2022

[6,1,7]Kévin Fossez [1,10]Chloë Hebborn [12]Sebastian König [14,15]Lucas
Platter

1]Facility for Rare Isotope Beams, East Lansing, 48824, MI, USA

2]Department of Physics and Astronomy, Michigan State University, East
Lansing, 48824, MI, USA

3]Department of Physics and Astronomy, Louisiana State University, Baton
Rouge, 70803, LA, USA

4]Institut für Kernphysik and PRISMA+ Cluster of Excellence, Johannes
Gutenberg-Universität, 55128 Mainz, Germany

5]Institute of Nuclear and Particle Physics, and Department of Physics and
Astronomy, Ohio University, Athens, OH 45701, USA

[6]Florida State University, Tallahassee, 32306, FL, USA

7]Physics Division, Argonne National Laboratory, Lemont, 60439, IL, USA

8]Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, São José dos Campos,
Brazil

9]Theory Center, Jefferson Lab, Newport News, 23606, VA, USA 10]Lawrence
Livermore National Laboratory, P.O. Box 808, L-414, Livermore, 94551, CA, USA

11]Department of Physics and Astronomy, Purdue University, West Lafayette,
47906, IN, USA

12]Department of Physics, North Carolina State University, Raleigh, 27695, NC,
USA

13]Department of Physics, Federal Fluminense University, Niterói, 24210-340,
R.J., Brazil

14]Physics Division, Oak Ridge National Laboratory, Oak Ridge, 37831 TN, USA

15]Department of Physics and Astronomy, University of Tennessee, Knoxville,
Knoxville, 37996,TN, USA

# Perspectives on few-body cluster structures in exotic nuclei

Daniel Bazin<EMAIL_ADDRESS>Kevin Becker<EMAIL_ADDRESS>Francesca
Bonaiti<EMAIL_ADDRESS>Charlotte Elster<EMAIL_ADDRESS>Tobias
Frederico<EMAIL_ADDRESS>Alex Gnech<EMAIL_ADDRESS><EMAIL_ADDRESS>Michael Higgins${}^{\text{11}}$<EMAIL_ADDRESS>Linda Hlophe
<EMAIL_ADDRESS>Benjamin Kay<EMAIL_ADDRESS><EMAIL_ADDRESS>Konstantinos
Kravvaris<EMAIL_ADDRESS>Jesus Lubian<EMAIL_ADDRESS>Augusto
Macchiavelli  Filomena Nunes<EMAIL_ADDRESS><EMAIL_ADDRESS>Gregory
Potel<EMAIL_ADDRESS>Xilin Zhang<EMAIL_ADDRESS>[  [  [  [  [
*  [  [  [  [  [  [  [  [  [

###### Abstract

It is a fascinating phenomenon in nuclear physics that states with a
pronounced few-body structure can emerge from the complex dynamics of many
nucleons. Such halo or cluster states often appear near the boundaries of
nuclear stability. As such, they are an important part of the experimental
program beginning at the Facility for Rare Isotope Beams (FRIB). A concerted
effort of theory and experiment is necessary both to analyze experiments
involving effective few-body states, as well as to constrain and refine
theories of the nuclear force in light of new data from these experiments. As
a contribution to exactly this effort, this paper compiles a collection of
“perspectives” that emerged out of the Topical Program “Few-body cluster
structures in exotic nuclei and their role in FRIB experiments” that was held
at FRIB in August 2022 and brought together theorists and experimentalists
working on this topic.

###### keywords:

clustering, halo structure, few-body, exotic nuclei, decay

###### Contents

  1. Preface
  2. 1 Building the nuclear theory to experiment pipeline
  3. 2 Emergence of few-body effects in exotic nuclei
  4. 3 Understanding few-body dynamics in knockout and transfer reactions
  5. 4 Continuum and cluster effects in reaction theory
  6. 5 Spectroscopic factors, experimental probes and perspectives
  7. 6 Universal corrections for few-body reaction models
  8. 7 Remarks on experimental and theoretical uncertainties in reaction studies
  9. 8 New opportunities for nuclear Efimov states
  10. Acknowledgements

## Preface

L. Platter, K. Fossez, C. Hebborn, and S. König

Nuclear physics is rich with complexity, emerging ultimately out of the
Standard Model of particle physics and in particular out of Quantum
Chromodynamics (QCD), the fundamental theory of the strong interaction.
Protons and neutrons that bind together to form atomic nuclei are the initial
effective degrees of freedom in nuclear physics and ab initio nuclear
structure theory uses these to compute observables for a wide range of nuclei.
However, interesting emergent phenomena appear near the boundaries of nuclear
stability, the so-called driplines, where many nuclei display a transition to
new effective degrees of freedom by hosting phenomena which indicate a
dominant few-body cluster or halo structure. This fascinating feature of
nuclear systems leads frequently to dynamics that experimentally are similar
to those of few-body systems in atomic and particle physics and are considered
to be signatures of low-energy universality. The signatures of cluster states
are also particularly pronounced in nuclear reactions that, in most cases,
cannot be accessed directly in ab initio approaches at present.

The impact of clustering in nuclei close to the driplines affects ongoing and
proposed experiments at the Facility for Rare Isotope Beams (FRIB) and other
radioactive beam facilities around the world. Halo nuclei, i.e., systems that
can be described as a tightly bound core nucleus with weakly bound valence
nucleons, are an important example of such clustered systems that have been
studied extensively both experimentally and theoretically. In these systems, a
clear separation of scales exists facilitating the construction of a model-
independent low-energy effective field theory (EFT), a systematic expansion
with quantified uncertainty estimates. However, clustering plays a role for a
much larger group of nuclei and for those it is at present not clearly
understood how to construct such EFTs.

An important task in this exciting time with new facilities starting to
operate is therefore to develop new models and EFTs that treat cluster effects
explicitly, consistently, and efficiently. These approaches should be
constrained by experimental data and be able to predict observables with
reliable uncertainty estimates. In addition, it is important to understand the
mechanisms underlying the transition from many-nucleon systems to new
effective degrees of freedom, and how the nuclear force gives rise to these
phenomena. However, the interplay between few-body effects and nuclear shell
structure is not yet well understood and requires a combined effort from
theory and experiment to advance our understanding.

Clearly the low-energy nuclear physics community needs to build bridges
between different groups, involving both theorists and experimentalists, to
join together in order to design, analyze, and explain current and future
experiments and to refine and extend nuclear theories. Experimental data can
guide theorists in constructing better models and approaches that improve our
understanding and interpretation of nuclear observables. In turn, a strong
theory program can assist experimental groups in identifying observables that
can provide new insights into the underlying mechanisms of emergent phenomena
in nuclear physics.

In this collection of “perspectives,” we give an overview of thoughts on the
requirements for effective progress in this critical time that will provide a
significant amount of new data. The articles compiled here are the result of
the FRIB Topical Program “Few-Body Clusters in Exotic Nuclei and Their Role in
FRIB Experiments” held at FRIB/Michigan State University in August 2022. This
program supported by the FRIB Theory Alliance brought together theorists and
experimentalists working on this topic. Each perspective was written by a
small subgroup of participants as indicated for each piece. It is our hope
that this collection will be useful to the broader community and serve as a
continuation of the stimulating discussions that we had during the program.

##  1 Building the nuclear theory to experiment pipeline

K. Fossez, C. Hebborn, D. Bazin, and A. Macchiavelli

The DOE-NP flagship FRIB has started operations at MSU and will push forward
the exploration of the nuclear landscape by giving access to more than a
thousand new exotic nuclei, mostly using nuclear reactions. Depending on the
reaction considered and the nuclear degrees of freedom to be probed, the
required beam energies can vary from keV/u to GeV/u. The different types of
reactions that can be used as a function of the energy are shown schematically
in Fig. 1. Reactions at energies up to about 200 MeV/u are already accessible
using the current nominal FRIB beam energy, but those up to 400 MeV/u will
require the so-called high-energy upgrade FRIBupgrade . In addition, in
anticipation of the coming online of FRIB, critical detector development
programs were started in the last decade Equ . These programs led to new
state-of-the-art detector systems having unparalleled resolving power, and
which will play a central role in addressing essentially all of the challenges
put forth in the National Academies Decadal Study NAS  and the 2015 NSAC Long
Range Plan LRP1 , i.e. by answering fundamental questions in both nuclear
structure and nuclear astrophysics such as “How does subatomic matter organize
itself and what phenomena emerge?” and “How did visible matter come into being
and how does it evolve”?

Figure 1: Schematic representation of the possible reaction mechanisms that
can be used to probe exotic nuclei at FRIB as a function of energy. Reactions
beyond 200 MeV/u will require the FRIB400 upgrade. Figure adapted from Ref.
FRIBupgrade

However, the ultimate success of the FRIB scientific program will require a
concerted effort between experiment and theory. Indeed, to infer accurate and
precise nuclear properties of exotic nuclei from experimental data, reliable
nuclear structure and reaction models will be needed. The schematic pipeline
shown in Fig. 2 depicts some of required theory inputs necessary to guide
experiments or even analyse data in some cases. Of course, data can then be
used to provide feedback on theory models.

For example, the Active Target Time Projection Chamber (AT-TPC) AT-TPC  is
ideally suited to probe the cluster structure of low-energy resonances since
it allows to perform resonant scattering reactions by scanning different beam
energies in a single experiment. In order to determine complementary
information such a the spins, parities, or widths of the resonances observed,
it might be necessary to directly compare with structure models, or to look at
transfer angular distributions which require the use of reaction models to be
analysed. Moreover, the absolute value of transfer cross sections is directly
related to the cluster content of the populated states, usually quantified in
terms of spectroscopic factors.

Figure 2: Schematic representation of (some of) the theory inputs required to
fully exploit experimental capabilities at FRIB.

On the theory side, confrontation with experimental data should ideally be
done with a theory using nucleons as degrees of freedom, and the nuclear
forces between nucleons should be derived from effective field theories (EFTs)
of the underlying theory of quantum chromodynamics (QCD) and electroweak
theory hergert20_2412 ; HammerReview2020 . However, solving this canonical ab
initio nuclear many-body problem exactly for the description of both nuclear
structure and reactions in a unified picture is a particularly daunting task,
which so far has only be achieved in selected light nuclei with $A<12$
Navratil_2016 ; quaglioni08_755 ; quaglioni09_756 ; lee09_949 ; nollett12_835
; flores22 . To make matters worse, issues remain in the construction of EFTs
of nuclear forces due to the lack of clear separation of scales, effectively
reducing the predictive power of any theory of the atomic nucleus beyond light
nuclei stroberg21_2483 . This situation leads to the conundrum of low-energy
nuclear theory: making accurate predictions by solving a difficult problem
using an uncertain input. In that regard, the use of simplified effective
approaches such as few-body reaction models or the widely successful nuclear
shell model caurier05_424  to go beyond light nuclei comes as no surprise.

In the present context of few-body cluster structures in exotic nuclei, one
important line of development is the move towards a unification of nuclear
structure and reactions, in which 1-, 2-, …, $A$-nucleons decay channels are
included in the description of the $A$-body problem rotter91_448 ; michel09_2
; volya05_470 ; baroni13_385 . It has been recognized early on that, far from
the valley of stability, weak binding and couplings to continuum states play
an important role in shaping the properties of exotic nuclei matsuo10_1238 ;
michel10_4 . As a consequence, closer to the drip lines, i.e. the regions of
greatest interest for FRIB, nuclear models have shortcomings due to, for
instance, the influence of couplings to continuum states okolowicz03_21 ;
michel10_4 , the increased number of particles involved, or the presence of
new emergent phenomena otsuka20_2383  (or interplays of them) having
collective degrees of freedom and presenting a complex few-body dynamics
fossez16_1335 ; fossez16_1793 ; fossez18_2171 ; kravvaris17_1960 ; wang18_2144
; fossez22_2540  (see also Sec. 2). Similar comments can also be made for
unbound excited states in well bound nuclei. As a result, it is not rare to
find halo states jensen04_233 ; frederico12_372 ; tanihata13_549 , clustered
states freer18_2138  often near decay thresholds ikeda68_2329 ; oertzen06_1017
; freer07_1018 ; okolowicz13_241 ; okolowicz12_998 , or the decay of two or
more nucleons guillemaud90_1761 ; blank08_449 ; pfutzner12_1169 ;
spyrou12_1216 ; thoennessen13_1776  in exotic nuclei. The need to unify
nuclear structure and reactions has been formally recognized during a previous
FRIB Theory Alliance Topical Program johnson20_2389 .

From the structure side of the problem, the inclusion of continuum states
within many-body calculations has essentially been achieved in, notably,
continuum shell model approaches rotter91_448 ; okolowicz03_21 ; volya06_94
and many-body methods generalized in the complex-energy plane using the
Berggren basis michel09_2 ; rotureau06_15 ; hagen07_976  or the uniform
complex-scaling technique carbonell14_1240 . However, describing the emergence
of collective degrees of freedom in exotic nuclei within such many-body
approaches is still a challenge (see more details in Sec. 2), and extracting
reaction observables requires additional treatment as explained hereafter.

On the reaction side, several strategies have been employed to include
structure information explicitly, for example by combining the resonating
group method with structure methods Navratil_2016 ; quaglioni08_755 ;
quaglioni09_756 ; jaganathen14_988 ; fossez15_1119 ; kravvaris17_1960  to
treat the relative motion of a target and a projectile directly within a many-
body framework. Such approaches can thus provide reaction cross sections in a
unified picture of nuclear structure and reactions, but so far their ab initio
formulation limits them to light nuclei and at best includes only selected
two- or three-body reaction channels. Other ab initio approaches that can
provide scattering cross sections in light nuclei include, for instance, the
lattice-EFT (L-EFT) lee09_949  and the quantum Monte Carlo (QMC) nollett12_835
; flores22  methods. Both techniques are formulated in radial space and thus
offer an alternative to traditional cluster approaches based on nucleons as
degrees of freedom ono92_2334 ; enyo01_2335 ; feldmeier00_2337  to study the
impact of clustering in exotic nuclei. In the near future, symmetry adapted ab
initio methods launey16_2403  will hopefully provide a way to extend ab initio
reactions to medium-mass nuclei while capturing nuclear shapes and collective
modes naturally. For all these methods, additional developments will be needed
to include more continuum couplings within the target and projectile, e.g.
through the inclusion of many-body reaction channels.

Such first-principles methods can provide and have provided valuable insight
about the emergence of few-body dynamics in nuclei, but few of these methods
can be used in a practical way to give reliable spin and parity assignments in
medium-mass nuclei due to their prohibitive computational cost.

An alternative to solving the full $A$-body problem for structure and
reactions consists in using effective degrees of freedom, as is illustrated in
Fig. 3 showing a schematic reduction of the many-body problem into a few-body
one, composed of tightly bound clusters assumed structureless. This strategy
has the obvious advantage of dramatically reducing the computational cost of
calculations, and thus to allow for studies beyond light nuclei.

Figure 3: Reduction of the many-body problem into a few-body one.

In this few-body picture, the structures of both the projectile and the target
are described using effective potentials reproducing their low-energy single-
particle spectra, and the cluster-cluster interactions are simulated using
optical potentials feshbach1958unified ; TN09 ; BC12 . Additionally, only
pairwise cluster interactions are usually considered and other irreducible
interactions beyond this level are neglected, even though there is some
evidence from deuteron-induced reactions that this approximation leads to
sizeable uncertainties Hlophe:2022fdn ; PhysRevC.74.044304 ;
PhysRevC.104.034614 ; PhysRevC.104.024612 . For that reason, efforts are
underway to quantify the effect of irreducible three-body cluster-cluster
interactions (see Sec. 6). In general, additional investigations on the
reduction of the many-body problem into a few-body one should be supported, as
they will provide paths to efficiently inform few-body reaction models by
many-body predictions.

For most reactions, the few-body problem is solved using approximate schemes
such as continuum-discretized coupled-channel methods AUSTERN1987125 ; Yetal86
; Kawai86 ; PhysRevC.68.064609 , the eikonal approximation G59 ; Hansen03 ;
PhysRevLett.95.082502 , the distorted wave Born approximation, or the
adiabatic distorted wave approximation JOHNSON197456 ; TIMOFEYUK2020103738 .
These models are usually applied to the three-body problem, but some have
recently been extended up to the five-body problem (see for example Ref.
PhysRevC.101.064611 ). Moreover, due to the high cost of treating the few-body
problem in the continuum, most of these models rely on additional assumptions
such as neglecting excitations of the clusters during the reaction,
considering only one-step processes, or they neglect excitations of the
projectile and the target during the reaction. These approximations limit the
accuracy of the information inferred from experiments (see for example Refs
PhysRevC.78.054602 ; CooperPairPotel ; PhysRevLett.109.232502 ) and it is thus
crucial to develop more complex reaction models treating dynamical excitations
explicitly (see more detailed discussion in Secs. 4 and 5).

Besides the treatment of the reaction dynamics, another important element is
the quality of the cluster-cluster effective potentials. For bound states,
these interactions are typically adjusted to reproduce properties of the low-
energy spectra of the relevant nuclei, i.e., the energy of the states, their
spins, parities, partial widths, and electromagnetic transition probabilities,
as well as information about the overlap functions, their asymptotic
normalization constants (ANCs), root mean square (rms) radii, and
spectroscopic factors (SFs) (see Fig. 2). The optical potentials between the
projectile’s and target’s clusters have been historically fitted on elastic-
scattering data on the same system koning03 ; becchetti69 . However, most of
the scattering data used to constrain these phenomenological interactions were
from stable targets, and therefore their quality for unstable systems is
uncertain. With the start of FRIB, it is thus urgent to develop optical
potentials that can reliably be extrapolated away from stability to the
driplines.

As a consequence, recent efforts have been undertaken to enhance our
predictive power for reactions involving exotic nuclei by strengthening the
connection between many- and few-body approaches. One promising avenue rests
on the EFT framework, which provides a rigorous way to construct theories. By
definition, EFTs are as good as the underlying theory within a limit range
defined by the effective separation of scales exploited, and they can be
improved systematically with quantified systematic errors at each order.
Examples of successful EFTs related to few-body physics are the so-called halo
EFTs bertulani02_869 ; ryberg14_997 ; ji14_1101 ; Hammer2017 ; Capel:2018kss
(see Sec. 3) and the EFT formulation of the particle-plus rotor model
papenbrock11_1310 ; papenbrock14_1312 ; papenbrock15_1311 . What makes EFTs
interesting in the reduction of the many-body problem is that when they are
applied to the few-body problem, their low-energy constants can be fixed using
results from experiments or many-body approaches. While this limits their
predictive power, in principle, it allows for the rigorous construction of
cluster-cluster interactions for few-body reaction models. For that reason, we
propose to prioritize two research programs to address this problem: i) to
reformulate phenomenological models into EFTs to ensure a proper connection
with the underlying theory, and ii) to match all the EFTs with each other when
appropriate, and with the underlying theory. Such programs will bring few-body
approaches to a new level of predictive power and ensure consistency across
the description of all few-body effects in exotic nuclei.

In situations where EFTs cannot a priori be developed, i.e., when they are no
clear separation of scales that can be exploited, cluster-cluster interactions
for few-body reaction models can be built from many-body theory using Watson
multiple scattering theory, the Feshbach projection operator formalism,
Green’s function theory, or by starting from nuclear matter predictions (see
recent white paper WhitePaperOptPot22  and references therein). Optical
potentials obtained from many-body approaches have the advantage of carrying
the correct isospin dependence and are therefore more likely to extrapolate
properly in the regions of greatest interest for FRIB away from stability. In
addition, when an optical potential obeys dispersion relations, bound and
scattering states are described on the same footing, providing that a
consistent description of the reaction (see discussion in Sec. 5).
Nevertheless, a common shortcoming of these microscopic potentials is their
lack of absorption strength, which can be improved by informing them elastic
and inelastic experimental data from, for example, transfer, charge-exchange
and breakup reactions. This experimental feedback will become particularly
important for experiments at FRIB using, for instance, the previously
mentioned AT-TPC detector, in which plenty of high-quality data for different
reaction channels and at different energies will be obtained (Fig. 4 shows an
example of trajectory in the AT-TPC chamber).

Figure 4: Example of a proton track recorded in the AT-TPC from a $(d,p)$
reaction between a 10Be beam and a deuteron from the deuterium gas filling the
active volume of the detector.

To conclude, in light of the experimental needs at FRIB and the state of
theoretical developments, the success of the FRIB scientific mission will
require a concerted effort between theory and experiment, and in particular
important developments to pursue the unification of nuclear structure and
reactions in exotic nuclei to provide reliable predictions during the
exploration of the drip lines. We believe that close collaborations between
experimentalists and theorists will benefit the planning of new experiments
and maximize our discovery power of new exotic phenomena in nuclei.

##  2 Emergence of few-body effects in exotic nuclei

K. Kravvaris, F. Bonaiti, A. Gnech, K. Fossez, and K. Becker

How do few-body effects emerge in atomic nuclei? This remains one of the most
intriguing open questions of low-energy nuclear physics LRP1 ; NAS .
Collective phenomena appear when a transition from (many) single-particle
degrees of freedom to (few) collective ones occurs, leading to the emergence
of few-body dynamics, which can have a direct impact on nuclear observables
and astrophysical processes. It is thus critical to develop a proper
theoretical understanding of collective phenomena and their associated few-
body effects. However, a comprehensive description of collectivity in nuclei
is challenging because it appears at various nearby energy scales comparable
to single-particle energies, and leads to a rich variety of phenomena such as
deformation, clustering, rotational and vibrational motions, or pairing
dynamics. In exotic nuclei near the drip lines, which the Facility for Rare
Isotope Beams (FRIB) will be exploring in the coming decades, couplings to the
continuum of scattering states and decay channels matsuo10_1238 ; michel10_4 ;
johnson20_2389  can dramatically impact collective phenomena, and lead to new
ones such as halo structures, the superradiance effect volya03_858 , or exotic
decay modes like two-neutron or two-proton decay guillemaud90_1761 ;
blank08_449 ; pfutzner12_1169 ; spyrou12_1216 ; thoennessen13_1776 . Moreover,
the delicate interplay between various collective phenomena can fundamentally
shape the properties of a system, as seen, for example, in the case of a halo
structure above a deformed core, or when a rotational band extends into the
continuum forming resonant rotational states garrido13_1171 ; fossez16_1335 .
In this perspective piece, we will mention some recent works related to the
emergence of few-body effects in exotic nuclei to illustrate more general
ideas which, we will argue, should be investigated to ensure the success of
the FRIB scientific program. Essentially, these ideas fall into two main
topics which are i) the proper theoretical description of collectivity in
exotic nuclei and the associated few-body dynamics, and ii) the connection
between nuclear forces and emergent properties in exotic nuclei.

### Deformation

Nuclear deformation is one of the most common emergent phenomena in atomic
nuclei. It has been shown to naturally arise from the underlying strong
nuclear force in large-scale ab initio calculations DytrychLDRWRBB20 , and yet
its fundamental origin and link to the elementary particle dynamics is still
not fully understood. While enhanced deformation is evident in both heavy
nuclei and those away from closed shells, as recognized early on Mottelson_NP
; Elliott58 , deformed configurations are present even in a nucleus such as
the doubly closed-shell 16O, which is commonly treated as spherical in its
ground state, and whose first excited state is the head of a strongly deformed
rotational band. This shape coexistence of states of widely differing
deformation in many nuclei is now well established HeydeW11  and it has been
argued that it occurs in nearly all nuclei Wood16 .

Moreover, while vibrational spectra are often associated with spherical
nuclei, the fact that a nucleus has a zero quadrupole moment in its $0^{+}$
ground state does not imply that it is spherical in its intrinsic frame; for
quantum mechanical reasons, it appears spherical in the lab frame. With an
expanding body of experimental evidence, it is becoming clear that non-zero
deformation is far more widespread than zero deformation RoweW2010book .
Deformation is also found in the so-called “islands of inversion”, the regions
of the nuclear chart where shell evolution brings two major shell blocks, such
as $sd$ and $fp$ shells, together (for further discussion, see, e.g.,
physics4030048 ). The preference of the nuclear force for spatially symmetric
and low-spin configurations, which are energetically favored, leads to an
enhancement of quadrupole couplings and a spontaneous symmetry breaking in the
intrinsic frame, as observed in Elliott’s model (similarly in the Jahn-Teller
effect). The emergence of deformation and the subsequent dominance of
collective rotational degrees of freedom in the nuclear spectrum reflects the
presence of an effective separation of scales, which can be exploited to
simplify the description of deformed nuclei.

Interestingly, unambiguously claiming the presence of deformation in an atomic
nucleus based on energy spectra can be challenging. For instance, phenomena
like shape coexistence can affect energy levels, and disentangling vibrational
and deformation effects can be difficult.

On the theory side, one of the best ways to detect deformation is to study the
neutron and proton quadrupole (or higher) moments. When such quantities can be
extracted from, for example, ab initio calculations, results can then be
matched with simpler collective models to extract additional information. Such
matching can be made rigorous using the effective field theory framework
weinberg1990 ; ordonez1992 ; HammerReview2020 , as was done, for instance, in
Ref. papenbrock11_1310 ; papenbrock2016  for the particle-plus-rotor model.
However, in exotic nuclei, and in particular in unbound states, measuring
quadrupole moments might not be an option, and one has to rely on the
observation of rotational bands extending in the continuum. The study of how
continuum couplings affect deformation luo21_2394 ; fossez22_2540  and
rotational motion fossez16_1335 ; fossez16_1793  will be critical for the
interpretation of experimental data at FRIB.

Apart from quadrupole moments, but directly related to them, certain
electromagnetic transitions–such as the electric quadrupole (E2)–between
states at low excitation energy are also typically considered signatures of
collective behaviour eisenberg1976 . However, such calculations remain
challenging in various first-principle approaches scaling polynomially with
model space sizes, as can be seen in, for instance, valence-space in-medium
similarity renormalization group stroberg2017  and coupled-cluster theory
hagen2014  calculations, which systematically underestimate experimental data
stroberg2022 . One of the culprits behind this discrepancy is the difficulty
to capture both ”static” and ”dynamical” correlations cordoba2016  and hence,
collectivity. Dynamical correlations involve the mixing of a dominant
configuration with small (but energetically significant) contributions coming
from other configurations. This is what ab initio many-body methods as the
ones mentioned above are designed to do. In this framework, in fact, one
starts from a reference state (usually a Hartree-Fock Slater determinant), and
includes dynamical correlations adding particle-hole excitations on top of the
reference. Static correlations, instead, are harder to capture: they emerge
when there are two or more equally relevant configurations, as, for instance,
the degenerate states produced by rotating a symmetry-breaking reference. Some
progress in this direction has already been made. As an example, static
correlations can be included in the reference state, as was done in the in-
medium generator coordinate method yao2018 . Restoration of the broken
rotational symmetry, e.g. via angular momentum projection is another strategy
that can be pursued to capture such correlations. For instance, recent
coupled-cluster computations, starting from axially-symmetric reference states
and including angular momentum projection, reproduced the ground-state
rotational band of 34Mg hagen2022 . This result is particularly relevant
considering the ongoing experimental investigations of magnesium isotopes,
e.g. 40Mg, at FRIB.

Figure 5: a. Emergence of almost perfect symplectic symmetry in nuclei from
first principles launey16_2403 ; LauneyMD_ARNPS21  (figure adapted from Ref.
DytrychLDRWRBB20 ), enabling ab initio descriptions of clustering,
collectivity, and continuum in (b) exotic nuclei as 10He and (c) intermediate-
mass nuclei as 20Ne. d. Experimental and theoretical $B(E2\mathchar
24635\relax\;5/2^{+}\to 9/2^{+})$ values for 21Mg and 21F; the ab initio SA-
NCSM calculation is shown without the use of effective charges (figure from
Ref. Ruotsalainen19 ). e. Energy and cluster structure of the lowest-lying
$0^{+}$ states in the challenging 8Be (figure adapted from Ref.
PhysRevLett.128.202503 ). Fig. (a), (d), and (e) are reproduced with
permission.

To capture both static and dynamical correlations in medium-mass ab initio
calculations, it is advantageous to use a physically motivated, correlated
basis to solve the many-body problem. This is the case in the symmetry-adapted
no-core shell model (SA-NCSM) (reviewed in launey16_2403 ; LauneyMD_ARNPS21 ),
which exploits symmetries of the nuclear dynamics (Fig. 5a). Specifically, ab
initio descriptions of spherical and deformed nuclei up through the calcium
region are now possible in the SA-NCSM without the use of interaction
renormalization procedures and effective charges. Using the SA basis, one can
reach higher shells necessary for the accurate description of clustering,
collective, and continuum degrees of freedom (dof’s), and to reach heavier
nuclei such as 20Ne, 21,22Mg, 28Mg, as well as 32Ne and 48Ti LauneyMD_ARNPS21
(e.g., Fig. 5b & c). These studies provide collective observables, such as E2
transition strengths and quadrupole moments, as well as $\alpha$-decay partial
widths and ANCs DreyfussLESBDD20 ; PhysRevLett.128.202503  (Fig. 5d & e). The
SA eigenfunctions are represented in terms of all possible deformed
microscopic configurations, including zero deformation and single-particle
dof’s, and directly reveal the deformation content of a nucleus (Fig. 5).
This, in turn, provides a straightforward way to detect deformation. Indeed,
any microscopic model through projections onto the SA basis can identify
deformation, even in $0^{+}$ states. The critical importance of dynamical
correlations is recognized when the SU(3) SA basis DraayerLPL89 ; DraayerW83
is further reorganized according to the symplectic $\text{Sp}(3,\mathbf{R})$
group RosensteelR77 , and highly regular patterns in wave functions become
strikingly clear. In particular, a single nuclear shape (or only a few shapes)
is found to dominate low-lying nuclear states (Fig. 5) DytrychLDRWRBB20 .
Collectivity is almost completely driven by these shapes, and typically only a
few hundred shapes or less are needed for convergence of various short- and
long-range observables. Remarkably, it is the dynamical deformations that,
when properly developed, drastically decrease the energy of a nuclear shape
(the 4p-4h enhanced static deformation of the Hoyle state lies at $\sim 30$
MeV excitation energy or higher, whereas the vibrations lower it to $\sim 7$
MeV); in addition, as shown in Ref. Rowe_book16 , without vibrations the
quadrupole moment is underestimated roughly by a factor of two. This factor is
often accommodated by various models by an effective charge or starting from
an effective static shape.

In short, state-of-the-art many-body methods can provide insight in how
deformation emerges in exotic nuclei and how the associated few-body dynamics
affect their properties. They can also provide inputs for effective approaches
with collective degrees of freedom. However, difficulties remain in either
capturing deformation from first principles, or in dealing with couplings to
continuum states near particle emission thresholds. In this regard, it would
be worth exploring new strategies to include static correlations in ab initio
calculations of deformation-relevant observables, as quadrupole moments and
B(E2) transitions, with an eye to the reduction of their computational cost.
This effort should be accompanied by robust estimates of theoretical
uncertainties, which play a key role in the comparison to experimental data.

### Clustering

Parallel to collectivity emerging in rotovibrational nuclei is the phenomenon
of nuclear clustering. Traditionally, clustering refers to the formation of
distinct substructures within atomic nuclei; for example, states in 20Ne
having a 16O+4He character. Naturally, such states are expected to form in the
proximity of the respective decay threshold ikeda68_2329 ; oertzen06_1017 ;
freer07_1018 ; okolowicz13_241 ; okolowicz12_998 , however they are also
observed at higher energies johnson09_1479 . An even more extreme example of
clustering is seen in halo nuclei, where the clustered state lies slightly
below the decay threshold, leading to spatially extended nucleus that can be
well described in a core plus particle(s) picture.

From a practical point of view, clustered states can significantly impact
astrophysical reaction rates since clustering typically translates to
relatively large decay widths Kubono1995 ; Descouvemont2008 ; Shen2020 ;
DreyfussLESBDD20 . This is perhaps most famously exemplified in the case of
the Hoyle state, which has a pronounced 3$\alpha$ cluster arrangement allowing
for the triple-$\alpha$ reaction to proceed in solar environments, bypassing
the mass-eight gap. Indeed, the population of nuclear states in, for example,
transfer reactions–one of the main tools to be employed at FRIB for the study
of exotic nuclei–is expected to be directly proportional to the degree of
clustering of a state (typically quantified in terms of spectroscopic factors)
endt77_2579 .

Thus, a concrete understanding of how clustering (and other types of
collectivity) emerge from single particle degrees of freedom and the nuclear
interaction, and how the resulting interplay of the bound and continuum parts
of the nuclear wave function affects observables is a key target of nuclear
theory in the coming years. However, building such an understanding requires
both advances in few- and many-body nuclear theory as well as a significant
amount of new experimental data, especially along isotopic chains and near the
driplines where cluster phenomena may dominate the spectrum.

On the theory side, first steps have already been made for the microscopic
description of clustered states in light nuclei, both in phenomenological and
ab initio approaches, for example Refs wiringa13_1182 ; kanadaenyo12_2568 ;
elhatisari15_2324 ; elhatisari16_2311 ; kravvaris17_1960 ;
PhysRevLett.128.202503  to name a few, and in intermediate-mass nuclei
LauneyMD_ARNPS21 . Despite this progress, a lot remains to be done.

In particular, the competition between various forms of clusterization remains
a pressing issue, as this type of shape coexistence can serve to quench or
amplify clustering effects Kravvaris2017 . Furthermore, while first ab initio
studies of how clustering evolves along an isotopic change have been performed
Elhatisari2017 , the pertinent question of how composite clusters manifest in
the nuclear medium remains. Moreover, as origins of $\alpha$-clustering have
been investigated in mean-field approaches Ebran2012 , it is relevant to draw
parallels to the fundamental nuclear interactions and map out the connection.
Last, but definitely not least, the observed signatures of $\alpha$-clustering
in heavier nuclei vonTresckow2021  suggest that there may still exist new
mechanisms, that are significantly different than the ones active in light
systems and cannot yet be accessed in ab initio approaches, marking the need
for further investigations.

As a particular example where collectivity manifests itself one can focus on
dipole-excited states. In such collective states the analysis of the so-called
“cluster sum rule” alhassid1982 ; hencken2004  can give some insight on how
clustering phenomena emerge from first principles. This sum rule becomes
particularly interesting when looking at halo nuclei, which are usually
modeled as a tightly-bound core, surrounded by one or more weakly-bound
nucleons. The ratio of this cluster sum rule with respect to the full energy-
weighted dipole sum rule of the nucleus provides an estimate of how much of
the dipole strength is given by the relative motion between the core and the
halo. Here it is worth pointing out that taking ratios of observables instead
of their absolute value allows for the suppression of systematic errors
PhysRevLett.128.202503 ; caprio2022 ; capel2011 , and thus leads to more
robust predictions. Such an analysis has been done in a recent work on the
halo nucleus 8He bonaiti2022 . There, the discretized dipole response of 8He
was computed in the framework of coupled-cluster theory, merged with the
Lorentz Integral Transform (LIT-CC) bacca2013 ; bacca2014 . The evaluation of
the cluster sum rule in this case pointed out that around 1/3 of the total
dipole strength in 8He is determined by the relative motion between the 4He
core and the four-neutron halo.

### Halo structures

The definition of a halo state, usually accepted as a state in which one or
few nucleons have a 50% probability of being found in the classically
forbidden region of the potential generated by the rest of the system, is not
without ambiguity jensen04_233 ; canham08_2450 ; frederico12_372 ;
tanihata13_549 . First, the classically forbidden region is not so clear for
more than one nucleon in the halo, and second, the rest of the system can be
itself made of more than one cluster. It could be argued that the latter
situation is found in the first excited state of 11Be fukuda91_1605 , which
can be seen as one neutron halo around ${}^{10}\text{Be}=2\alpha+2n$. Another
challenge to the accepted definition of a halo state is the presence of
deformation in the core, as mentioned previously. While it can be shown that
the deformation of the core becomes irrelevant in the limit of a vanishing
binding energy between the core and halo nucleon misu97_1181 , in practice
many weakly bound nuclei have binding energy sufficiently large for couplings
between the deformed core and the halo nucleon to play a role as, for example,
in 17C suzuki08_1557  or 31Ne nakamura09_1513 ; takechi12_1512 . When
deformation matters, the deformed Nilsson orbits in which halo nucleons live
can be represented as linear combinations of spherical shells, and can include
a mix of $\ell=0$ or $\ell=1$ shells with $\ell>1$ shells. The question then
becomes whether or not we still have a halo system if $\ell>1$ shells have a
significant occupation and do not extend far into the classically forbidden
region. In a naive classical picture halo nucleons would be on an elliptical
orbit passing close to the core and extending far from it. From a theoretical
point of view, clarifying the definition of a halo structure is critical as
FRIB will allow pursuing the exploration of drip lines into the medium-mass
region and is likely to uncover many deformed halo states, potentially
involving several neutrons in the halo. Clustering is usually expected to play
a smaller role in medium-mass nuclei based on the dominance of the mean-field,
but this is not necessarily true in excited states and especially if states
are found just below decay thresholds as in 11Be. To summarize, testing the
boundaries of what is considered a halo state should be an experimental
priority to stimulate progress in our understanding of halo physics. On the
theory side, priority should be given to the development of more comprehensive
models of halo structures through the development of effective fields theories
going beyond existing halo EFTs bertulani02_869 ; ryberg14_997 ; ji14_1101 ;
Hammer2017 , and of many-body method capable of describing such physics from
nucleons as degrees of freedom more efficiently than current approaches.

### Making the connection with nuclear forces

Besides the challenge of understanding how few-body effects emerge from the
many-body dynamics, there is the question of which specific components of
nuclear forces drive collective phenomena. For example, sensitivity studies on
the low-energy constants (LECs) of a two-body chiral EFT have been carried out
for the quadrupole moments of 6Li and 12C using the SA-NCSM BeckerLED22 ;
Becker23 . This analysis demonstrated, on the one hand, that the quadrupole
moments are predominantly sensitive to only two of the LECs parameterizing
chiral interactions at NNLO. On the other hand, simultaneous variation of all
14 LECs at this order has little effect on the dominant nuclear shape(s) and
the associated symplectic symmetry. It remains to be seen if the extension to
other observables, such as quadrupole and dipole transitions, will lead to the
same outcome. Some recent studies on 6Li based on NCSMC PhysRevLett.129.042503
and HH PhysRevC.102.014001  showed the strong dependence of clustering
observables to the tensor terms of the two and three-body nuclear
interactions. In Ref. PhysRevC.102.014001  it has been shown that the strength
of 6Li quadrupole moment is directly connected to the strength of the tensor
forces in the NN nuclear interaction. A similar effect is expected to be
determined also by three-nucleon interactions. Indeed in Ref.
PhysRevLett.129.042503  the comparison of the ratio of the ANCs obtained with
the NN interaction only and NN+3b interaction shows an increase of the
strength of the $D$-wave component in 6Li which suggests that three-nucleon
interactions play an important role too in determining cluster and deformation
observables. Clearly, these studies are limited only to 6Li and should be
extended to more systems and chiral interactions to define a general trend.
Therefore, in the future, it would be interesting to extend the LECs
sensitivity studies of the SA-NCSM, which have the possibility to deal with
larger and more deformed systems, also to individual terms of the two and
three-body nuclear interaction, i.e. single diagrams of the chiral expansion,
in particular the ones that give rise to tensor forces. In addition, this type
of investigation should be performed on other systems with different
properties to better understand how individual interaction terms give rise to
collective behavior across the Segre chart. We note that the problem of the
prohibitive computational cost of such calculations can be contained thanks to
the reduced basis method known as eigenvector continuation (see for example
the symmetry-adapted eigenvector continuation (SA-EVC) case BeckerLED22 ),
which provides the basis to build efficient and precise emulators of the many-
body method Frame:2017fah ; Konig:2019adq ; Ekstrom:2019lss .

Another way to study the impact of specific components of nuclear forces on
emergent properties in exotic nuclei is to disentangle the role of two- and
three-body forces. It appears that two-body forces alone seem to be able to
generate most emergent phenomena observed in exotic nuclei, which is a
consequence of the fact that many of these systems live close to the unitary
limit and manifest signs of universal behavior (see Sec. 8). However, as
discussed earlier, in exotic nuclei where an interplay between continuum
couplings and standard phenomena such as deformation and clustering is
present, the situation becomes more complex (for instance in so-called
deformed halo states where one or more nucleons form a halo around a deformed
core whose rotational motion can have an impact on the dynamics of the halo,
as is the case in 40Mg crawford14_1607 ; crawford19_2359 ; refId0 ). Indeed,
ab initio calculations suggest the need of three-body forces in specific two-
body parameterizations to obtain an accurate description of nuclear structure
in general, but even more so in nuclei where clustering is present such as 10B
navratil07_733 ; mccutchan12_2321 ; freer18_2138 , where it was found that
using ${(n,\alpha)}$ scattering data to determine three-body forces improved
the ordering of the low-lying states carlson15_1610 . Unfortunately the role
of three-body forces in collective phenomena has not been fully investigated
yet, but this knowledge will be necessary to understand many exotic nuclei at
the drip lines.

Finally, we can also turn the tables and use emergent phenomena in exotic
nuclei to constrain nuclear forces. For example, widths of resonances depend
exponentially on the associated decay thresholds, making them difficult to
describe and fairly sensitive to details of the interaction. This sensitivity
can be exacerbated when considering multi-nucleon resonances, like the famous
two-neutron decay of 26O guillemaud90_1761 ; lunderberg12_556 ; kohley13_1541
; caesar13_1765 ; kondo16_1439 , which only happens because the energy pattern
of the ground states of 24O and 25O, confined within a few keV, allows it.
Even more exotic patterns can be observed in neutron-rich helium isotopes
where the presence of a large number of neutrons in 9He kalanee13_1909  leads
to a situation where it is still unclear whether or not the lowest state
observed, with $J^{\pi}=1/2^{+}$ and strongly impacted by $s_{1/2}$ partial
waves, is a resonance or a virtual state and can thus be considered as the
ground state vorabbi18_1977 ; fossez18_2171 ; votaw20_2353 . While testing
nuclear interactions with precision on such phenomena remains challenging,
they can shed some light on which parts of the nuclear forces we have not yet
fully under control and give us some hints on how to constrain them.
Therefore, when it is possible, we should post-select interactions reproducing
data on exotic phenomena.

##  3 Understanding few-body dynamics in knockout and transfer reactions

T. Frederico, B. Kay, Ch. Elster, L. Hlophe, M. Higgins, and S. König

### Overview and motivation

Low-energy universal dynamics are present in a nuclear system when the degrees
of freedom for the driving physics are configurations characterized by length
scales larger than the range of the forces involved in the process
jensen04_233 ; frederico12_372 . In EFT this is associated with long
wavelength scales in weakly bound systems and a short wavelength scale that
allows a systematic expansion of the scattering amplitude, including in the
LECs, for example, effects from the effective ranges, besides the scattering
lengths or low-lying resonances taken into account directly at leading order
(see e.g. Hammer2017 ; HammerReview2020 ). A manifestation of this
universality are so-called “halo nuclei” states, which can be thought of one
or more valence nucleons loosely bound to a core with lowest excitation energy
much smaller than the separation energy of the core valence nucleons. The low-
energy continuum of such few-body halo states may exhibit the properties that
are model independent and closely related to the bound halo formation, namely
excitation energies of few hundreds of keV. These properties can be probed in
$(d,p)$ and $(t,p)$ reactions, and other processes, where the projectile gets
one or two neutrons and eventually breaks. In this section we discuss
possibilities to study universality from that angle, suggesting new
experiments and perspectives for accompanying theoretical efforts.

The study of $(d,p)$ and $(t,p)$ reactions, and also of nucleon or nucleons
removal processes, will allow to learn about the evolution of universality in
the formation of neutron halos along isotopic chains and their final state
interaction (FSI) as the continuum counterpart to the bound halo state. The
outcome of such investigation will be the determination of the minimal
conditions from the nuclear force that allows to form the neutron halos and
their continuum counterparts. Possibly, much more relying on the on-shell
properties of the neutron-core interaction, which may be determined as
essential for the halo formation along the isotopic chain and for the
continuum properties at low excitation energies, as long as universality and
model independence holds. We will be able to provide the conditions for the
stability of neutron-halos in the presence of a core, and also the continuum
features, as for example, how the tetraneutron properties manifests in the
presence of a core and its relation to the formation of neutron rich nuclei.
At the same time, the involved few-body reaction mechanism will challenge our
methods due to the explicit presence of the halo neutrons, which will require
further theory developments.

### Suggested experimental studies

Various known halo nuclei are interesting candidates to study final-state
universality, like for example 6He$(t,p)$8He∗, with
8He${}^{*}\,=\,\alpha$+4$n$, or other situations as 9Li$(t,p)$11Li∗,
11Li${}^{*}=\,^{9}$Li+2$n$, where we are interested in the physics revealed by
the fragments FSI, which comes from the transfer to the continuum. These final
states with low-excitation energies can show up a diverse range of universal
phenomena, which we discuss further using some typical examples: (i) 11Li∗,
(ii) 19B∗ and (iii) 8He∗ (the ∗ indicates excitation to the continuum).

In the case of 11Li${}^{*}\,=\,^{9}$Li+$n$+$n$, the dominant interactions will
be on the s-wave. Indeed for 10Li${}^{*}\,=\,^{9}$Li+$n$ already the virtual
$s_{1/2}$ state has been studied BarrancoPRC2020  in a 9Li$(d,p)$ reaction,
where the scattering length of $-8\,$fm was found. The 9Li+$n$+$n$
characterizes a three-body system in the continuum, which is dominated at
hundreds of keV by the s-wave physics, considering in addition to the 10Li
s-wave virtual state the dominant 1S0 neutron-neutron channel. It is expected
that the final state interaction within the 9Li+$n$+$n$ system is dominated by
the Efimov physics Efimov:1990rzd , when the interaction is resonant in the
s-wave, namely the scattering lengths have large magnitudes with respect to
the interaction range. Although there is not yet a two-neutron halo system,
Borromean or Samba (20C), which can have an Efimov excited state AmorimPRC1997
; frederico12_372 ; Hammer:2022lhx , still would be possible that Borromean
systems present an Efimov-like continuum resonance for large and negative
scattering lengths Bringas:2004zz ; Deltuva:2020sdd ; Dietz:2021haj , which on
the theory side has to be extended to mass-imbalanced systems. It could well
be that these continuum resonances, associated with Borromean states, would be
broad and hard to be seen, although the few-body universal Efimov dynamics
will drive the FSI in core+$n$+$n$ systems with the quantum numbers of the
ground state. We return to discuss in Sect. 8 the possibility to form an
excited Efimov state by removing nucleons from the core, while keeping the
neutron halo intact during some time.

In particular, we should mention the 17B+$n$+$n$ system, which has a shallow
Borromean bound state of 19B HiyamaPRC2019  and a large n$-\,^{17}$B
scattering length. The transfer to the continuum in the reaction
17B$(t,p)$19B∗ close to the threshold could eventually populate a nearby
Efimov continuum resonance. In this case it will be desirable to have
measurements for 17B$(d,p)$18B∗ to achieve a better accuracy in the scattering
length, as has been done for 9Li$(d,p)$ BarrancoPRC2020 . Increasing the
number of neutrons in the final state, for example with the transfer reaction
19B$(t,p)$17B+4$n$, at low relative energies, the FSI in 17B+4$n$ system,
should be determined by $S_{2n}$ in 19B, the $nn$ and $n-^{17}$B scattering
lengths, as suggested by the dominance of the universal Efimov physics.
Curiously enough, when the nn system is at unitarity the neutron cloud turns
to be a system within “unnuclear physics” Hammer:2022lhx . Therefore, it seems
quite interesting the situation where the nuclear core is in the presence of a
neutron cloud with 2$n$, 3$n$, 4$n$,… dominated by “unnuclear” FSI effects at
low relative energies, which correspond to small excitation energies in the
continuum of the core+$Nn$ $(N\geq 4)$, going beyond the drip line. Indeed the
much debated tetraneutron, which is now understood as a strong correlation
between the free neutrons DuerNat2022  could reveal itself in the presence of
the core, when the core+4$n$ is at low excitation energies. The detection of
the charged core can provide further information of the interaction of a quasi
“unnuclear” system, i.e. the neutron cloud, with the core nucleus.
Furthermore, it may well be possible that a core+$n$+$n$ continuum Efimov
resonance can be formed in the presence of the two other neutrons at low
relative energies from the $(t,p)$ reaction when the n-core scattering length
is large and negative.

Among other possibilities which are not Efimov-physics dominated we mention
the 6He$(t,p)$4He+4$n$ reaction close to the scattering threshold where, the
low-lying p-wave 5He resonance is the main interaction channel of the $\alpha$
with the neutron cloud. The model independent and universal correlations are
enriched with low-lying resonances in the subsystems, like the 5He $p$-wave
one necessary to establish the properties of 6He. Furthermore, such a low-
lying p-wave resonance may possibly dominate the 8He properties in a
$\alpha+$4$n$ description.

### Overview of available theory approaches

The theory should be able to compute the inclusive cross-sections in $A(d,p)$,
$A(d,n)$, $A(t,p)$ and $A(t,n)$, where for example the final states $(A+Nn)$∗
or $(A+n+p)$∗ can be achieved. If the projectile is a weakly bound one- or
two-neutron halo, the different processes are possible $(A+n)(d,n)(A+2n)$∗,
$(A+n)(t,n)(A+3n)$∗, $(A+2n)(d,n)(A+3n)$∗, $(A+2n)(t,n)(A+4n)$∗, and
eventually it will require up to six-body calculations. The inclusive cross-
sections where the proton is observed and the excitation spectrum is obtained,
is one of the possibilities, and if in addition the charged core is detected,
further information on the FSI of the core and neutrons can be revealed. The
angular distribution of the core as well as the proton at low excitation
energies of the system ($A+Nn$), can be a tool to select the angular momentum
states of the neutron cloud and the core nucleus. It is interesting to observe
that at non-zero total orbital angular momentum of the core+$Nn$ system the
effect of the FSI at low excitation energies is even less sensitive to the
details of the potential, beyond the low-energy on-shell pairwise observables,
enhancing the universality of the expected results. Developments were done
incorporating the halo-EFT in reaction models for one-body halos Capel:2018kss
; Moschini:2018lwh ; Moschini:2019tyx ; Capel:2020obz ; Hebborn:2021mzi ,
which has to be generalized to two or more neutron halos, to allow the
investigation of the N-neutron cloud properties in the presence of the core.
Effects as few-body forces induced by the core excitation can be for example
incorporated by the theory Capel:2020obz , however the systems worked out
recently within EFT are restricted to one-body halos, as has also been done in
extracting the virtual state of 10Li from the excitation spectrum in a $(d,p)$
reaction BarrancoPRC2020 .

The s-wave interactions in the limit of zero range collapse the three-body
system, and therefore a three-body information has to be included as a short-
range scale. The low-energy observables will be correlated in an universal
form and independent on the details of the short-range interaction once the
two-neutron separation energy and the scattering lengths are given. This, for
example, provides the different radii in the core-n-n weakly bound nuclei
frederico12_372 . Corrections due to the effective range are expected and
already studied HammerReview2020 . These universal correlations are
represented by scaling functions, which are limit cycles from the breaking of
the continuous scale symmetry to a discrete one frederico12_372 ;
HammerReview2020 . Systems including more neutrons will not exhibit further
sensitivity to short-range effects beyond the three-body one, as the Pauli
principle does not allow more than two neutrons be on the top of the other.
Therefore, at leading-order a system like 18C+4$n$ for low excitation energies
will only depend on the two neutron separation energy in 20C and the nn
scattering length, in a model independent way. One can be reminded of the
Bertsch unitary Fermi gas (see BakerPRC1999 ), for which the only scale is the
Fermi momentum.

Theory has to evolve to treat the five-body system at low excitation energies
to unravel the universal physics of the configuration of an unbound cloud of
four neutrons and a core nucleus. Of course, this should not be restricted to
light nucleus as the core, but to other nuclei, which the low lying excited
state can be a MeV or so above the ground state, without being restrictive.
The reaction mechanism has also to be taken care, as the quasi-free scattering
can be distorted by the remaining proton, in $(d,p)$ or $(t,p)$ reactions, as
FSI have to be accounted for. The elastic breakup and the incomplete fusion
processes may overlap in the kinematical region of low excitation energies.

#### Hyperspherical expansion methods

Hyperspherical methods have been extensively used in studying few-body physics
in a wide variety of atomic and nuclear systems, such as cold and ultracold
atomic collisions and low-energy states in light nuclei. Some of the types of
systems studied include three-body recombination PhysRevA.65.010705 , some
four-body recombination rates (i.e. the formation of Efimov trimers in the
recombination process $B+B+B+B\rightarrow B_{3}+B$ StecherNature ;
NatureKramer2006 ), and even five-body recombination Zenesini_2013 , dimer-
dimer collisions PhysRevA.79.030501 , universality in trapped two-component
fermionic systems PhysRevA.77.043619 , the BCS-BEC crossover problem
PhysRevLett.99.090402 , among other physical processes. In the nuclear
context, hyperspherical methods have been used to study bound states of light
nuclei up to $A=7$ PhysRevC.102.014001 ; KievskyARNP2021 , and few–nucleon
scattering processes such as $n+d$ PhysRevA.86.052709 , $t+p$ and $d+d$
elastic and inelastic collisions PhysRevC.102.034007 ; PhysRevC.105.014001 .
The hyperspherical approach has even treated scattering in the full $N$-body
continuum, as is the case for the three and four neutron systems
PhysRevLett.125.052501 ; PhysRevC.103.024004 . Existing formalism can also
accommodate systems of a core plus a few neutrons (i.e. processes where the
final state includes core+$Nn$), using model core+nucleon interactions.

The adiabatic hyperspherical treatment can treat local two- and three-body
interactions including but not limited to central, spin-orbit, and tensor
interactions. However, the adiabatic hyperspherical formalism in its current
development is incompatible with interactions that are non-local. A
Hamiltonian containing non-local interactions would lead to non-locality in
the hyperradial wave function and thus in the adiabatic hyperspherical
potential curves. It is unclear how to interpret non-local adiabatic
hyperspherical potential curves, which would actually be potential surfaces in
hyperradii $\rho$ and $\rho^{\prime}$. While the current treatment of the
hyperspherical method in an adiabatic representation is incompatible with non-
local interactions, semi-local $l$-dependent potentials in a partial-wave
expansion can be used, like in the case of the ${}^{10}\text{Be}+n+n$ system
to study states of ${}^{12}\text{Be}^{*}$ PhysRevC.77.054313 ;
PhysRevC.86.024310 .

With the adiabatic hyperspherical approach, there are a vast number of nuclear
systems and reaction processes that can be studied. A class of reactions that
would be of interest to study are knockout reactions such as the four-body
processes ${}^{6}\mathrm{He}(p,p\alpha)nn$ and
${}^{10}\mathrm{Be}(^{3}\mathrm{H},p){{}^{12}\mathrm{Be}^{*}}$ to probe two-
neutron halo dynamics in
${}^{6}\mathrm{He}+p\rightarrow({{}^{4}\mathrm{He}}+n+n)+p$ and
${}^{10}\mathrm{Be}+{{}^{3}\mathrm{H}}\rightarrow({{}^{10}\mathrm{Be}}+n+n)+p$.
In principle adiabatic hyperspherical methods can be pushed to more than four
particles. In this case, the computational limitations of this method would be
put to a rigorous test since the vast majority of studies using this approach
are for systems with $N=3$ or $N=4$ Rittenhouse_2011 . For systems larger than
$N=4$, only the lowest hyperradial channel has been studied (i.e. in studies
of alpha clusters up to $N=10$ BlumeGreene2000 ; PhysRevA.105.022824 ). There
are many nuclear reactions that are modeled as reactions of nucleons and
clusters with $N>4$ that are of experimental and theoretical interest. Such
systems include studying five-body systems like ${}^{8}\mathrm{He}$ to probe
the final state interaction of an alpha core interacting with four neutrons,
and five- and six-body reaction processes ${}^{8}\mathrm{He}(p,p\alpha)4n$,
and ${}^{4}\mathrm{He}(^{8}\mathrm{He},^{8}\mathrm{Be})4n$ (by modeling
${}^{8}\mathrm{Be}$ as an $\alpha\alpha$ cluster). The reactions
${}^{8}\mathrm{He}(p,p\alpha)4n$ and
${}^{4}\mathrm{He}(^{8}\mathrm{He},^{8}\mathrm{Be})4n$ would be of particular
importance to better understand the nature of the low-energy correlation of
four–neutrons seen in both the Kisimori et al. experiment in (2016)
PhysRevLett.116.052501  and subsequently the M. Duer et al. experiment in
(2022) DuerNat2022 .

#### Faddeev equations framework

The Faddeev formalism faddeev1960mathematical  provides a rigorous
mathematical formulation of the quantum mechanical three-body problem in the
framework of non-relativistic dynamics. Alt, Grassberger, and Sandhas
alt1967reduction  (AGS) introduced a momentum space formulation of the Faddeev
equations for scattering in terms of three-particle transition amplitudes. The
latter are convenient since they are smooth functions of the momentum contrary
to the original Faddeev wave function components. For a long time applications
of the Faddeev-AGS formalism to the nuclear scattering problem with realistic
potentials were limited by the difficulty of treating the long-ranged Coulomb
potential. A breakthrough was achieved deltuva2005momentum ;
deltuva2005benchmark  by developing the screening and renormalization
procedure for the Coulomb interaction in momentum space using a smooth but at
the same time sufficiently rapid screening. This approach has been
successfully applied to obtain ab initio solutions for the three-body
scattering problem (below and above the breakup threshold) for particles with
arbitrary masses deltuva2006coulomb ; deltuva2007three . Following the success
of the Faddeev-AGS approach, coordinate space techniques based on the Faddeev-
Merkuriev MERKURIEV1980395:1980  (FM) were developed and successfully applied
to the three-particle scattering problem lazauskas2011application ;
Deltuva2014  using complex scaling Nuttall:1969nap ; Balslev:1971vb  to handle
the complicated boundary conditions above the breakup threshold.

On one hand, the FM method has been extensively applied to lighter systems
(i.e. three nucleons or a light core plus two nucleons). On the other hand,
the Faddeev-AGS equations have been successfully used to describe nuclear
reactions involving a broader range of nuclei, from 4He to 58Ni. Limitations
are primarily due to the utilization of the screening and renormalization
procedure needed to treat the long-ranged Coulomb potential. Numerical
instabilities limit the applicability of this approach to lighter systems
where the Coulomb potential is not too strong. An alternative approach to
incorporating the Coulomb potential was already suggested by AGS
alt1967reduction , which involved reformulating the Faddeev-AGS equations in a
basis of Coulomb scattering wave functions by adopting a separable expansion
of the pairwise potentials. A more rigorous development of the method was
presented in Refs. Mukhamedzhanov:2000qg ; Mukhamedzhanov:2000nt  and was
later extended to incorporate core excitations Mukhamedzhanov:2012qv . This
approach has the advantage that the Rutherford scattering amplitudes are
isolated and treated analytically so that only the well-behaved short-ranged
three-body scattering amplitudes are computed numerically. The need for
screening the Coulomb potential is thus eliminated allowing the approach to be
applied to heavier systems (and lower energies) where the Coulomb effects are
very strong. Recently, a practical implementation of this formulation has been
carried out Hlophe:2022fdn  and extensions above the breakup threshold are
nearing completion. Further, the Faddeev-AGS equations are suitable for
generalizations to systems with more than three clusters and a formulation for
four clusters was already developed in the seventies Alt:1970xe . The Faddeev-
AGS approach is thus suitable for computing the aforementioned deuteron-
induced transfer and breakup cross sections for processes, namely, $A(d,p)$,
$A(d,n)$, and $A(d,np)$.

Following their successful application to three-cluster systems, the Faddeev-
AGS equations have been used to provide an exact description of four nucleon
reactions Deltuva:2007xv ; Deltuva:2006sz . Application of the Faddeev-AGS
formalism to systems with four clusters, e.g., $A$+$n$+$n$+$n$, is thus
feasible in principle. Since the number of two-body and three-body partial
waves increases with the size of the system, the feasibility of computing
four-cluster reactions involving heavy nuclei still needs to be tested. An
implementation of the four-particle Faddeev-AGS equations would enable the
description of reactions involving, e.g., one-neutron halos such as
$(A+n)+d\longrightarrow A+2n+p$. The Faddeev-AGS equations can in principle be
generalized to an even larger number of clusters (e.g., five or six clusters).
Such calculations maybe feasible especially for systems of the type
$A$+$3n$+$p$ and core+$3n$+p, where the number of coupled equations are
reduced considerably due to the identical nature of the external nucleons.

The generalization of Faddeev-AGS to more complex systems, four and five-
particles, is crucial to understand/describe exotic nuclear structures close
to the driplines and central to FRIB science. However, such developments have
not yet occurred and the Faddeev-AGS description of reactions involving five
and six clusters remains a subject for future investigations. We note that a
four and five-particle generalization of the FM equations, the coordinate
counterpart of the Faddeev-AGS equations, has been successfully applied to
four- and five-nucleon scattering Lazauskas:2012jc ; Lazauskas:2017paz ,
although the five-nucleon scattering has been limited to low energy elastic
scattering.

##  4 Continuum and cluster effects in reaction theory

G. Potel, L. Hlophe, and J. Lubian Rios

Not surprisingly, our experimental understanding of the physics of clusters
and halos in nuclei derives mostly from nuclear reactions. Clustering effects
can be probed, for example, in cluster transfer reactions, resonant
scattering, and through the interplay of incomplete fusion (ICF) and complete
fusion (CF).

When a clustered nucleus collides with a target, the fusion process can
proceed both through CF, where the whole projectile fuses with the target, or
through ICF, where only one of the fragments undergo fusion, while the rest of
the projectile flies away. The relative importance of these two contributions
depend strongly on the cluster dynamics of the process, as well as on the
degree of clustering of the projectile. This problem gathered attention in the
80’s, when the Ichimura Austern Vincent (IAV) IAV85  model was developed. This
model was recently revived by the Sevilla group to determine the CF for the
6,7Li +209Bi LMo15 ; LMo17 ; LMo19 , and has also been applied to the study
inclusive breakup cross sections in reactions induced by 6He and 6,7Li in Ref.
SCC21  and by 17F in Ref. YLY21 .

The Rio group was also able to compute the CF as well as the ICF of each
fragment for the case of projectiles breaking into two fragments. This model
has been applied to the study of fusion of 6,7Li with heavy targets (209Bi,
197Au, 198Pt, and 124Sn) with success RCL20 ; CRF20 ; LFR22 . The effect of
the breakup channels (continuum) on the fusion cross section has been
discussed in detail, and it has been shown that the CF is hindered above the
barrier due to the dynamic effects of the couplings of the elastic channel
with the continuum, that produces a repulsive polarization. It was also shown
that the cluster structure of these projectiles produces a static effect that
reduces the Coulomb barrier (VB) and makes it thinner, producing an
enhancement at energies above and below the barrier. The net effect is a
competition between static and dynamic effects. The total fusion (defined as
the sum of CF and ICF) is enhanced at energies below VB because the ICF
becomes dominant and larger than the CF. However, it is essential to have more
exclusive and inclusive cross sections for CF, ICF, TF, elastic breakup and
nonelastic breakup that would put strong constrains on the theoretical
calculations.

Breakup effects are particularly important for weakly bound nuclei, to the
extent that clustered configurations are often present in weakly bound and
halo nuclei. However, present theoretical models for the derivation of CF and
ICF were developed for two-body projectiles, and the extension to projectiles
that break into more than two fragments is still pending.

The effects of the breakup channel on other reaction mechanisms are not the
same at different energy regimes and mass ranges. In the case of light
projectiles the CF and ICF cannot usually be separated because they have
similar decay modes, calling, therefore, for the development of new detection
techniques. In a more general case, where the target is deformed, the models
have to be improved in order to account for inelastic channels. The
competition of the breakup channel with transfer reactions, as well as core
excitation and sequential processes (breakup triggered by transfer) might also
be relevant and should be included in the present models.

These reaction calculations rely on a variety of nuclear structure
ingredients. Optical potentials are needed to describe the interaction between
the fragments and the target, and the double-folding São Paulo has been
successfully used in this context (SP) CCG02 . Its first version uses a
systematic parametrization for matter density distributions for the projectile
and target, based on experimental data and Dirac-Hartree-Bogolyubov
calculations. In order to extend the applicability away from stability, a new
version of the SP potential was published CCG21  where the specific matter
densities for each specific nucleus near the drip-line were included. Within
this context, fragment-target elastic scattering angular distributions could
be useful to constrain nucleus-nucleus optical potentials. In order to
generate the cluster-cluster and cluster-particle continuum states, Woods-
Saxon potentials are usually used. The parameters of these simple potentials
(depth, reduced radius, and diffuseness) are adjusted to describe experimental
binding energies, position and width of the resonances, root mean square
radius of the composite nucleus, reduced transition probabilities, multipole
moments, etc. The use of more microscopic wavefunctions would provide further
tests to the nuclear structure models in reaction calculations, as they can be
introduced as external information in some reaction codes. In order to
implement the aforementioned coupled channel calculations for deformed nuclei,
deformation parameters of collective motions are usually taken from systematic
such as Ref. RNT01  for $\beta_{2}$ and Ref. KiS02  for $\beta_{3}$. Transfer
reactions are also good candidates to study the cluster configurations of
exotic nuclei and pairing correlations, and the main ingredients and
experimental inputs needed for the corresponding theoretical description are
similar to the ones mentioned above.

When one or two neutrons are transferred to/from a one- or two- neutron halo
system, one has to add to the above ingredients a good description of the
continuum. In particular, an accurate quantitative account of the two-neutron
transfer cross section associated with Borromean two-neutron halo systems
requires the consideration of the successive transfer contribution through the
intermediate, one-neutron transfer, unbound channel Potel:10 ; Descouvemont:21
. The role of continuum is, unarguably, even more conspicuous when the one- or
two-neutron state populated in the final channel is unbound BarrancoPRC2020 ;
Cappuzzello:15 ; Barranco:17b . Among the techniques employed to describe
virtual or real states in the continuum in the contexts describe above, we
mention dicretization in a finite spherical box Potel:10 , Continuum
Discretized Coupled Channels (CDCC) calculations Descouvemont:21 ;
Cappuzzello:15 , and the Green’s Function Transfer (GFT) formalism
Barranco:17b ; BarrancoPRC2020 . Let us finally mention that the accuracy of
reaction calculations in these contexts relies on a good description of the
structure associated with the relevant unbound states. In particular, one
needs to describe one- and two-neutron resonances, and describe pairing
correlations in the continuum. An excellent example of such combined structure
and reaction challenges is given by the quest for the Giant Pairing Vibration
(GPV) Cappuzzello:15 ; Cavallaro:19 ; Assie:19 .

##  5 Spectroscopic factors, experimental probes and perspectives

C. Hebborn, G. Potel, D. Bazin, and A. Macchiavelli

The quantitative signature of clustering in nuclei is often expressed making
use of the concept of spectroscopic factors, which are also widely applied to
the study of single-particle degrees of freedom AUMANN2021103847 . Within this
context, transfer processes, quasi-free scattering, and knockout reactions,
which either add or remove one nucleon, play a prominent role in experiments
GM18 ; LeePRC06 ; Tetal09 ; PhysRevC.103.054610 . The standard theoretical
approach to describe these processes consists in combining the overlaps
computed within some structure formalism of choice with a description of the
collision process making use of appropriate reaction theory.

Systematic measurements of quasi-free scattering, transfer and knockout have
been done for nuclei exhibiting different binding energies from the valley of
stability to extremely exotic regions of the nuclear chart. The difference
between the theoretical cross section and the experimental one has been
interpreted as a quantitative measure of the missing many-body correlations in
the structure model. Arguably, the most striking feature of this analysis is
that the ratio between the experimental and theoretical knockout cross
sections exhibits a systematic trend as a function of the neutron-proton
asymmetry of the studied nucleus (see right panel of Fig. 6), which is still
not well understood PhysRevC.103.054610 ; AUMANN2021103847 . Since similar
analyses on electron-induced KRAMER2001267 , quasi-free GM18  and transfer
reactions Tetal09 ; LeePRC06 ; PhysRevLett.129.152501  do not show this trend
(see three left panels of Fig. 6), a possible deficiency in the theoretical
description of knockout reactions has been put forward AUMANN2021103847 . The
analysis of knockout reactions relies on shell model spectroscopic factors and
the high-energy eikonal approximation Hansen03 ; PhysRevC.103.054610 . In
order to understand the asymmetry dependence of the knockout cross section,
the validity of these approaches have been thoroughly analyzed for the last
two decades.

Figure 6: Quenching of the spectroscopic factor for (from left to right):
electron-induced knockout reactions, transfer reactions, quasifree knockout
reactions, and knockout reaction on composite target (typically ${}^{9}\rm Be$
or ${}^{12}\rm C$). This figure is taken from Ref. AUMANN2021103847 .

In the shell model, nucleons filling completely shells are treated as an inert
core, the rest of the nucleons are considered as active, and excited states
are obtained by elevating a nucleon to a higher shell B88 . With this
approximation, the nuclear structure is approximated by Slater determinants
built from various single-particle states, describing the active nucleons as
independent particles subject to a mean field generated by the inert core.
This simplistic view therefore approximates the correlations associated with
coupling of nucleons to high-momentum states arising from short-range nucleon-
nucleon interactions and long-range couplings involving collective
excitations, through the use of effective shell-model interactions.

Different studies have quantified correlations beyond the independent-particle
motion. In particular, microscopic structure studies PhysRevLett.107.032501 ;
PhysRevLett.103.202502 ; PhysRevC.92.014306 ; PhysRevC.104.L061301  showed
that many-body correlations arising from a combination of configuration mixing
at the Fermi surface and coupling to the continuum induce a reduction—so
called quenching—of the spectroscopic strength for deficient nucleon species.
Other studies PhysRevC.92.034313 ; PhysRevC.104.034311 ; PhysRevC.106.024324 ;
PhysRevC.106.024616  have also pointed out that spectroscopic factors are not
observables as they run with the resolution scale of the interaction.
Moreover, systematic studies of knockout cross sections have shown that these
observables are not sensitive to the interior of the single-particle wave
functions, and scale with the square of the asymptotic normalization constants
for halo nuclei and mean square radii for more bound systems
PhysRevC.100.054607 ; Hebborn:2021mzi ; HCinprep ; PhysRevC.77.044306 ;
PhysRevLett.119.262501 . Both the scheme dependence and these sensitivity
analyses highlight the need of a consistent description of structure and
reaction observables.

Other groups using phenomenological approaches have also studied the isospin
dependency of the spectroscopic strengths. In particular, the Dispersive
Optical Model (DOM), which treats consistently structure and reaction by
constraining an optical potential with both bound and scattering data, also
observes a quenching of the spectroscopic strengths for more deeply bound
nucleons ATKINSON2019135027 . Another phenomenological study
PASCHALIS2020135110  showed that the quenching of the spectroscopic factor in
the knockout of a deficient species is consistent with the recent results on
short-range correlations from high-energy electron scattering experiment.
However, all of the microscopic and phenomenological studies predict a rather
mild dependence of the quenching of the spectroscopic factor for different
proton-neutron asymmetry of nuclei, suggesting that the reaction theory
involved in the analysis of knockout data are also at stake.

The typical observables for knockout reactions are inclusive with respect to
the target states, i.e., they are summed over all possible target-nucleon
final states. The experimental cross sections are usually analyzed with the
eikonal approximation, which assumes that the projectile is only slightly
deflected by the target and thus that the projectile-target relative motion
does not differ much from the initial plane wave G59 ; Hansen03 . This model
is valid only at high enough energy, i.e., beam energy above 50 MeV/nucleon.
We want to stress here that two subsequent approximations are made in the
traditional eikonal description of knockout reactions Hansen03 : the adiabatic
and the core-spectator approximations.

The adiabatic approximation implies that the projectile’s intrinsic energy and
its velocity stay constant during the collision and, as a consequence,
violates energy conservation. The resulting parallel momentum distributions
are symmetric with respect to the maximum at zero momentum transfer, while
experimental ones exhibit some degree of asymmetry that becomes very apparent
for knockout of more deeply bound nucleon, for which the ratio between the
single-nucleon separation energy and the beam energy is smaller
PhysRevLett.108.252501 . Some works including in their analysis this mismatch
between initial and final energies obtained theoretical asymmetric momentum
distributions which reproduce very well the experimental shapes
PhysRevLett.108.252501 . However, recent knockout measurements done at higher
energies, at which the adiabatic approximation is expected to be more
accurate, exhibit the same quenching of the ratio of the experimental to
theoretical cross sections PhysRevC.103.054610 . This suggests that the cross
section integrated over the whole distorted momentum distribution is
essentially identical to the one obtained from the adiabatic, symmetric one.

The core-spectator approximation assumes that the core degrees of freedom are
“frozen” during the collision process, thus ignoring dissipation effects
associated with the extraction of the nucleon. A recent work
PhysRevC.83.011601  suggests that these dissipation effects might populate
excited states above the particle emission thresholds, leading to particle
evaporation and a loss of flux that could explain part of the quenching of
spectroscopic factors. This has stimulated efforts in the theory community to
revisit the reaction theory at the basis of the description of knockout
experiments. Recently, Gómez Ramos and Moro studied the effects of absorption
in the final channel GOMEZRAMOS2022137252 , while Hebborn and Potel went also
beyond the core-spectator approximation within the Green’s Function Knockout
(GFK) formalism HP2022 , in order to account for dissipative effects in the
nucleon extraction process. None of these approaches use the adiabatic
approximation. A systematic revisiting of the available knockout data making
use of these and other improved reaction formalisms is desirable and might
contribute to elucidate the quenching puzzle.

The above parlance highlights the general need to integrate in a consistent
theoretical framework the structure and dynamics of many nucleons in a
nucleus, and the description of reactions used to study them in an
experimental context. A possible approach is suggested in the GFK, where the
dynamics (driven by the optical potential) and the structure (embodied in the
Green’s function) are consistently integrated making use of the DOM.

Experimentally, the possibility to disentangle the reaction model and
structure contributions to the quenching trend observed with knockout
reactions has been put forward using mirror reactions. In this type of study,
knockout cross sections between mirror initial and final states are compared
to each other, with the assumption that the relevant spectroscopic factors are
identical due to isospin symmetry. Pioneering works have been performed on
some p-shell nuclei GRI11 ; KUC22 , but need to be extended to more cases and
to heavier nuclei.

It is clear that more experimental data is needed to improve the quenching
comparisons between different reaction probes. This is particularly true for
transfer and quasi-free scattering for which the few existing data sets only
cover a limited range of the neutron-proton asymmetry energy. Progress is
being made in that direction, for instance using high luminosity detectors
such as the AT-TPC AT-TPC , that enable to perform transfer reactions with
lower beam intensities. Reactions such as (d,p) on proton-rich nuclei or
(3He,d) on neutron-rich nuclei would probe the quenching effect on deeply
bound nucleons, for which the knockout data shows largest effect.

Finally, theoretical uncertainties, coming mainly from the choice of the
optical potentials (discussed more extensively in Sec. 7), are usually not
quantified. Recent works on transfer reactions have shown that these
uncertainties introduce an error from $\sim 20\%$ to $100\%$ on the
spectroscopic strength inferred from transfer data (see Ref. Lovell_2020  for
a recent review). To be able to make a robust comparison between different
experimental probes, all the theoretical uncertainties should be quantified. A
first step in that direction would be to quantify the errors introduced by the
choice of optical potentials, which strongly influence the magnitude of these
cross sections and hence the extracted spectroscopic factors.

Ultimately, the quantification of clustering or halo effects in nuclei relies
on the determination of spectroscopic factors of the corresponding states or
resonances. As these can be best populated via reactions, it becomes essential
to have the most accurate theoretical tools to extract the spectroscopic
factors from measured cross sections. The understanding of the different
quenching trends observed between various reaction probes is therefore
paramount to achieve this objective.

##  6 Universal corrections for few-body reaction models

L. Hlophe and Ch. Elster

Experiments at FRIB are poised to produce and study the properties of a wide
variety of exotic isotopes, often using light probes such as the deuteron or
the triton. Such nuclear reactions generally constitute a complex many-body
scattering problem for which exact solutions are typically not feasible.
Traditional models rely on identifying the relevant degrees of freedom and
projecting the many-body system onto the few-body space, leading to a system
of few clusters interacting via pairwise forces. The latter are usually
constructed by fitting phenomenological (Woods-Saxon) forms to (elastic)
scattering data. A few-body Hamiltonian ($H_{FB}$) is then constructed using
the effective pairwise potentials and observables are computed from
eigenstates of $H_{FB}$. However, a formal reduction of the many-body problem
onto the few body space generally leads to ‘irreducible” few-body forces
beyond the direct sum of the pairwise potentials. A very common case is that
of three-body reaction model which is employed to described processes such as
the transfer of a cluster from one nucleus to another as well as three-body
breakup processes. In this case, the irreducible effective three-body
potential arises from antisymmetrization of the many-body problem and
excitation of the clusters. In the past, attempts based on simplified nuclear
structure and reaction models Ioannides19781331 ; Johnson1981750 ;
Johnson1982348 ; TOSTEVIN198783  were made to quantify the effects of the
irreducible three-body force (3BF) on deuteron-induced reaction observables.
Such works already gave indications that the impact of the 3BF on reaction
observables was significant, necessitating more thorough investigations using
modern techniques for describing nuclear structure and reactions. Recently a
study of the effects of the 3BF arising from core excitations based in
deuteron-induced reactions has been carried out Dinmore:2019ouu . The study
utilizes multiple scattering theory to estimate leading order contribution to
the 3BF, which is expressed in terms of the phenomenological nucleon-nucleus
potentials. This work indicates that the impact of the 3BF on transfer cross
sections can be substantial and should be accounted in order to reliably
extract structure information from reaction measurements.

Quantifying effects of the 3BF arising from exchange of nucleons between
clusters requires a microscopic theory that is valid for the description of
both structure and reaction dynamics. In Ref. Theeten:2007  several bound
systems (6He=$\alpha$+$n$+$n$, 9Be=$\alpha$+$\alpha$+$n$,
12C=$\alpha$+$\alpha$+$\alpha$) were studied using the no-core shell model
coupled with the resonating group method (NCSM/RGM) starting from
phenomenological nucleon-nucleon (NN) forces. The bound state energies for
each nucleus were computed using two approaches, namely, (1) a fully
microscopic NCSM/RGM calculation and (2) a three-body calculation using
effective nucleon-nucleus potentials evaluated using the NCSM/RGM starting
from the same NN potentials used in (1). To ensure that differences only
reflect the effects of nucleon exchanges between clusters, all clusters were
fixed in their ground states. The result showed that the three-body model
consistently leads to smaller binding energies when compared to the fully
microscopic calculation. The work of Ref. Hlophe:2022fdn  investigated the
effects of the 3BF arising from antisymmetrization in deuteron ($d$)+nucleus
systems, using as a test case the $d$+$\alpha$. Using a similar strategy as in
Ref. Theeten:2007 , fully microscopic NCSM/RGM calculations for the
6Li=$\alpha$+$p$+$n$ bound state were compared to three-body Faddeev
calculations based on microscopic NCSM/RGM nucleon-$\alpha$ potentials. Both
the fully microscopic NCSM/RGM calculation and the NCSM/RGM nucleon-$\alpha$
were based on the same chiral effective theory ($\chi$EFT) NN potentials. The
three-body model was found to be underbound by approximately $600$keV,
consistent with the observations in Ref. Theeten:2007 . In addition to
comparing the bound state energies, the study of Ref. Hlophe:2022fdn  was
extended to the scattering regime. The fully microscopic NCM/RGM was used to
compute phase shifts and cross sections for deuteron-$\alpha$ scattering below
the deuteron breakup threshold, starting from the same $\chi$EFT NN potential
adopted for the bound-state calculations. The results were compared against
three-body Faddeev calculations using the aforementioned NCSM/RGM microscopic
nucleon-$\alpha$ potentials. It was observed that the two approaches yield
significantly different cross sections. Particularly, the $3^{+}$ resonance
peak is found shifted to a higher energy in the three-body (see Fig. 7). Both
the bound state and scattering results suggest that the contribution to the
3BF due to antisymmetrization is attractive. Owing primarily to the difference
in the resonance position, the $d$+$\alpha$ angular distributions computed
using the two methods also differ considerably. While the contribution to the
binding energy was shown to increase with system size in Ref. Theeten:2007 , a
similar investigation of the effects on reactions observables is still
outstanding and should be a subject for future studies. In addition, a study
of the full effects of the 3BF (antisymmetrization and core excitations) can
be quantified by adopting the state-of-the-art ab initio reaction theories
such as the no-core shell model with continuum in place of the NCSM/RGM.

Figure 7: The differential cross section for elastic $d+\alpha$ scattering as
a function of the center-of-mass energy $E_{\rm c.m.}$ at the scattering angle
$\theta_{\rm c.m.}=38.7$ deg. The dashed lines shows phase shifts computed
using the NCSM/RGM while the Faddeev results are depicted by solid lines. The
model space for the Faddeev calculation is restricted to a total two-body
angular momentum of ${\cal I}_{n+p}\leq 3$ and ${\cal I}_{N{\rm+}\alpha}\leq
9/2$ for the $n$+$p$ and $N$-$\alpha$ subsystems. The NCSM/RGM model space is
truncated at $N_{\rm max}=12/13$ for the positive/negative parity partial
waves and $\hbar\Omega=14$ MeV.

To take full advantage of the capabilities at FRIB, accurate few-body reaction
models are necessary as they provide means to translate observed experimental
yields into useful structure information. In particular, single nucleon
transfer reactions are typically described within a three-body model where the
primary input are the nucleon-nucleus and nucleon-nucleon potentials. While a
lot of progress has been made in ensuring that the three-body dynamics in
these reactions are treated correctly, questions pertaining to the three-body
Hamiltonian remain. Currently there is a considerable effort being dedicated
to improving the effective nucleon-nucleus potentials (optical potentials)
with a particular focus on exotic nuclear systems. While this is an essential
step, it should be coupled with universal corrections to the three-body model
itself through the quantification of irreducible three-body effects. As
already indicated above, such corrections alter the theoretical energy and
angular distributions and can thus potentially distort the extracted structure
quantities such as spectroscopic factors. Moreover, some of the FRIB transfer
measurements will use heavier probes such as 3H, 3He, and 4He. Although
approximate solutions of the few-body problem provide means for analyzing
those experiments, exact methods will need to be extended to enable a
treatment of more degrees of freedom. This is necessary not only for verifying
the accuracy of the approximate methods but also make reliable predictions for
other exotic nuclei.

##  7 Remarks on experimental and theoretical uncertainties in reaction
studies

X. Zhang, F. Nunes, and B. P. Kay

Reactions are often used to populate, dissect or modify few-body cluster
states in nuclei. The understanding from a measurement requires the confluence
of a detailed account on the experimental errors but also quantification of
model uncertainties used in the interpretation. Arguably as compared to the
experimental counterpart, the theoretical community has lagged behind in
scrutinizing theory uncertainties. It is very important to make a significant
investment to first understand all sources of uncertainty and then devise
strategies to minimize the uncertainties so that meaningful physics can be
extracted. It is common for experimental data to be plotted including
statistical uncertainties only, while the information regarding systematic
uncertainties are mentioned in a less prominent way in the text of a
publication. Theorists have traditionally published predictions without
uncertainties (recent theory efforts on uncertainty quantification are
mentioned briefly in the last section of this contribution). Estimates of
variability are provided by showing results with two different models (or
parameterizations). Only recently has reaction theory begun to perform a
quantitative evaluation of uncertainties, particularly due to the effective
interactions used in models. Ultimately, final results should contain the full
uncertainty, combining experimental and theoretical contributions.

### Experimental uncertainties

Experimentally, there are many parameters to consider when assessing the
uncertainties of, or in designing, a measurement AUMANN2021103847 . Relevant
to the studies of halo and cluster states is being able to infer the relative
and absolute cross sections, to assign spin and parity of the final state and
to compare the cross section to a reaction and nuclear–structure model. Each
of these can be impacted by the experimental approach, such as detector
system, the reaction used, the incident beam energy, and the angular coverage
available. Some of these factors are commented on below in a general manner.

In some measurements the experimental circumstances are such that relative
cross sections are obtained with good accuracy over the excitation energy
range of interest, while the absolute uncertainties are substantially greater.
Relative cross sections for a broad class of spectrometer techniques are often
better than a few percent, though caution has to be taken when probing the
unbound region in coincidence with recoil fragments, which may have different
acceptances. Such data can be valuable in exploring a broad range of nuclear
structure properties, but to make meaningful comparisons to theory
predictions, absolute cross sections are necessary.

The absolute value of the cross section is most reliably determined by
minimizing systematic errors, for instance using the same apparatus for
multiple similar measurements. Accurate determination of the number of
incident beam ions on target, precise definitions of apertures and acceptance,
intrinsic detector efficiency, etc., are all needed. Challenges often arise in
determining the target thickness precisely. Ideally, but not always possible,
is measuring the elastic scattering cross section on the same target as the
direct reaction of interest but with a beam energy well below the Coulomb
barrier, where the scattering is Rutherford. Other factors to consider are
target uniformity and target degradation, which can be concerns when using
heavier beams on thin plastic targets. Some detector systems, such as active
targets can have the advantage of being self-normalizing in that the beam dose
is determined absolutely, in the same way that the reaction yield is (for
example, Ref. Avila17 ).

In the treatment of cross sections, once the quantum numbers for a transition
are well established, it is apparent that the comparison between a reaction
model and the experimental data is best done in the angular region where the
cross section is at its maximum. The shape of angular distributions may not be
well described at larger angles, where cross sections can be orders of
magnitude lower than the primary maxima.

The range of uncertainties in absolute cross sections cannot be easily
generalized as it depends on many experimental parameters. With that said, it
is not uncommon for measurements with stable light-ion beams to be better than
10% and many radioactive ion beam measurements are now around 20-30%, though
in some cases much better.

### Interface between experimental data and model interpretation

The choice of beam energies can have a significant impact not on the
uncertainty of the measured cross sections (they are what they are), but in
the reliability of the model used in the interpretation of the cross sections.
It has been well-documented Schiffer12 ; Schiffer13 ; Wimmer18 ;
AUMANN2021103847  that for direct reaction studies, modeled as peripheral,
single-step reactions, energies a few MeV/u above the Coulomb barrier in both
the incident and outgoing channels are the most reliable regime (largest cross
sections) in which to determine absolute cross sections for comparison to
models like the distorted-wave Born approximation.

However, experiments have different goals. If the objective is to determine
the asymptotic normalization of the wave function, which is needed in order to
estimate astrophysical cross section for astrophysical purposes (mostly for
unbound resonances), then sub-Coulomb energies are generally the most
reliable. At these sub-Coulomb energies, the incoming wave is affected mostly
by the Coulomb barrier and the transfer cross sections become proportional to
the square magnitude of the tail of the wave function, the Asymptotic
Normalization Coefficient (ANC) (a quantity that is relevant for both bound
and resonance states). Note that the ANC can also be extracted using breakup
or knockout reactions at high energy (e.g. capel2007 ).

Of course, there are a whole class of intermediate and high-energy reactions,
where different models have been developed and again, in those cases, strong
arguments are made for specific energy ranges, such as quasi-free scattering
and so on. The choice of beam energy and reaction then depends on which
quantity is of primary interest.

Related to the topic above is that of momentum matching, which is modified by
the choice of reaction used. For the study of halo states and many clusters,
often these are low angular momentum states, being $\ell=0$, 1, etc. Reactions
such as ($d$,$p$) and ($t$,$p$) are well suited to probing such excitations,
in the nominal energy range of 10 MeV/u. It is worth remembering that these
reactions do not reveal the parity of the final state necessarily, though it
can often be easy to infer using the shell-model or from complementary
experimental probes.

Other reactions, such as ($\alpha$,3He) will add a neutron much like the
($d$,$p$) reaction, but are not suitable for low-$\ell$ transfer due to the
often large $Q$ values involved.

Each detector set up and experimental arrangement, will have a certain angular
coverage. A challenge with some systems commonly used for the study of e.g.,
transfer reactions in inverse kinematics, is angular coverage which can impact
the uncertainties associated with fitting of angular distributions and thus
the physics interpretation.

Typically, one would like to cover the primary maximum and minima of angular
distributions. When considering the canonical 10 MeV/u, and reactions like
($d$,$p$) and ($t$,$p$), one often sees these distinct features in the 0-50∘
center of mass range. Of particular importance, but also often difficult to
access, are the most forward center-of-mass angles where $\ell=0$ angular
distributions are peaked. Particular effort in the experiment community to
design detector systems that can explore this region more reliably is ongoing.

### Theory uncertainty quantification

Bayesian statistics has attracted much attention in theoretical low-energy
nuclear physics Phillips:2020dmw ; Zhang:2015ajn ; Iliadis:2016vkw ;
Acharya:2016kfl ; Wesolowski:2018lzj ; Zhang:2019odg ; Premarathna:2019tup ;
Acharya:2019fij ; Melendez:2019izc ; Maris:2020qne ; Higa:2020pvj ;
Drischler:2020hwi ; Drischler:2020yad ; Poudel:2021mii ; Acharya:2021lrv ;
Odell:2021nmp ; Wesolowski:2021cni ; Acharya:2022drl . Recently, Bayesian
analyses of the optical model have been performed lovell2018 ; king2018 ;
catacora2019 ; Lovell_2020 ; catacora2021 . In these studies, nucleon elastic
scattering data is used to calibrate the nucleon optical potential, which is
then propagated to transfer (d,p) reactions. An illustration is shown in Fig.
8 where two standard reaction models for (d,p) transfer are compared, with the
shaded areas corresponding to the 95% confidence interval resulting from the
uncertainty in the optical potentials in each model.

A detailed comparison between the Bayesian approach and the traditional
frequentist approach king2019  demonstrated that the traditional approach
underestimates uncertainties and that uncertainties from the optical model are
in general larger than previously thought. New statistical tools catacora2021
can help in identifying which measurements are more promising to reduce these
uncertainties.

Figure 8: Comparison between the 95% confidence intervals for
40Ca(d,p)41Ca(g.s.) cross section using two standard reaction models: DWBA
(blue) and ADWA (red). Figure taken from Lovell_2020 .

The optical models considered in the aforementioned studies are the simplest
available. That choice was specifically due to computational limitations. For
a Bayesian analysis, large number of samples are needed to explore the model’s
parameter space. A full continuum state solution of a few-body Hamiltonian at
a single point in parameter space can be expensive, e.g. the Continuum
Discretized Coupled Channel (CDCC) Method, the method of choice when studying
breakup reactions with halos. Repeating such calculations for thousands or
even millions of times, as required in Bayesian analysis, becomes
computationally infeasible.

Emulators are developed to solve this bottleneck issue. They can be considered
as fast and accurate ways to interpolate and extrapolate either the full
solutions or a particular set of observable predictions of a given complex
model. Generally speaking, there are data-driven and model-driven emulators
Melendez:2022kid ; Bonilla:2022rph . An example of a data-driven emulator is
the recent work on quantifying the uncertainty in the breakup of 8B surer2022
. In that work, off-the-shelf emulators were trained on cross section breakup
data to replace the full computationally intensive CDCC calculations.

For model-driven emulators, the so-called eigenvector continuation (EC)
method111The EC method was recently recognized as one of the reduced basis
methods (RBM) developed in the field of model order reduction Melendez:2022kid
; Bonilla:2022rph . was initially developed for many-body bound state
(discrete level) calculations Frame:2017fah ; Konig:2019adq ; Ekstrom:2019lss
, and later generalized to two-body scattering Furnstahl:2020abp ;
Drischler:2021qoy ; Melendez:2021lyq ; Bai:2021xok , and lately to three-body
scattering Zhang:2021jmi . Compared to the data-driven emulators which learn
physics from data, the model-driven ones incorporate the relevant physics and
therefore require generally less training data and have better emulation
accuracy. Some exemplary comparisons can be found in Konig:2019adq .

Effort is ongoing to further generalize the RBM/EC emulation to higher-
energies and N-body systems (with N larger than 3), and its implementation in
a Bayesian framework for uncertainty quantification. It is worth noting that
the applications of these emulators are not limited to uncertainty
quantification. A scheme depicting their roles in research workflow is
presented in Fig. 9. They can serve effectively as the interfaces for
expensive calculations. Through these interfaces, users can easily access the
expensive calculations without spending significant computing resources. As a
result, new interactions between different research areas can be created, and
novel studies might then emerge.

Figure 9: The emulator’s role in coupling different research areas. The dashed
arrows indicate desirable connections enabled by the emulators. Figure taken
from Zhang:2021jmi .

##  8 New opportunities for nuclear Efimov states

S. König, T. Frederico, and F. Nunes

Searches for the Efimov effect in atomic nuclei have by now a relatively long
history (see e.g. jensen04_233 ; Macchiavelli:2015xml ), and yet these states
remain elusive in nuclear systems. Weakly bound two-neutron halo nuclei are in
principle excellent candidates to observe this effect, the minimum signature
of which would be two levels apart in energy by a universal geometric scaling
factor. The scattering length between the neutrons and the core $a_{nc}$ and
the neutron-neutron scattering length $a_{nn}$ determine the Efimov spectrum:
in the unitarity limit, where both these scattering lengths are infinite, the
Efimov effect in its purest form implies the existence of infinitely many
geometrically spaced energy levels, that is, the ratio of energies of two
successive states is a constant. This has been presented in Efimov’s
pioneering work for three identical bosons Efimov:1970zz ; Efimov:1971zz ,
where he showed that the reason for such a remarkable phenomenon is the
emergence of an effective hyper-radial long-range inverse square potential,
with strength large enough to allow Landau’s “fall to the center”
Landau:1991wop . For finite scattering lengths, only a limited number of bound
states exists, the precise number of which depends on the actual values of the
scattering lengths.222There is additionally a dependence on the mass ratio
between the halo constituents and the core, but its influence is rather weak.
Alternatively, one can express the Efimov criterion in terms of one- and two-
neutron separation energies, and in this parameter space one can map out a
critical region for the existence of a bound excited Efimov state above the
halo ground state. This is shown in Fig. 10 and, as can be seen in this
figure, essentially all candidates considered to date elude the critical
region.

Efimov states move when the system moves away from unitarity limit to finite
scattering lengths, and they gradually disappear one by one beyond the lowest
scattering threshold Efimov:1970zz ; Efimov:1971zz . In the situation where at
least one of the subsystems becomes bound, an Efimov state approaches the
continuum threshold and eventually becomes a virtual state by decreasing the
scattering length or increasing the subsystem binding energy Adhikari:1982zz .
In other words, an Efimov state migrates to the second complex energy sheet of
the S-matrix, moving through the lowest two-body cut as the corresponding
subsystem binding energy increases. This is the case, for example, for the
triton Adhikari:1982zz  (more recently studied in EFT Rupak:2018gnc ), 20C
(see e.g. frederico12_372 ), and possibly for 12Be and 62Ca Hammer2017 , all
of which are located in the second quadrant of Fig. 10. If we had a knob to
play with the strength of the $n$-core interaction, the virtual states of
these nuclei might be pulled down into the Efimov region, by increasing the
neutron-core scattering length. We propose to explore this scenario
experimentally by suddenly stripping out nucleons from the core of a halo
nucleus.

Figure 10: Critical region for the Efimov excited state above a bound one. The
axis are the ratios of the neutron-core energy $(S_{1n})$ and the two-neutron
$(S_{2n})$ separation energies. $E_{nn}$ the neutron-neutron virtual or bound
energies. The positive axis means a bound n-core s-wave system and the
negative one a virtual s-wave state. $A$ is the core mass number. The physical
region of the two-neutron halo nuclei are the left-upper quadrant (samba
nuclei) and the left-lower quadrant (Borromean nuclei). The triangles
illustrate the possible configurations, all-bound, tango, samba and Borromean,
where the continuous line corresponds to a bound s-wave state of the pair and
the dashed a virtual state. The quoted nuclei ground states with a
core+$n$+$n$ structure are 62Ca, 20C, 12Be, 3H in the samba configuration and
22C and 11Li in the Borromean one. Thus figure has been adapted from a similar
one that appeared in Ref. Hammer2017 . We thank Hans-Werner Hammer for making
the original material available to us and for giving us permission to adapt
it.

All the candidates included in Fig. 10 are bound nuclei, classified as halos
in their ground states. Rare isotope facilities provide a fascinating
opportunity to look at more exotic scenarios where the halo configuration
occurs as an excited state. If the lifetime is sufficiently long compared to
the typical time scale of the halo dynamics, these states may be present as
resonances. For example, one might start with a beam of known halo states,
falling short of exhibiting the Efimov effect, and use reactions that remove
one or more nucleons _from the core_ , leaving, potentially, a new halo state
where the parameters are shifted in such a way that it is nudged inside the
critical region shown in Fig. 10. Very recently, an experiment performed at
TRIUMF Kuhn:2021dvu  achieved exactly such a stripping process to study
excited states in 10Be through the 11Be$(p,d)$ reaction at low energies. In
that work, the authors showed that the data is consistent with a one-step
process that removes a neutron from the core while leaving the one-neutron
halo in the original nucleus intact.

Due to the decoupling of core and halo degrees of freedom, we consider that,
for a short period of time after removing the core nucleons, the valence
nucleons remain in their extended halo configuration (where the fall-off is
still ruled by the same three-body binding energy). Naively, we can assume
that immediately after the removal of the nucleons from the core, the two-body
scattering length increases due to the change in mass as
$a^{\prime}_{nc}\approx\sqrt{M_{\text{final}}/M_{\text{initial}}}\,a_{nc}$,
reducing the corresponding two-body separation energy, and decreasing the
ratio $S_{1n}/S_{2n}$ towards the Efimov region, particularly in the second
quadrant shown in Fig. 10. Based on this expectation, we suggest that new
experiments utilize the technique of removing multiple neutrons/protons from
the core, with the goal of modifying the neutron-excited-core scattering
length so that it is more favorable to populate excited Efimov states in
nuclei.

## Acknowledgements

This work is supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under the FRIB Theory Alliance award no. DE-
SC0013617. This work is supported by the National Science Foundation under
Grant Nos. PHY-1555030, PHY-2111426, PHY-1913728, PHY-2209060, PHY-2044632,
PHY-1912350, OAC-2004601 and the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics under Contract Nos. DE-AC52-07NA27344, DE-
AC05-00OR22725, DE-SC0021422, DE-AC02-06CH1135, DE-FG02-93ER40756, DE-
SC0020451, and DE-AC05-06OR23177. GP’s work is supported by the LLNL-LDRD
Program under Project No. 21-ERD-006. KSB greatly appreciates the financial
support of a research fellowship from the Louisiana Board of Regents; it
benefited from computing resources provided by the National Energy Research
Scientific Computing Center NERSC (under Contract No. DE-AC02-05CH11231),
Frontera computing project at the Texas Advanced Computing Center (under
National Science Foundation award OAC-1818253) and LSU (www.hpc.lsu.edu). FB’s
work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through Project-ID 279384907 - SFB 1245. FB would like to
acknowledge Sonia Bacca for useful discussions. TF’s work is partially
supported by Fundação de Amparo à Pesquisa do Estado de São Paulo
(2017/05660-0, 2019/07767-1), Conselho Nacional de Desenvolvimento Científico
e Tecnológico (308486/2015-3) and the INCT-FNA project No. 464898/2014-5.

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