (2023) 059.
  * (8) Zack Carson and Kent Yagi, Probing Einstein-dilaton Gauss-Bonnet gravity with the inspiral and ringdown of gravitational waves, Phys. Rev. D101, 104030 (2020).
  * (9) K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916 189 (1916).
  * (10) B.P. Abbott, et al., [LIGO/Virgo Scientific Collaborations], Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016).
  * (11) B.P. Abbott, et al., [LIGO/Virgo Collaborations], GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X9, 031040 (2019).
  * (12) B.P. Abbott, et al., [LIGO/Virgo Collaborations], Open data from the first and second observing runs of Advanced LIGO and Advanced Virgo, SoftwareX, Volume 13, 100658 (2021).
  * (13) B.P. Abbott, et al., [LIGO/Virgo Collaborations], GW190425: Observation of a Compact Binary Coalescence with Total Mass $\sim 3.4M_{\bigodot}$, ApJL 892 L3 (2020); https://www.ligo.caltech.edu.
  * (14) B.P. Abbott, et al., [LIGO/Virgo/KAGRA Collaborations], GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run, arXiv:2111.03606v1 [gr-qc].
  * (15) C. J. Moore, R. H. Cole and C. P. L. Berry, Gravitational-wave sensitivity curves, Class. Quantum. Grav. 32, 015014 (2015).
  * (16) Y. Gong, J. Luo and B. Wang, “Concepts and status of Chinese space gravitational wave detection projects,” Nature Astron. 5, no.9, 881-889 (2021).
  * (17) https://cosmicexplorer.org.
  * (18) ET Steering Committee Editorial Team, ET design report update 2020, ET-0007A- 20 (2020); https://www.et-gw.eu/.
  * (19) https://www.lisamission.org.
  * (20) S. Liu, Y. Hu, et al., Science with the TianQin observatory: Preliminary results on stellar-mass binary black holes, Phys. Rev. D101, 103027 (2020).
  * (21) C.-F. Shi, et al., Science with the TianQin observatory: Preliminary results on testing the no-hair theorem with ringdown signals, Phys. Rev. D100, 044036 (2019).
  * (22) W.-H. Ruan, Z.-K. Guo, R.-G. Cai, Y.-Z. Zhang, Taiji Program: Gravitational-Wave Sources, Int. J. Mod. Phys. A 35, No. 17, 2050075 (2020).
  * (23) S. Kawamura, et al., Current status of space gravitational wave antenna DECIGO and B-DECIGO, arXiv:2006.13545.
  * (24) E. Berti, V. Cardoso and A. O. Starinets, Quasinormal modes of black holes and black branes, Class. Quantum. Grav. 26, 163001 (2009).
  * (25) E. Berti, K. Yagi, H. Yang, N. Yunes, Extreme gravity tests with gravitational waves from compact binary coalescences: (II) ringdown, Gen. Relativ. Grav. 50, 49 (2018).
  * (26) C. Zhang, Y. Gong, D. Liang and B. Wang, Gravitational waves from eccentric extreme mass-ratio inspirals as probes of scalar fields, JCAP 06 (2023) 054.
  * (27) Ze-Yi Tu, Tao Zhu and Anzhong Wang, Periodic orbits and their gravitational wave radiations in a polymer black hole in loop quantum gravity, Phys. Rev. D108, 024035 (2023).
  * (28) C. Zhang, T. Zhu, X. Fang and A. Wang, Imprints of dark matter on gravitational ringing of supermassive black holes, Physics of the Dark Universe 37, 101078 (2022).
  * (29) Z. Shen, A. Wang, Y.-G. Gong, S.-Y. Yin, Analytical models of supermassive black holes in galaxies surrounded by dark matter halos, Phys. Lett. B855 (2024) 138797.
  * (30) M.H.Y. Cheung, et al., Nonlinear effects in black hole ringdown, Phys. Rev. Lett. 130, No. 8 (2023) 081401.
  * (31) K. Mitman, et al., Nonlinearities in Black Hole Ringdowns, Phys. Rev. Lett. 130, 081402 (2023).
  * (32) T. Regge and J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063.
  * (33) S. A. Teukolsky, Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations, Astrophys. J. 185, 635 (1973).
  * (34) B. F. Schutz and C. M. Will, BLACK HOLE NORMAL MODES: A SEMIANALYTIC APPROACH, Astrophys. J. 291, L33-L36 (1985).
  * (35) E. W. Leaver, An analytic representation for the quasi-normal modes of Kerr black holes, Proc. R. Soc. Lond. A. 402, 285-298 (1985).
  * (36) S. Iyer and C. M. Will, Black-hole normal modes: A WKB approach. I. Foundations and application of a higher-order WKB analysis of potential-barrier scattering, Phys. Rev. D35, 12 (1987); S. Iyer, Black-hole normal modes: A WKB approach. II. Schwarzschild black holes, Phys. Rev. D35, 3632 (1987).
  * (37) S. Chandrasekhar, the mathematical theory of black holes, Oxford classic texts in the physical sciences (Oxford Press, Oxford, 1992).
  * (38) K.D. Kokkotas, B.G. Schmidt, Quasi-normal modes of stars and black holes. Living Reviews in Relativity 2 (1999) 1.
  * (39) R. Konoplya, A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory. Reviews of Modern Physics 83 (2011) 793.
  * (40) Gregorio Carullo, et al., Empirical tests of the black hole no-hair conjecture using gravitational-wave observations, Phys. Rev. D98, 104020 (2018).
  * (41) M. Isi, M. Giesler, W. M. Farr, M. A. Scheel and S. A. Teukolsky, Testing the no-hair theorem with GW150914, Phys. Rev. Lett. 123, 111102 (2019).
  * (42) J.C. Bustillo, P.D. Lasky, and E. Thrane, Black-hole spectroscopy, the no-hair theorem, and GW150914: Kerr versus Occam, Phys. Rev. D103, 024041 (2021).
  * (43) E. Finch and C. J. Moore, Searching for a ringdown overtone in GW150914, Phys. Rev. D106, 043005 (2022).
  * (44) S. Ma, L. Sun, and Y. Chen, Using rational filters to uncover the first ringdown overtone in GW150914, Phys. Rev. D107, 084010 (2023).
  * (45) M. Isi and W. M. Farr, Comment on “Analysis of Ringdown Overtones in GW150914”, Phys. Rev. Lett. 131, 169001 (2023).
  * (46) G. Carullo, R. Cotesta, E. Berti, and V. Cardoso, Reply to Comment on “Analysis of Ringdown Overtones in GW150914”, Phys. Rev. Lett. 131, 169002 (2023).
  * (47) C. D. Capano, M. Cabero, J. Westerweck, J. Abedi, S. Kastha, A. H. Nitz, Y.-F. Wang, A. B. Nielsen, and B. Krishnan, Multimode Quasinormal Spectrum from a Perturbed Black Hole, Phys. Rev. Lett. 131, 221402 (2023)
  * (48) C.-F. Shi, Q.-F. Zhang, and J.-W. Mei, On the detectability and resolvability of quasi-normal modes with space-based gravitational wave detectors, arXiv:2407.13110 [gr-qc].
  * (49) J. Lewandowski, Y. Ma, J. Yang and C. Zhang, Quantum Oppenheimer-Snyder and Swiss Cheese Models, Phys. Rev. Lett. 130 (2023) 101501.
  * (50) Luca Cafaro and Jerzy Lewandowski, Status of Birkhoff’s theorem in polymerized semiclassical regime of Loop Quantum Gravity, arXiv:2403.01910.
  * (51) R. Gambini, J. Olmedo, J. Pullin, Quantum geometry and black holes, in C. Bambi, L. Modesto, I. Shapiro, (eds) Handbook of Quantum Gravity (Springer, Singapore (2023)).
  * (52) A. Ashtekar, J. Olmedo, P. Singh, Regular black holes from Loop Quantum Gravity, in C. Bambi, L. Modesto, I. Shapiro, (eds) Handbook of Quantum Gravity (Springer, Singapore (2023)).
  * (53) G. Ongole, P. Singh, A. Wang, Revisiting quantum black holes from effective loop quantum gravity, Phys. Rev. D109, 026015 (2024); W.-C. Gan, G. Ongole, E. Alesci, Y. An, F.-W. Shu, A. Wang, Understanding quantum black holes from quantum reduced loop gravity, Phys. Rev. D106, 126013 (2022); W.-C. Gan, N.O. Santos, F.-W. Shu, A. Wang, Properties of the spherically symmetric polymer black holes, Phys. Rev. D102, 124030 (2020).
  * (54) W.-C. Gan, X.-M. Kuang, Z.-H. Yang, Y.-G. Gong, A. Wang, B. Wang, Nonexistence of quantum black and white hole horizons in an improved dynamic approach, Science China: Phys., Mech. $\&$ Astron. 67, No. 8, 280411 (2024); H.-C. Zhang, W.-C. Gan, Y.-G. Gong, A. Wang, On the improved dynamics approach in loop quantum black holes, Commun. Theor. Phys. 76, 035401 (2024); G. Ongole, H.-C. Zhang, T. Zhu, A. Wang and B. Wang, Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes, Universe 2022, 8, 543 (2022).
  * (55) Jian-Ming Yan, et al., Observational tests of a quantum extension of Schwarzschild spacetime in loop quantum gravity with stars in the Galactic Center, Phys. Rev. D 107, 084043 (2023).
  * (56) J.-M. Yan, Q. Wu, C. Liu, T. Zhu, A. Wang, Constraints on self-dual black hole in loop quantum gravity with S0-2 star in the galactic center, JCAP, 09 (2022) 008.
  * (57) Y.-C. Liu, J.-X. Feng, F.-W. Shu, A. Wang, Extended geometry of Gambini-Olmedo-Pullin polymer black hole and its quasinormal spectrum, Phys. Rev. D104 (2021) 106001\.
  * (58) Ramin G. Daghigh, Michael D. Green and Gabor Kunstatter, Scalar perturbations and stability of a loop quantum corrected Kruskal black hole, Phys. Rev. D103, 084031 (2021).
  * (59) A. Arbey, J. Aufffnger, M. Geiller, E.R. Livine, F. Sartini, Hawking radiation by spherically-symmetric static black holes for all spins: I – Teukolsky equations and potentials, Phys. Rev. D103, 104010 (2021).
  * (60) E.R. Livine, C. Montagnon, N. Oshita, H. Roussille, Scalar Quasi-Normal Modes of a Loop Quantum Black Hole, [arXiv:2405.12671 [gr-qc]].
  * (61) Z.-W.Xia, H. Yang, Y.-G. Miao, Scalar ffelds around a rotating loop quantum gravity black hole: Waveform, quasi-normal modes and superradiance, Class. Quantum Grav. 41 165010 (2024).
  * (62) G. Fu, D. Zhang, P. Liu, .-M. Kuang, J.-P. Wu, Peculiar properties in quasi-normal spectra from loop quantum gravity effect, arXiv:2301.08421 [gr-ac].
  * (63) C.-Y. Shao, C. Zhang, W. Zhang, and C.-G. Shao, Scalar ffelds around a loop quantum gravity black hole in de Sitter spacetime: Quasinormal modes, late-time tails and strong cosmic censorship, Phys. Rev. D109, 064012 (2024).
  * (64) J. Yang, C. Zhang, Y. Ma, Shadow and stability of quantum-corrected black holes, Eur. Phys. J. C83 (2023) 619\.
  * (65) M. Okounkova, L.C. Stein, J. Moxon, M.A. Scheel, S.A.Teukolsky, Numerical relativity simulation of GW150914 beyond general relativity, Phys. Rev. D101, 104016 (2020).
  * (66) C. Liu, T. Zhu, Q. Wu, et al., Shadow and Quasinormal Modes of a Rotating Loop Quantum Black Hole, Phys. Rev. D101, 084001 (2020); Phys.Rev. D103 (2021) 089902 (erratum).
  * (67) G.A. Mena Marugán, A. Mínguez-Sánchez, Axial perturbations in Kantowski-Sachs spacetimes and hybrid quantum cosmology, Phys.Rev.D109, No. 10 (2024), 106009.
  * (68) Dah-Wei Chiou, Phenomenological loop quantum geometry of the Schwarzschild black hole, Phys. Rev. D78, 064040 (2008).
  * (69) Krzysztof A Meissner, Black-hole entropy in loop quantum gravity, Class. Quantum. Grav. 21, 5245–5251 (2004).
  * (70) The LVK Scientific Collaboration, Observation of Gravitational Waves from the Coalescence of a 2.5-4.5 $M_{\odot}$ Compact Object and a Neutron Star, arXiv:2404.04248 (2024).
  * (71) Richard Haberman, APPLIED PARTIAL DIFFEREN- TIAL EQUATIONS: with Fourier Series and Boundary Value Problems (5th ed.) (Pearson Education, Inc., One Lake Street, New Jersey 07458, USA, 2013).
  * (72) R. A. Konoplya, Quasinormal behavior of the D-dimensional Schwarzschild black hole and the higher order WKB approach, Phys. Rev. D68, 024018 (2003).


# Cluster structure of 3$\alpha$+p states in ${}^{13}\mathrm{N}$

J. Bishop  Cyclotron Institute, Texas A&M University, College Station, TX
77843, USA  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, United Kingdom  G.V. Rogachev  Cyclotron
Institute, Texas A&M University, College Station, TX 77843, USA  Department of
Physics & Astronomy, Texas A&M University, College Station, TX 77843, USA
Nuclear Solutions Institute, Texas A&M University, College Station, TX 77843,
USA  S. Ahn  Center for Exotic Nuclear Studies, Institute for Basic Science,
34126 Daejeon, Republic of Korea  M. Barbui  Cyclotron Institute, Texas A&M
University, College Station, TX 77843, USA  S.M. Cha  Center for Exotic
Nuclear Studies, Institute for Basic Science, 34126 Daejeon, Republic of Korea
E. Harris  Cyclotron Institute, Texas A&M University, College Station, TX
77843, USA  Department of Physics & Astronomy, Texas A&M University, College
Station, TX 77843, USA  C. Hunt  Cyclotron Institute, Texas A&M University,
College Station, TX 77843, USA  Department of Physics & Astronomy, Texas A&M
University, College Station, TX 77843, USA  C.H. Kim  Department of Physics,
Sungkyunkwan University, Suwon 16419, Republic of Korea  D. Kim  Center for
Exotic Nuclear Studies, Institute for Basic Science, 34126 Daejeon, Republic
of Korea  S.H. Kim  Department of Physics, Sungkyunkwan University, Suwon
16419, Republic of Korea  E. Koshchiy  Cyclotron Institute, Texas A&M
University, College Station, TX 77843, USA  Z. Luo  Cyclotron Institute, Texas
A&M University, College Station, TX 77843, USA  Department of Physics &
Astronomy, Texas A&M University, College Station, TX 77843, USA  C. Park
Center for Exotic Nuclear Studies, Institute for Basic Science, 34126 Daejeon,
Republic of Korea  C.E. Parker  Cyclotron Institute, Texas A&M University,
College Station, TX 77843, USA  E.C. Pollacco  IRFU, CEA, Université Paris-
Saclay, Gif-Sur-Yvette, France  B.T. Roeder  Cyclotron Institute, Texas A&M
University, College Station, TX 77843, USA  M. Roosa  Cyclotron Institute,
Texas A&M University, College Station, TX 77843, USA  Department of Physics &
Astronomy, Texas A&M University, College Station, TX 77843, USA  A.
Saastamoinen  Cyclotron Institute, Texas A&M University, College Station, TX
77843, USA  D.P. Scriven  Cyclotron Institute, Texas A&M University, College
Station, TX 77843, USA  Department of Physics & Astronomy, Texas A&M
University, College Station, TX 77843, USA<EMAIL_ADDRESS>
###### Abstract

Background

Cluster states in 13N are extremely difficult to measure due to the
unavailability of 9B+$\alpha$ elastic scattering data.

Purpose

Using $\beta$-delayed charged-particle spectroscopy of 13O, clustered states
in 13N can be populated and measured in the 3$\alpha$+p decay channel.

Method

One-at-a-time implantation/decay of 13O was performed with the Texas Active
Target Time Projection Chamber (TexAT TPC). 149 $\beta 3\alpha p$ decay events
were observed and the excitation function in 13N reconstructed.

Results

Four previously unknown $\alpha$-decaying excited states were observed in 13N
at an excitation energy of 11.3 MeV, 12.4 MeV, 13.1 MeV and 13.7 MeV decaying
via the 3$\alpha$+p channel.

Conclusion

These states are seen to have a
[${}^{9}\mathrm{B}(\mathrm{g.s})\bigotimes\alpha$/
$p+^{12}\mathrm{C}(0_{2}^{+})$],
[${}^{9}\mathrm{B}(\frac{1}{2}^{+})\bigotimes\alpha$],
[${}^{9}\mathrm{B}(\frac{5}{2}^{+})\bigotimes\alpha$] and
[${}^{9}\mathrm{B}(\frac{5}{2}^{+})\bigotimes\alpha$] structure respectively.
A previously-seen state at 11.8 MeV was also determined to have a
[$p+^{12}\mathrm{C}(\mathrm{g.s.})$/ $p+^{12}\mathrm{C}(0_{2}^{+})$]
structure. The overall magnitude of the clustering is not able to be extracted
however due to the lack of a total width measurement. Clustered states in 13N
(with unknown magnitude) seem to persist from the addition of a proton to the
highly $\alpha$-clustered 12C. Evidence of the $\frac{1}{2}^{+}$ state in 9B
was also seen to be populated by decays from 13N⋆.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

The most well-known instance of $\alpha$ clustering in light nuclei is perhaps
that of the Hoyle state in 12C Freer and Fynbo (2014). When additional protons
or neutrons are added to this system, the propensity for clustering is of
interest to study to understand the phenomenon of clustering. While clustering
in 13C and 14C has been examined through resonant scattering - the unbound
nature of 9B means that these data for 13N are not accessible. Instead, one
must use different mechanisms to populate these highly-excited exotic states.
Combined with the experimental difficulties of observing the 3$\alpha$+p decay
of these states (as the characteristic clustered decay mode), a high-
sensitivity experimental approach is required.

To probe these $\alpha$-clustered states in 13N, $\beta$-delayed charged-
particle spectroscopy was used to populate states in 13N via 13O and decays to
a final state of 3$\alpha$+p were then measured. To achieve this, the Texas
Active Target Time Projection Chamber (TexAT TPC) was used to perform one-at-
a-time implantation and decay which has been demonstrated previously to have a
very high sensitivity to rare decays due to the absence of background Bishop
_et al._ (2020a, b, 2021). This paper provides more details on this approach
and deeper insight into the states observed in a previous paper detailing the
first observation of the $\beta 3\alpha p$ decay channel Bishop  _et al._
(2023).

##  II Experimental setup

The experimental setup utilized for this experiment follows that successfully
applied to studying the rare decay modes of near-threshold $\alpha$-clustered
states in 12C Bishop  _et al._ (2020a, b, 2021) via the $\beta$-delayed
3$\alpha$-particle decay of ${}^{12}\mathrm{N}$ via 12C⋆. Instead, the
$\beta$-delayed 3$\alpha p$-particle decay of ${}^{13}\mathrm{O}$ via 13N⋆ is
studied here.

The K500 Cyclotron at Texas A&M University was used to produce a beam of 14N
which was incident on a 3He gas cell to produce a secondary beam of 13O via
the ${}^{14}\mathrm{N}(^{3}\mathrm{He},xn)^{13}$O reaction. The 13O of
interest was then selected using the MARS (Momentum Achromat Recoil Separator)
Tribble  _et al._ (1989) with a typical intensity of 5 pps and an energy of
15.1 MeV/u. The beam was then degraded to 2 MeV/u using an aluminum foil in
order to stop the beam inside of the sensitive area of the TexAT TPC Koshchiy
_et al._ (2020) in 50 Torr of CO2 gas. The one-at-a-time $\beta$-delayed
charged-particle spectroscopy technique requires the implantation of the
$\beta$-decaying nucleus 13O into the active area of the TexAT TPC (with
$t_{\frac{1}{2}}$ = 8.58 ms). The implanted ion then subsequently decays with
the TPC volume being insensitive to the $\beta^{+}$ in the gas (and the
subsequent $\beta^{+}$ from the decay of 13N for some events). When states in
13N are populated above the particle decay threshold (1.944 MeV for the
p-threshold and 9.496 MeV for the $\alpha$-threshold), the daughter nucleus
can undergo particle decay and the recoil products are measured inside the
TPC.

As the ion is incident into the TPC active volume, the implantation beam track
triggers the GET electronics setup Pollacco  _et al._ (2018) which sends a
30-ms-long busy signal to the K500 phase shifter which blocks the transmission
of the primary 14N through the cyclotron and therefore prevents any subsequent
implantations. During the 30 ms busy signal, the data acquisition then awaits
an additional event corresponding to the decay of 13N⋆ through the proton or
$\alpha$ decay channels. For the majority of events, this second trigger is
not generated as the 13O decays to states in 13N that are particle bound.
After either 30 ms has elapsed or a decay event has registered, the data
acquisition is ready for a new implantation event and the K500 phase shifter
signal is disabled. This setup is known as the ‘2p-mode’ in the GET system
Mutant module and allows for the beam implant and decay events to be
correlated. The implantation from the beam is referred to as the L1A trigger
and requires 10 channels in TexAT to be above threshold. The decay event is
known as the L1B trigger and requires only a multiplicity of 1 to allow for
very low-energy recoils to trigger the data acquisition. The time between the
L1A and L1B event (known as the d2p time) is also recorded in the GET system
and corresponds to the decay time of the $\beta$-delaying particle. Partial
events where an L1A trigger was not followed by an L1B trigger were also
recorded for normalization and beam characterization. An overview of this
experimental setup is shown in Fig. 1.

Figure 1: An overview of the experimental setup showing how the K500 Cyclotron
phase shifter inhibits the ${}^{14}\mathrm{N}$ primary beam following an
implantation event from the ${}^{13}\mathrm{O}$ secondary beam (L1A).
Following this, for a fraction of events a subsequent decay (within 30 ms) of
the ${}^{13}\mathrm{N}\rightarrow 3\alpha+p$ (or more-likely, a single proton)
provides a second L1B (decay) trigger and the decay products from can be
reconstructed inside the TPC. A silicon detector at zero degrees was used for
tuning the radioactive beam and providing a veto signal for beam events that
do not stop inside of the TexAT active region.

##  III Methodology

To select events of interest, the full L1A (implant) + L1B (decay) events were
selected where the time between the two was between 1 and 30 ms (with small
times omitted to remove double trigger events due to sudden beam-induced
noise). In addition to the secondary 13O of interest, there were also some
other beam contaminants. Therefore, to ensure the implanted ion corresponded
to ${}^{13}\mathrm{O}$, the energy deposited by the beam implant event in the
Micromegas “Jr” (MM Jr) beam tracker Holmes  _et al._ (2020) at the entrance
to the TexAT chamber was recorded. The beam contaminants were
${}^{7}\mathrm{Be}$ and ${}^{10}\mathrm{C}$, dominated by ${}^{7}\mathrm{Be}$
at $\approx$ 28% of the beam intensity. The energy spectrum from the
Micromegas Jr is shown in Fig. 2.

Figure 2: Energy deposition (dE) within the Micromegas Jr detector to
differentiate the beam species. The contaminants to the ${}^{13}\mathrm{O}$
were dominated by ${}^{7}\mathrm{Be}$ and they total $\approx$ 28% of the
total beam intensity of 5 pps.

Figure 3: Decay time between the implant of ${}^{13}\mathrm{O}$ and decay
overlaid with a background free exponential fit yielding a value of
$t_{1/2}$=8.55$\pm$0.09 (stat.) ms compared to the adopted value of
8.58$\pm$0.05 ms Wang  _et al._ (2012). $\tilde{\chi}^{2}$ = 1.26 from log-
likelihood minimization.

Following a selection of ${}^{13}\mathrm{O}$ implants, the stopping position
of the beam was evaluated event-by-event and events where the beam stopped
inside the active area of the Micromegas were taken, events where a signal was
observed in the zero-degree silicon detector were automatically vetoed as
signifying the beam did not stop in the gas. Because of the proximity of some
events to the edge of the active region, only events which were more than 31.5
mm from the edge of the Micromegas were taken for further analysis. Due to the
large energy degradation required to stop inside TexAT, the spread in the
${}^{13}\mathrm{O}$ stopping position was 67.5 mm.

To ensure the implant and decay events are correlated, the 2D distance (in the
plane of the Micromegas) between the stopping position of the implanted beam
and the decay vertex (identified by the highest energy deposition inside the
active volume) is then evaluated. Following previous results Bishop  _et al._
(2020a), the separation for correlated events is given by:

$\displaystyle f(r)=re^{-\frac{(r-r_{0})^{2}}{2\sigma^{2}}},$ (1)

with $\sigma$ = 4.9 mm for these data which agrees well with a predicted
diffusion value of $\approx 4$ mm from the ideal gas law. A selection of
events with a displacement of $<5$ mm were taken as correlated. To determine
the purity of this selection, the time between the implantation and decay for
these events was accumulated (the d2p time). A background-free exponential fit
yielded a half-life value of 8.55$\pm$0.09 (stat.) ms compared to the adopted
value of 8.58$\pm$0.05 ms ($\tilde{\chi}^{2}$ = 1.26) which is shown in Fig.
3. The excellent agreement without the need for any background demonstrates
the purity of the selection of ${}^{13}\mathrm{O}$ in the system.

Tracks from these events were fit with a single track segment using a
randomly-sampled $\chi$-squared minimization algorithm (more detail on this
fitting technique is covered in Sec. V). For those tracks for which the
reduced $\chi$-squared was good, these events were identified as single proton
events and are discussed in Sec. IV. For those which gave a poor reduced
$\chi$-squared, these events were fit with four track segments as candidate
3$\alpha$+p events using randomly-sampled $\chi$-squared minimization and were
visually inspected to evaluate the quality of fit and also identify any events
which were unable to be fit with the algorithm (given the complexity of the
fit required). These 3$\alpha$+p events are discussed in Sec. V.

##  IV Single proton events

Due to the low gas pressure used, the majority of proton event tracks escape
the TPC active volume. In order to evaluate the state populated in
${}^{13}\mathrm{N}^{\star}$, the recoiling ${}^{12}\mathrm{C}$ was therefore
used instead which has 1/12th of the proton energy and a much higher dE/dx
therefore stops safely inside the active volume.

Figure 4: Selection of (${}^{13}\mathrm{N},p_{0,1})$ events. Top: Experimental
data of 12C recoil range in TPC versus energy. The expected result from TRIM
Ziegler  _et al._ (2010) is overlaid as a dotted red line. Bottom: Proton
energy spectrum from these data obtained from the ${}^{12}\mathrm{C}$ recoil
(red histogram). The expected yield using previously-obtained branching ratios
Knudsen  _et al._ (2005) is overlaid as a black solid line after being
convoluted with a Gaussian profile to best replicate the data.

In order to calculate the ${}^{12}\mathrm{C}$ recoil energy, the decay event
is fit with a single line segment defined by two points: the end point of the
proton track and the end point of the ${}^{12}\mathrm{C}$ track. The decay
vertex then dissects this line segment and the energy in the point cloud that
is on the recoil side of the fit was also added to the ${}^{12}\mathrm{C}$
recoil energy. Fig. 4a shows the relationship between the recoil energy and
range from the track overlaid with the expected TRIM result which allowed for
an additional cut on the data. The events that lie outside of this cut
constitute events where the separation between the proton and
${}^{12}\mathrm{C}$ recoil could not be accurately attained. The number of
events lost through this cut is $<4\%$ which has little effect on the
branching ratios but removes all background at higher excitation energies.
Given the typical energy of the recoil is from 0.12 to 1.1 MeV (corresponding
to track lengths of 6.7 to 30.5 mm), the resolution of the energy spectrum
using this technique is poor and the proton energies and relative intensities
from previous studies Knudsen  _et al._ (2005) are used to demonstrate the
compatibility of our result with those previously observed. The excitation
function obtained is shown in Fig. 4b with the expected yield overlaid from
previous studies Knudsen  _et al._ (2005) convoluted with a Gaussian response.
Any deviation can be attributed to the dynamics of the gas avalanche that is
better characterized as a Breit-Wigner convoluted with a Gamma distribution
(representing the gas gain dynamics), convoluted with a Gaussian (representing
the resolution associated with noise and difficulties associated with fitting
the 12C recoil track). This also confirms the implanted number of
${}^{13}\mathrm{O}$ that make it through our cuts is 1.90$\times 10^{5}$ from
1.86$\times 10^{4}$ proton decay events. An independent branching ratio
measurement from the number of implants was not reliable during this
experiment due to a sizeable noise contribution which adversely affected the
L1A/L1B ratio (corresponding to the branching ratio to particle-unbound
states) but was necessary to ensure 100% trigger efficiency on genuine proton
events.

##  V 3$\alpha$+proton events

A total of 149 3$\alpha$+p events were identified. Due to the size of the TPC
and limitations on reconstruction in parts of the TexAT TPC, only 102 out of
149 of these events could be fully-reconstructed. These events that are lost
are almost exclusively from $\alpha$-decay as this produces a high-energy
$\alpha$-particle that may escape. The efficiency for the $\alpha_{0}$ decay
starts to deviate from 100% at $E_{x}$ = 10 MeV, slowly drops to around 60% at
$E_{x}$ = 14 MeV. The efficiency for $\alpha_{1}$ and $\alpha_{3}$ are less
affected and only decrease to 70% at $E_{x}$ = 14 MeV. For events that proton
decay to the Hoyle state, the majority of the energy is taken by the proton
which is not required for reconstruction. Corrections to the yields obtained
during this work are made to account for this effect in Table 1.

In order to accurately fit the four-track events, a highly-robust fitting
technique is required. In order to achieve this, the ansatz for the reaction
vertex is identified by the stopping position of the beam and then the point
cloud for the decay events is fit with 15 parameters: the decay vertex and the
endpoint of each of the four tracks (with each being a 3D vector). Due to the
highly complex nature of the fit with many free parameters and a ‘noisy’
fitting space, a modified version of RANSAC Fischler and Bolles (1987) was
used that has been successfully employed in several other TexAT experiments
whereby the four parameters are randomly selected from the set of points in
the pointcloud (as per RANSAC) and then a goodness of fit is evaluated by the
sum of the distance-squared for all points to the nearest line defined by the
vertex to each of the three endpoints. It is better characterized as RANdomly-
Sampled Chi-Squared Minimization, referred to here as RANSChiSM. To reduce the
influence of outliers, where the distance to all three lines exceeded 10 mm,
the distance was saturated to be 10 mm. This functional form is given by:

$\displaystyle\tilde{\chi}^{2}=\frac{\Sigma_{i}^{N_{\texttt{points}}}}{N_{\texttt{points}}}\left(\texttt{min}_{j=1,2,3,4}\left[\frac{|(\vec{P_{i}}-\vec{V_{j}})\times\vec{L_{j}}|}{|\vec{L_{j}}|},10\right]^{2}\right),$
(2)

where $\vec{P_{i}}$ is the ith point in the pointcloud, $\vec{V_{j}}$ is the
vector for the endpoint of the $j^{th}$ track, and $\vec{L_{j}}$ is the vector
from the decay vertex to the endpoint of one of the three tracks. The vector
product here merely calculates the shortest 3D distance of the point
$\vec{p_{i}}$ to the line $\vec{L_{j}}$ To ensure the point that is chosen is
at the end of the track, if the distance between the point and the vertex
exceeds the distance between the vertex and the end of the arm, then the
distance between the point and the end of the arm is taken instead, weighted
by a factor of 10. This was seen to be very effective at ensuring the proper
selection of the end of the tracks which is necessary for the conversion
between range in the gas and the energy. To reduce the number of random
samples of endpoints and the decay vertex, one endpoint was defined as that
which had the largest distance from the decay vertex and a weighting scheme
was used where two of the remaining three endpoints were randomly sampled with
a weight given by the distance of the point in the pointcloud to the randomly-
chosen decay vertex squared. Similarly, only points within 5 mm of the known
beam stopping point were allowed to be selected for the decay vertex point. A
schematic of this fitting is shown in Fig. 5. For the $\beta 3\alpha p$
events, 20,000 iterations were sufficient to converge on the best fit. Each
decay was examined manually and those that were not perfectly fitted were
reanalyzed and, if needed, the parameters for certain tracks were constrained
manually and the fit reevaluated. For some extremely short tracks, this was
necessary due to the presence of many ‘fake’ minima in the chi-squared phase
space.

Figure 5: Schematic showing the basis of the RANSChiSM fit by selection of
five points to parameterize the four-track fit. Any points in the pointcloud
more than 10 mm from the nearest track line are counted as if they are 10 mm
away to reduce the influence of outliers.

Figure 6: Relative energy spectrum for pairs of $\alpha$-particles (taking the
smallest relative energy) showing the ${}^{8}\mathrm{Be}$(g.s) energy of 92
keV is well-reproduced in our data.

Figure 7: For events that do not decay via the Hoyle state, the relative
energy spectrum is shown here which is generated by selecting the two
$\alpha$-particles that produce the ${}^{8}\mathrm{Be}$(g.s) and then
reconstructing the ${}^{9}\mathrm{B}$ relative energy with the proton.
Overlaid in dashed red are simulated data for the ground state contribution
and in solid red are the $\frac{1}{2}^{+}$ and $\frac{5}{2}^{+}$ states from
single channel R-Matrix calculations convoluted with a Gaussian with $\sigma$
= 0.23 MeV. The $\frac{1}{2}^{+}$ parameters are those obtained by Wheldon
Wheldon  _et al._ (2015) which show excellent agreement.

Figure 8: Invariant mass spectrum for ${}^{12}\mathrm{C}$ from
3$\alpha$-particles. A peak at 7.65 MeV is seen, well reproducing the Hoyle
state energy and a broad peak is seen at higher excitation energies which
correspond to events that decay via ${}^{9}\mathrm{B}+\alpha$.

In order to identify the parent state in ${}^{13}\mathrm{N}^{\star}$, the
lowest energy deposition arm was identified as the proton track and the
momentum of the 3 $\alpha$-particles was determined by the length and
direction of the track in the gas. As the proton almost always escapes the TPC
sensitive volume, the proton momentum is reconstructed from momentum
conservation. The decay energy is then the sum of the three $\alpha$-particles
and proton energy. From here, the ${}^{8}\mathrm{Be}$ (Fig. 6),
${}^{9}\mathrm{B}$ (Fig. 7) relative energies and ${}^{12}\mathrm{C}$ (Fig. 8)
excitation energies were determined from the invariant mass. This allowed for
a selection of events which proceeded to decay via
p+${}^{12}\mathrm{C}(0_{2}^{+})$ [$p_{2}$], $\alpha$+${}^{9}\mathrm{B}$(g.s)
[$\alpha_{0}$], $\alpha$+${}^{9}\mathrm{B}(\frac{1}{2}^{+})$ [$\alpha_{1}$]
and $\alpha$+${}^{9}\mathrm{B}(\frac{5}{2}^{+})$ [$\alpha_{3}$]. An example
$p_{2}$ event is shown in Fig. 9 and an example $\alpha_{0}$ event is shown in
Fig. 10. It is remarkable to see in Fig. 7 evidence of strength in
${}^{9}\mathrm{B}$ between 1 and 2.4 MeV that cannot be explained without the
long-sought after $\frac{1}{2}^{+}$ state in ${}^{9}\mathrm{B}$ that is the
mirror of the well-studied ${}^{9}\mathrm{Be}$ $\frac{1}{2}^{+}$. The ground
state contribution (from Monte Carlo simulations) and the higher-lying states
from a single-channel Breit-Wigner convoluted with a Gaussian ($\sigma$=0.23
MeV) were fitted to the spectrum. Attempting to fit the spectrum with and
without the $\frac{1}{2}^{+}$ contribution shows the probability such a peak
can occur by chance is 0.01% demonstrating this contribution can only occur
due to a contribution from the $\frac{1}{2}^{+}$ due to the absence of any
background. The $\frac{1}{2}^{+}$ state in ${}^{9}\mathrm{B}$ was selected by
taking an excitation energy of between 1.4 and 2.4 MeV in ${}^{9}\mathrm{B}$
(following the centroid and width as observed via
${}^{9}\mathrm{Be}(^{3}\mathrm{He},t)$ Wheldon  _et al._ (2015) which is
consistent with our current results) and the $\frac{5}{2}^{+}$ was taken as
having an excitation energy of above 2.4 MeV. A more recent experimental study
has suggested the possibility that discrepancies in experimental results are
the result of a doublet of $\frac{1}{2}^{+}$ states Odsuren  _et al._ (2023).
Any contribution from the relatively-narrow 2.345 MeV $\frac{5}{2}^{-}$ is not
present in the presented plots as this state decays almost exclusively via
${}^{5}\mathrm{Li}$ and therefore would not correspond to a peak in the 8Be
spectrum. There were only 3 events associated with this decay to
${}^{5}\mathrm{Li}$ hence the statistics were insufficient to incorporate into
the analysis.

Figure 9: An example $p_{2}$ (${}^{12}\mathrm{C}(0_{2}^{+})+p$) event where
the energy deposition as a function of distance in the TPC is shown projected
in 2D.

Figure 10: An example $\alpha_{0}$ (${}^{9}\mathrm{B}(\mathrm{g.s.})+\alpha$)
event where the energy deposition as a function of distance in the TPC is
shown projected in 2D. The track going downwards can be identified as the
proton by its lower energy deposition. Also shown in Bishop  _et al._ (2023).

Following the channel selection, the excitation energy in ${}^{13}\mathrm{N}$
was calculated and is shown in Fig. 11. Despite low statistics, a number of
states can be seen and will be discussed individually. A summary of the
properties of these states observed is then shown in Table 1. A GEANT4
simulation was performed to test the variation in experimental resolution as a
function of excitation energy for the $\alpha_{0}$ channel which, as shown in
Fig. 12, is typically around $\sigma$ = 200 keV. The $p_{2}$ channel
resolution is almost entirely dominated by discrepancies between the expected
and real stopping powers for the $\alpha$-particles and therefore cannot be
accurately determined and is extremely sensitive to small changes. For all
excitation energies, it is realistically greater than $\sigma$ = 160 keV
however.

Figure 11: Excitation spectrum in ${}^{13}\mathrm{N}$ for 3$\alpha+p$
separated by channels. Dashed vertical arrows show previously-known states
populated by $\beta$-decay in black and new states observed are shown by solid
magenta arrows. Also shown in Bishop  _et al._ (2023).

Figure 12: Variation of experimental resolution from a GEANT4 simulation of infinitely narrow states decaying via 9B(g.s)+$\alpha$ as a function of excitation energy. The statistical error bars are not shown as the error for each point is dominated by systematic errors via the stopping power uncertainties for $\alpha$-particles and may be as large as 0.1 MeV (towards larger peak $\sigma$ values). Table 1: States in ${}^{13}\mathrm{N}$ observed in the current work with a preferred (but not definite) spin-parity assignment along with the decay properties of the states, calculated from the yields for the state in several different decay channels. The fractional reduced width is also efficiency corrected. State | Counts | Efficiency-corrected $\bar{\gamma}^{2}$
---|---|---
$E_{x}$ | $J^{\pi}$ | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{3}$ |  $p_{0}$ Knudsen  _et al._ (2005) |  $p_{1}$Knudsen  _et al._ (2005) | $p_{2}$ | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{3}$ | $p_{0}$ | $p_{1}$ | $p_{2}$
11.3(1) | 3/2- | $18(4.4)$ | 0 | 0 | $6(2.6)$ | $<3$ | $7(2.8)$ | 67(21)% | 0% | 0% | 4(2)% |  $<$1% | 29(13)%
11.8(1) | 3/2- | $<1.8$ | 0 | 0 | $28(14)$ | $<4$ | $4(2.2)$ |  $<$12% | 0% | 0% | 50(30)% | 0% | 38(25)%
12.4(1) | 3/2- | $22(4.8)$ | $4(2.2)$ | 0 | $<3$ | $<10$ | $5(2.5)$ | 6(2)% | 88(49)% | 0% |  $<$0.1% |  $<$2% | 2(1)%
13.1 | 1/2- | 0 | $3(2)$ | $5(2.5)$ | $21(6)$ | $<10$ | 0 | 0% | 1(1)% | 98(48)%111 Here the $\alpha_{3}$ channel is assumed to be through the $J^{\pi}=\frac{1}{2}^{-}$ channel in 9B rather than the $J^{\pi}=\frac{5}{2}^{+}$ state. | 0% |  $<$0.4% | 0%
5/2- | 0% | 10(10)% | 89(44)% | 0.7(0.2)% |  $<$0.2% | 0%
13.7(1) | 3/2- | $1(1.4)$ | $3(2)$ | $4(2.2)$ | $<3$ | $<10$ | $6(2.7)$ | 1(1)% | 8(8)% | 75(42)% |  $<$0.5% |  $<$7% | 8(3)%

###  V.1 11.3 MeV state

The first peak in the spectrum corresponds to an excitation energy of 11.3 MeV
in ${}^{13}\mathrm{N}$. The strength is almost entirely dominated by the
${}^{9}\mathrm{B(g.s)}+\alpha$ channel with a small fraction of
${}^{12}\mathrm{C}(0_{2}^{+})$+p. The yield in the $p_{0}$ from the previous
Knudsen data Knudsen  _et al._ (2005) shows a small very narrow peak at the
energy associated with this potential state ($E_{p}$(lab) = 8.64 MeV) and is
taken as $6(2.6)$. The yield in the $p_{1}$ channel is harder to estimate due
to the larger background from other states in this region but also shows no
evidence of a peak and is also taken to be negligible. Fitting this peak in
conjunction with neighboring peaks, the yield in the $\alpha_{0}$ channel is
$18(4.4)$ and yielding $\sigma$ = 280(80) keV and $E_{x}=11.3(1)$ MeV. In the
$p_{2}$ channel, the yield is $7(2.8)$ with $\sigma$= 220(100) keV and $E_{x}$
= 11.0(1) MeV. These widths are commensurate with the experimental resolution
therefore $\Gamma$ is expected to be relatively small ($\Gamma<200$ keV). This
is also corroborated by the small peaks seen in the Knudsen data where the
counts appear to lie within one energy bin (width=40 keV) suggesting the width
of this state may be even be $\Gamma<40$ keV but the significance of this peak
means this more strict width limit cannot be taken as definite. Given the
yields for $\alpha_{0}$ and $p_{2}$ are both strong, the spin-parity
assignment is favored towards $J^{\pi}=\frac{3}{2}^{-}$ where the angular
momentum transfer is L=0 and L=1 respectively. A choice of
$J^{\pi}=\frac{1}{2}^{-}$ or $J^{\pi}=\frac{5}{2}^{-}$ would require L=2 for
the $\alpha_{0}$ channel which should heavily suppress the yield and
$J^{\pi}=\frac{5}{2}^{-}$ would correspond to L=3 for $p_{2}$ so these options
are strongly disfavored. From Table 1, when taking the yield of the states and
correcting for the different channel penetrabilities, $P_{L}$, and
efficiencies, one can determine the structure of the measured states without a
measurement of the width of the state to compare to the Wigner limit. Many of
the states in ${}^{9}\mathrm{B}$ are very broad and the extreme simplification
of calculating the penetrability to the resonant energy is made. In reality,
the average penetrability will be higher. The structure is therefore
determined by the fractional reduced-width,
${\bar{\gamma}^{2}_{i}}=\frac{\gamma^{2}_{i}}{\sum_{j}\gamma^{2}_{j}}$ where
$\gamma^{2}_{i}=\frac{\Gamma_{i}}{2P_{iL}}$. This variable shows the type of
clustering but not the magnitude of the clustering. This state has
considerable strength in both $\alpha_{0}$ and $p_{2}$ with
${\bar{\gamma}^{2}_{i}}$ as 67% and 29% respectively. Taking the assumption
that the total width, $\Gamma$, of the state is $<200$ keV, one may compare to
the Wigner limit, $\gamma_{W}^{2}=\frac{\hbar^{2}}{\mu a^{2}}$ which is 0.57
and 2.1 MeV for $\alpha$-decay and $p$-decay respectively. The ratio to the
Wigner limit assuming $\Gamma$ = 200 keV is then $\theta_{W}^{2}$ $<28$ $\%$
and $<4\%$ for $\alpha_{0}$ and $p_{2}$ respectively. The former of these
(while notably only an upper limit) constitutes a well-clustered state.

###  V.2 11.8 MeV state

In the $p_{2}$ channel, the yield is $4(2.2)$ with $\sigma$= 170(110) keV and
$E_{x}$ = 11.8(1) MeV. Counts in the $\alpha_{1}$ channel are from higher
excitation energies extending down as the $P_{l}$ for $\alpha_{1}$ is
extremely suppressed prohibiting any strength. Due to the strength of the two
nearby states in the $\alpha_{0}$ channel, the yield in the $\alpha_{0}$
channel has very large uncertainties and can only be limited to be less than
$1.8$. There are two states previously known at this energy, a
$\frac{5}{2}^{-}$ and a $\frac{3}{2}^{-}$ with widths of 115(30) and 530(80)
keV respectively. Our data are more consistent with the $\frac{3}{2}^{-}$
assignment which was ascribed as $\frac{5}{2}^{-}$ in previous work Knudsen
_et al._ (2005). A $\frac{5}{2}^{-}$ assignment is the least favored from an
angular momentum perspective (L=3 vs L=1 for $\frac{1}{2}^{-}$ or
$\frac{3}{2}^{-}$) and this state is seen to populate the $p_{2}$ channel
reasonably well. This ambiguity suggests a reexamination of the total width of
each of these two states is needed in future work. From previous work assuming
that the strength seen was due to the $\frac{3}{2}^{-}$, the yield in the
$p_{0}$ was determined to be 28$(14)$. Making the same corrections for
penetrabilities as above, this state shares strength in the $p_{0}$ and
$p_{2}$ channels with ${\bar{\gamma}^{2}_{i}}$ $>50\%$ and $>38\%$
respectively with the remaining $\alpha_{0}$ component being $<12\%$. The
width for this state is perhaps poorly-known and the reduced width for $p_{2}$
can be compared to the Wigner limit and is $\sim 1\%$. Therefore, this state
is not strongly ${}^{12}\mathrm{C}(0_{2}^{+})\bigotimes p$ clustered (due to
the considerable $p_{0}$ branching ratio).

###  V.3 12.4 MeV state

Fitting this peak in conjunction with neighboring peaks, the yield in the
$\alpha_{0}$ channel is $22(4.8)$, yielding $\sigma$ = 310(90) keV and
$E_{x}=12.4(1)$ MeV. The corresponding yield of $\alpha_{1}$ is $4(2.2)$. In
the $p_{2}$ channel, the yield is $5(2.5)$ with $\sigma$= 110(70) keV and
$E_{x}$ = 12.5(1) MeV. Despite the relatively small yield in the $\alpha_{1}$
channel, when correcting for penetrability, the $\alpha_{1}$ dominates the
strength with ${\bar{\gamma}^{2}_{i}}=88\%$ with $\alpha_{0}$ and $p_{2}$
sharing the remainder with 6% and 2% respectively. The strong nature of the
${}^{9}\mathrm{B}(\frac{1}{2}^{+})\bigotimes\alpha$ suggests this is some kind
of near-threshold p-wave state.

This energy regime enters the region where existing
${}^{9}\mathrm{Be}(\alpha,\alpha_{0})$ Goss  _et al._ (1973); Lombardo, Ivano
_et al._ (2017) and ${}^{9}\mathrm{Be}(\alpha,n_{0})$ Obst  _et al._ (1972)
are available and one may look for analogous states in ${}^{13}\mathrm{C}$.
Given this state is in the s-wave in the entrance channel (assuming
$J^{\pi}=\frac{3}{2}^{-}$) and is expected to be relatively narrow, and
previous data seem to have a very large experimental width, it is perhaps
possible to explain that such a state has not been observed in this excitation
energy in ${}^{13}\mathrm{C}$ in the ${}^{9}\mathrm{Be}(\alpha,\alpha_{0})$
channel. It can be seen however that the $\alpha_{0}$ is rather weak and
therefore this state may not be strongly populated in this way. The sole
dominant feature in this region is a strong $\frac{5}{2}^{+}$ state at 11.95
MeV.

It is worth noting that the $\alpha_{1}$ channel is sub-threshold in
${}^{13}\mathrm{C}$ and the $n_{2}$ channel is heavily-suppressed until
${}^{13}\mathrm{C}$ excitation energies of above 13 MeV Obst  _et al._ (1972).
There are many states in this region ($E_{\alpha}>2$ MeV) visible in the
${}^{9}\mathrm{Be}(\alpha,n_{0})$ channel but the data are insufficient
resolution to provide spin-parity and width assignments.

This perhaps motivates a more extensive investigation of near-threshold states
in ${}^{13}\mathrm{C}$ from the ${}^{9}\mathrm{Be}+\alpha$ channel with higher
resolution and angular coverage. It is also worth noting in the previous
proton data Knudsen  _et al._ (2005) that there is a peak at this
corresponding energy for the $p_{1}$ channel ($E_{p}$(lab) = 5.55 MeV) where a
peak with a yield of $\approx 6$ can be seen above a considerable background.
The conservative limit of $<10$ for $p_{1}$ is therefore taken. The width in
this spectrum is also seen to be small which agrees with our results.

###  V.4 13.1 MeV state

Around 13.1 MeV, there is a relatively strong component seen in the
$\alpha_{3}$ channel where decays occur through the 2.75 MeV
$\frac{5}{2}^{+}$. There is only a very small contribution from the
$\alpha_{1}$ channel at this excitation energy so this state is almost
exclusively ${}^{9}\mathrm{B}(\frac{5}{2}^{+})\bigotimes\alpha$. Given the
dominance of $\alpha_{3}$, this suggests a spin-parity of
$J^{\pi}=\frac{5}{2}^{-}$ which suppresses the other channels.

In ${}^{9}\mathrm{B}$, there is also the extremely-broad 2.78 MeV
$\frac{1}{2}^{-}$ with $\Gamma$ = 3.13 MeV which cannot be excluded as the
source of the $\alpha_{3}$ strength. The reason for this is because we are
decaying into a broad intermediate state, the penetrability to lower
excitation energies in 9B is exponentially enhanced therefore meaning the
width-dependent Breit-Wigner shape is hugely distorted towards lower
excitation energies. Simple R-Matrix calculations show that the
$\frac{1}{2}^{-}$ being fed by an excitation energy of around 13 MeV in 13N
with L=0 may produce a yield in the 9B relative energy space that looks
similar to the narrower 2.75 MeV $\frac{5}{2}^{+}$ in 9B. Our data do not have
sufficient statistics to exclude this possibility and the $\frac{1}{2}^{-}$
decays primarily through ${}^{8}\mathrm{Be}$ via proton-decay. In this
possibility, the preferred spin-parity assignment is obviously
$J^{\pi}=\frac{1}{2}^{-}$ corresponding to L=0 $\alpha_{3}$ decay. The results
for both spin parities assignments are included in Table 1.

As with the 12.4 MeV state, there is evidence of a peak in previous data at
the correct energy in the $p_{1}$ channel ($E_{p}$(lab) = 6.20 MeV) which is
given a similar limit of $<10$.

###  V.5 13.7 MeV state

There is a collection of strength in the $p_{2}$, $\alpha_{0}$, $\alpha_{1}$
and $\alpha_{3}$ channel. With a yield of $6(2.7)$, the state is dominated by
$p_{2}$ and has parameters of $\sigma$= 260(70) keV and $E_{x}$ = 13.7(1) MeV.
Given the large $\bar{\gamma^{2}}$ in the $\alpha_{3}$ channel, this state can
be assigned as either $\frac{3}{2}^{-}$ or $\frac{5}{2}^{-}$. A
$\frac{5}{2}^{-}$ would correspond to L=3 for the $p_{2}$ channel so a
$\frac{3}{2}^{-}$ assignment would be more commensurate with the reasonable
$p_{2}$ yield. This state also exhibits a
${}^{9}\mathrm{B}(\frac{5}{2}^{+})\bigotimes\alpha$ structure.

Examining the previous work for evidence of a peak in the $p_{1}$ is not
possible for this state due to the presence of a strong $p_{0}$ branch from a
lower-lying state at the same energy. A similar limit of $<10$ is therefore
placed on this state.

Figure 13: Level scheme of measured 3$\alpha$+p states in ${}^{13}\mathrm{N}$
in the central column with the proposed spin-parity assignments. The location
of the thresholds for proton and $\alpha$ decay are shown in red with the
equivalent excitation energy shown. The corresponding states in the daughter
nuclei (${}^{12}\mathrm{C}$ and ${}^{9}\mathrm{B}$) are similarly displayed.
Also shown in Bishop  _et al._ (2023).

##  VI Conclusions

Three new states and a previously-tentative state have been observed with a
strong $3\alpha+p$ nature with their excitation energies relative to the
thresholds shown in Fig. 13. The first is a narrow potential $\frac{3}{2}^{-}$
state at $E_{x}$ = 11.3(1) MeV with mixed
${}^{9}\mathrm{B}(\mathrm{g.s})\bigotimes\alpha$ and
$p+^{12}\mathrm{C}(0_{2}^{+})$ nature.

Another previously-observed $\frac{3}{2}^{-}$ was seen to have mixed
$p+^{12}\mathrm{C}(\mathrm{g.s.})$ and $p+^{12}\mathrm{C}(0_{2}^{+})$ nature
at 11.8 MeV with around half of the total strength as
$p+^{12}\mathrm{C}(\mathrm{g.s.})$.

At higher excitation, another strong $\alpha$-decaying state was seen at
$E_{x}$ = 12.4(1) MeV although this state has a much stronger
${}^{9}\mathrm{B}({\frac{1}{2}^{+}})\bigotimes\alpha$ nature.

A revised excitation energy of 13.1(1) MeV is suggested for a previously-seen
state at 13.26 MeV. The ${}^{9}\mathrm{B}({\frac{5}{2}^{+}})\bigotimes\alpha$
structure dominates in this state and a spin assignment of $J^{\pi}$ =
${\frac{1}{2}^{-}}$ or ${\frac{5}{2}^{-}}$ are therefore preferred.

Finally, another $\frac{3}{2}^{-}$ is seen at 13.7 MeV which is also dominated
by ${}^{9}\mathrm{B}({\frac{5}{2}^{+}})\bigotimes\alpha$.

The inability to extract the width of these narrow states means that the
magnitude of clustering cannot be fully evaluated however the type (channel)
of clustering can be determined without this information. Higher resolution
data focusing on the proton channel may provide further information on the
magnitude of this clustering phenomenon. From our current data, one may
conclude however when comparing the reduced widths that the clustered channels
are very competitive against the single-particle $p_{0}$ channel.
Configuration mixing may however be very strong in these states and therefore
quench the single-cluster nature of these resonances.

One can compare these states with $3\alpha$+p events observed via other
reactions such as those populated by single proton decay from highly-excited
states in 14O⋆ in complete kinematics Charity  _et al._ (2009). Due to the
spin-parity selectivity of $\beta$-decay and the unusual population method of
states in 13N⋆ in the previous work, it is perhaps not surprising that the
same states are not observed in the current work.

Work using 13C(3He,t) by Fujimura et al. has also been performed which, at
high energy, is a better analogue to inverse $\beta$-decay and should
preferentially populate similar states although the ground state spin-parity
of 13C is $J^{\pi}=\frac{1}{2}^{-}$ rather than $J^{\pi}=\frac{3}{2}^{-}$ as
for 13O Fujimura  _et al._ (2004). This work relies on the excitation energy
being extracted by missing mass and the decay channels tagged solely by
protons to separate $p_{0}$, $p_{1}$, $p_{2}$ etc. A broad peak at $E_{x}$ =
13.5 MeV can be seen in the $p_{2}$ channel. This lies close in energy to our
observed peak at 13.8 MeV which, with a preferred $J^{\pi}=\frac{3}{2}^{-}$
agrees with the state being observed with (3He,t). The 11.8 MeV states, also
observed previously by Knudsen was also seen to be very strongly populated in
the Fujimura data. Their data show a smaller $p_{0}$ and $p_{2}$ yield and a
dominant $p_{1}$ yield which is at odds with the Knudsen result which ascribed
the yield in this region to the $J^{\pi}=\frac{5}{2}^{-}$ which would not be
populated strongly by Fujimara. Is it therefore possible that in the current
work and that of Knudsen, the $J^{\pi}=\frac{3}{2}^{-}$ and
$J^{\pi}=\frac{5}{2}^{-}$ are both contributing strength here in different
channels. Alternatively, in the work by Knudsen the $p_{1}$ yield could have
been partially obscured by the strong $p_{0}$ yield for the 7.376 MeV state.
Further studies with higher spin-sensitivity are necessary to disentangle
these two contributions however.

We hope this experimental work will motivate further theoretical studies for
13N including using the Algebraic Cluster Model (D${}^{{}^{\prime}}_{3h}$) as
performed for 13C Bijker and Iachello (2019) and AMD calculations.

##  VII Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Science under Award No. DE-FG02-93ER40773 and by the
National Nuclear Security Administration through the Center for Excellence in
Nuclear Training and University Based Research (CENTAUR) under Grant No. DE-
NA0003841. G.V.R. also acknowledges the support of the Nuclear Solutions
Institute. S.A., S.M.C., C.K., D.K., S.K. and C.P. also acknowledge travel
support from the IBS grant, funded by the Korean Government under grant number
IBS-R031-D1. C.H.K acknowledges travel support from the National Research
Foundation of Korea(NRF) grant, funded by the Korea government(MSIT) (No.
2020R1A2C1005981 and 2013M7A1A1075764).

## References

  * Freer and Fynbo (2014) M. Freer and H. Fynbo, Progress in Particle and Nuclear Physics 78, 1 (2014).
  * Bishop  _et al._ (2020a) J. Bishop, G. Rogachev, S. Ahn, E. Aboud, M. Barbui, P. Baron, A. Bosh, E. Delagnes, J. Hooker, C. Hunt, H. Jayatissa, E. Koshchiy, R. Malecek, S. Marley, R. O’Dwyer, E. Pollacco, C. Pruitt, B. Roeder, A. Saastamoinen, L. Sobotka, and S. Upadhyayula, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 964, 163773 (2020a).
  * Bishop  _et al._ (2020b) J. Bishop, G. V. Rogachev, S. Ahn, E. Aboud, M. Barbui, A. Bosh, C. Hunt, H. Jayatissa, E. Koshchiy, R. Malecek, S. T. Marley, E. C. Pollacco, C. D. Pruitt, B. T. Roeder, A. Saastamoinen, L. G. Sobotka, and S. Upadhyayula, Phys. Rev. C 102, 041303 (2020b).
  * Bishop  _et al._ (2021) J. Bishop, G. V. Rogachev, S. Ahn, E. Aboud, M. Barbui, A. Bosh, J. Hooker, C. Hunt, H. Jayatissa, E. Koshchiy, R. Malecek, S. T. Marley, M. Munch, E. C. Pollacco, C. D. Pruitt, B. T. Roeder, A. Saastamoinen, L. G. Sobotka, and S. Upadhyayula, Phys. Rev. C 103, L051303 (2021).
  * Bishop  _et al._ (2023) J. Bishop, G. V. Rogachev, S. Ahn, M. Barbui, S. M. Cha, E. Harris, C. Hunt, C. H. Kim, D. Kim, S. H. Kim, E. Koshchiy, Z. Luo, C. Park, C. E. Parker, E. C. Pollacco, B. T. Roeder, M. Roosa, A. Saastamoinen, and D. P. Scriven, Phys. Rev. Lett. 130, 222501 (2023).
  * Tribble  _et al._ (1989) R. Tribble, R. Burch, and C. Gagliardi, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 285, 441 (1989).
  * Koshchiy  _et al._ (2020) E. Koshchiy, G. Rogachev, E. Pollacco, S. Ahn, E. Uberseder, J. Hooker, J. Bishop, E. Aboud, M. Barbui, V. Goldberg, C. Hunt, H. Jayatissa, C. Magana, R. O’Dwyer, B. Roeder, A. Saastamoinen, and S. Upadhyayula, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 957, 163398 (2020).
  * Pollacco  _et al._ (2018) E. Pollacco, G. Grinyer, F. Abu-Nimeh, T. Ahn, S. Anvar, A. Arokiaraj, Y. Ayyad, H. Baba, M. Babo, P. Baron, D. Bazin, S. Beceiro-Novo, C. Belkhiria, M. Blaizot, B. Blank, J. Bradt, G. Cardella, L. Carpenter, S. Ceruti, E. D. Filippo, E. Delagnes, S. D. Luca, H. D. Witte, F. Druillole, B. Duclos, F. Favela, A. Fritsch, J. Giovinazzo, C. Gueye, T. Isobe, P. Hellmuth, C. Huss, B. Lachacinski, A. Laffoley, G. Lebertre, L. Legeard, W. Lynch, T. Marchi, L. Martina, C. Maugeais, W. Mittig, L. Nalpas, E. Pagano, J. Pancin, O. Poleshchuk, J. Pedroza, J. Pibernat, S. Primault, R. Raabe, B. Raine, A. Rebii, M. Renaud, T. Roger, P. Roussel-Chomaz, P. Russotto, G. Saccà, F. Saillant, P. Sizun, D. Suzuki, J. Swartz, A. Tizon, N. Usher, G. Wittwer, and J. Yang, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 887, 81 (2018).
  * Holmes  _et al._ (2020) J. Holmes, E. Galyaev, R. Alarcon, R. Acuna, D. Blyth, B. Fox, N. Mullins, and K. Scheuer, Journal of Instrumentation 15, T05001 (2020).
  * Wang  _et al._ (2012) M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer, Chin.Phys.C 36, 1603 (2012).
  * Ziegler  _et al._ (2010) J. F. Ziegler, M. Ziegler, and J. Biersack, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 268, 1818 (2010), 19th International Conference on Ion Beam Analysis.
  * Knudsen  _et al._ (2005) H. H. Knudsen, H. O. U. Fynbo, M. J. G. Borge, R. Boutami, P. Dendooven, C. A. Diget, T. Eronen, S. Fox, L. M. Fraile, B. Fulton, J. Huikary, H. B. Jeppesen, A. S. Jokinen, B. Jonson, A. Kankainen, I. Moore, A. Nieminen, G. Nyman, H. Penttilä, K. Riisager, S. Rinta-Antila, O. Tengblad, Y. Wang, K. Wilhelmsen, and J. Äystö, Phys. Rev. C 72, 044312 (2005).
  * Fischler and Bolles (1987) M. A. Fischler and R. C. Bolles, in  _Readings in Computer Vision_, edited by M. A. Fischler and O. Firschein (Morgan Kaufmann, San Francisco (CA), 1987) pp. 726–740.
  * Wheldon  _et al._ (2015) C. Wheldon, T. Kokalova, M. Freer, J. Walshe, R. Hertenberger, H.-F. Wirth, N. I. Ashwood, M. Barr, N. Curtis, T. Faestermann, R. Lutter, J. D. Malcolm, and D. J. Marín-Lámbarri, Phys. Rev. C 91, 024308 (2015).
  * Odsuren  _et al._ (2023) M. Odsuren, T. Myo, and K. Katō, Phys. Rev. C 107, 044003 (2023).
  * Goss  _et al._ (1973) J. D. Goss, S. L. Blatt, D. R. Parsignault, C. D. Porterfield, and F. L. Riffle, Phys. Rev. C 7, 1837 (1973).
  * Lombardo, Ivano  _et al._ (2017) Lombardo, Ivano, Dell´Aquila, Daniele, and Vigilante, Mariano, EPJ Web Conf. 165, 01036 (2017).
  * Obst  _et al._ (1972) A. W. Obst, T. B. Grandy, and J. L. Weil, Phys. Rev. C 5, 738 (1972).
  * Charity  _et al._ (2009) R. J. Charity, T. D. Wiser, K. Mercurio, R. Shane, L. G. Sobotka, A. H. Wuosmaa, A. Banu, L. Trache, and R. E. Tribble, Phys. Rev. C 80, 024306 (2009).
  * Fujimura  _et al._ (2004) H. Fujimura, H. Akimune, I. Daito, M. Fujiwara, K. Hara, K. Y. Hara, M. N. Harakeh, F. Ihara, T. Inomata, K. Ishibashi, T. Ishikawa, T. Kawabata, A. Tamii, M. Tanaka, H. Toyokawa, T. Yamanaka, and M. Yosoi, Phys. Rev. C 69, 064327 (2004).
  * Bijker and Iachello (2019) R. Bijker and F. Iachello, Phys. Rev. Lett. 122, 162501 (2019).