\begingroup

%% --- Kirscher macros ---
\newcommand{\refGuan}{Qingze Guan}
\newcommand{\refCapitani}{Ylenia Capitani}
\newcommand{\refGlazek}{Stanis\l{}aw G\l{}azek}
\newcommand{\refNaidon}{Pascal Naidon}
\newcommand{\refDawid}{Sebastian Dawid}
\newcommand{\refDeltuva}{Arnoldas Deltuva}
\newcommand{\refHiyama}{Emiko Hiyama}
\newcommand{\refMadeira}{Lucas Madeira}
\newcommand{\refKievsky}{Alejandro Kievsky}
\newcommand{\refNicholson}{Amy Nicholson}
\newcommand{\refKolck}{Ubirajara van Kolck}
\newcommand{\refPederiva}{Francesco Pederiva}
\newcommand{\refDavoudi}{Zohreh Davoudi}
\newcommand{\refHGrie}{Harald Grie\ss{}hammer}

%% --- Body ---
I must stress from the onset the subjective character of this introduction and summary
of topics and problems presented, discussed, and (re)solved during the meeting. I will find
it impossible to give due attention to all ideas present, and my knowledge and research activity
will bias the following and might even put emphasis on subjects the community rates less significant
while it considers other topics of broader interest. The contributions to this volume will remedy
this flaw while I believe the reader will find some benefit in reflecting back on what is highlighted
in this introduction when reading through the individual reports.

As a common ground, I would like to start with a list of observables that were presented as theoretical
predictions:

\vspace{0.4em}
\begin{itemize}
    \item Bethe's universal parameter for the description of a short-range 2-body interaction,
    i.e. the scattering length.
    \item\label{item:energy_spectra} Energy spectra.
    \item\label{item:spatial_distributions} Species-(in)dependent spatial distributions of particles within a bound state.
    \item Renormalization-group trajectories of coupling strengths.
\end{itemize}
\vspace{0.3em}

The apparent question we need to answer is about legitimacy: there were no experimental findings
presented at the workshop in imminent need for a theoretical interpretation.
In all cases where data was shown, an impressive agreement between theory and experiment was
obtained (see, e.g.~\refGuan's~but also~\refCapitani's--considering the expected uncertainty-- contributions):
theories seem to work, maybe not as accurately as desired but there was no serious challenge to the
referral to ``higher orders''.
So, why do we invest our resources in refining and testing the theories summarized at the beginning
of the meeting
by \refGlazek~and \refKolck? We do so primarily in the believe to discover something
as revolutionary as the seemingly mathematical exercise of V.~Efimov when he studies the predictions
of a zero-range interaction in a 3-body system.

With a set of contemporary numerical techniques, namely,

\vspace{0.4em}
\begin{itemize}
        \item Faddeev-Yakubovsky/Alt-Grassberger-Sandhas decompositions (\refKievsky,~\refDeltuva),
        \item variational, finite-basis methods (hyper-spherical:~\refKievsky, correlated-Gaussian:~\refHiyama),
        \item Monte-Carlo sampling (\refMadeira),
    \item neural-network approaches (\refPederiva),
    \item quantum computation (\refDavoudi),
\end{itemize}
\vspace{0.3em}

one aims to find peculiar features of quantum systems which Efimov was able to predict analytically in the
3-body system. This identification demands a sufficiently broad and diverse sampling of the reaction of
a generic system with respect to different parametrization of unobservable short-distance structure
innate to \emph{any} interaction theory. Also during this meeting, this sampling mostly translated
into the jargon:\\[1ex]
\emph{Have you varied your ``cutoff'' enough in order to exclude a potential diverging, non-universal
behavior?}\\[1ex]
All the phenomenal advances in the numerical techniques, on which the participants reported, and
on which they represent the state of the art, were not bespoke to the problem of sampling the space of
short-distance interactions comprehensively in a practical way
\footnote{The models presented by~\refGlazek~and the cyclic dependence of the 3-boson spectrum on a
sharp momentum cutoff on, respectively, an effective 2-body momentum and the relative momentum between a,
e.g., PDS-regulated dimer and a single boson exemplify a parametrization of the short-distance structure
which exhibits non-universal behavior within a practically accessible range of the cutoff. In contrast, Gaussian
regulation of the 2-body interaction, as used in the results shown by \refHiyama~and \refKievsky,
is impractical to recover limit cycles.}
.
For all disciplines, atomic, nuclear, and particle, the development of such a sampling technique would enable
significant new insights and expose phenomena, while it entails intriguingly non-trivial numerical and conceptual
problems.
A possible solution follows the understanding that
in order to assess RG (in)variance taking the, mostly
computationally costly, zero-range limit of a 2-body potential is not necessarily an
efficient approach to evoke those short-distance structures that lead to non-universal behavior;
imposing qualitatively different
renormalization constraints, e.g., instead of fitting to a bound-sate pole, use alternatives like
residue fits~\cite{Phillips_2000}~to calibrate potentials for few-body analyses,
could expose RG variance much faster and convincingly.
I chose this example of an alternative renormalization strategy because it does not require
a, mostly tedious, change of the interaction's operator structure. All potential forms and regulators
which were tailored to the above-listed techniques can incorporate such an alternative strategy with
a set of parameters whose regulator dependence encodes the chosen candidate of a qualitatively new
short-distance structure.
The practical and conceptual benefit of a refinement of effective-field-theory-based potentials
in this sense should not be neglected but combined with the parallel evolution to higher orders and precision
through, e.g., alternative leading-order starting points
(cf.~so-called \emph{improved actions}~as touched upon by \refKolck). The intense discussion ensuing different
incarnations of the above-cited question called the community to reflect on this point in the development
of the next generation of potentials.

We dedicated one day on how the phenomenon of a resonance, in particular, and a pole of the
S-matrix, more generally, is observed, understood, extracted, and analyzed in various fields.
The topical complex was opened by \refNaidon~with an analysis on the mass-ratio dependence of the resonance
width in a three-body system.
The presented interpretation of the observed nodes of that dependence as
bound-states-(embedded)-in-the-continuum (BIC~\cite{Newton1982})
has, to my knowledge, not been considered in the context of the desired stabilization of nuclear systems.
In the latter, this question presents a challenge to EFT potentials which predict certain systems unbound
at some low order and thus must stabilize them at higher orders (if nature tells us so).
It is not so much the relation to truly bound states which intrigues more thought on the BIC phenomenon
in nuclear physics but to states which assume numerically characteristics of bound states despite the
existence of deeper thresholds in the spectrum. For instance, the characterization of eigenstates
with eigenvalues above composite-fragment-breakup thresholds as resonances, for example,
in finite-volume or variational-basis calculations is non-trivial. The ideas advanced by \refNaidon~
for atomic systems, the stabilization by means of magnetic fields, are thus very pertinent to
the renormalization of nuclear interactions, as touched in the previous paragraph, when often an
interpretation of states higher in the spectrum with natural ground states is desired but impractical
(cf.~renormalization for a short-distance structure which exhibits enough spacing between thresholds
such that the coupling to the break-up channel is insignificant). This strongly motivates further
exchange between the two seemingly different problems: stabilization of cold-atom systems through
mass and magnetic-field tuning and stabilization of nuclear systems.

In two ensuing sessions,
the analysis of S-matrix-pole trajectories for three- and four-body systems
was discussed. Also here, the migration to different sheets, the crossing of cuts, and their
understanding as passages from bound to virtual to resonance states, was the focus.
\refDawid~ highlighted the problem of a seeming disappearance of poles
while inducing their already phenomenal cyclic behavior by tuning the two-body scattering
length out of unitarity using
\emph{relativistic} dynamics. 
Since then, he and his group have further investigated this problem, and a solution is sketched
in his contribution below.
\refDeltuva~demonstrated the cyclic transformation from a resonance to a bound state
close to a four-body threshold. At least with a naive glance, both works appear related, and at least
epitomize the many complexities that arise in even the fewest-body systems due to a slight
modification of the way two bodies interact.
The most impressive program of simulating hadronic reactions with three asymptotic fragments with
standard-model degrees of freedom employs the techniques of tracing poles and analytic continuation of
a three-body scattering amplitude after obtaining part of its momentum dependence from lattice QCD.

I would like to make an effort, now, to relate all other contributions by their importance in the
discovery of specific universal few-body phenomena. To make this latter notion less uselessly
abstract, I chose the results presented by \refHiyama~on the $3\leq A\leq 6$ equimassive,
 non-relativistic systems of bosons
. The presented spectra seemed to corroborate an old hypothesis on the correlation
of at least one universal $\qty(A+1)$-body bound state to a universal $\qty(A)$-body state, starting,
of course, with $A=3$. Even with the presented, highly-sophisticated and accurate Gaussian-basis-expansion
technique, the energies of states close to the $\qty(A+1)$ breakup thresholds are difficult to assess.
The meeting thus reminded us of the need for a numerical tool tailored to expansions of states of this peculiar type--
single atom loosely bound to an itself loosely bound core by means of an interaction short-ranged relative
to the core and the core-atom systems. Such a method would provide the empirical base for a
rigorous answer to puzzles like:

\vspace{0.4em}
\begin{itemize}
    \item $\exists A_{\text{max}}$ s.t.
    $\lim\limits_{\text{unitarity}/\text{zero-range}}B_n\qty(A_{\text{max}}) \to \pm\infty/0$ (critical particle number for stability),
    \item the number and degeneracy of the universal clusters,
    \item the geometry of the universal clusters,
    \item $\pm\infty\to a<0$ pushing the almost-bound dimer further into the continuum.
\end{itemize}
\vspace{0.3em}

\refMadeira~presented the implications of being close to unitarity for clusters
comprising up to 15 $^4$He atoms from which one attempts to extrapolate to atomic $^4$He matter.
Although not named as such, \refHiyama's approach of a set of two-Gaussian interactions whose
strength parameters were fit to the same $^4$He dimer properties used to renormalize  the EFT-derived
interactions used by L.~Madeira overlapped by more than just ground-state spectra for $A\leq6$
but must be understood similarly as much more than predictions with yet another model.
Owed to the employed numerical method, \refMadeira~analyses solely ground states, and it will be
fascinating to observe how the results for those states, whose binding momenta seemingly would preclude
a fast convergence or usefulness of the employed leading-order interaction, are affected by higher-order
corrections parametrized, e.g., by the effective range of the $^4$He dimer-dimer potential. This
study of the predictions of an EFT expansion at a higher-than-leading order as a function of the
particle number will provide the right lense to understand the transition from few-body dynamics to
matter and density with unprecedented rigour.

The above already mentioned interpretation of the Gaussian interactions as representatives of the
same class of interactions was elaborated on by \refKievsky. This contribution highlighted
the shared 3- and 4-body scales predicted by 2-body-scattering-length-equivalent Gaussian 2-body ($^4$He atoms)
interactions.
From a nuclear-physics perspective, this is remarkable and it
sets these potentials apart from those phase-equivalent 2-nucleon
interactions which led to the so-called Phillips and Tjon lines.
In a detailed analysis, A.~Kievsky showed that all the 2-atom interactions
can be mapped onto Gaussian 2-body potentials with an appropriately tuned range. In the EFT context, this range
assumes the role of a cutoff regulator, and the close agreement in the 3- and 4-body systems is rephrased as
choosing cutoff value and regulator-shape in a region of the RG flow of the 3-body coupling where the latter is practically
zero and remains so over the analyzed range of cutoffs/ranges.
The answer to the obvious question of why all those potentials--to be concrete, HFD-HE2, LM2M2, SAPT96, PCKLJS--
seemingly miraculously choose such an advantageous range is suggested in~\cite{Kievsky_2020}, and combines
the constraint all those potentials impose on the number of dimer,trimer, and tetramer states with the
insight that the typical regulator range over which the spectrum fluctuates wildly is narrow compared with
the interval over which it is almost constant (cf. the limit-cycle running of the 3-boson contact~\cite{Bedaque:1998km}).
Regrettably, the discussion during this session diverged, and I would hereby like to draw the attention
back to this point as it adds to the need to simulate few-body systems over a much wider range of ranges/cutoffs.

Another approach by \refNicholson~obtains an analytic expression for the
ratio between the ground-state energy of an even number of bosons at unitarity and that of
the tetramer using two assumptions: first,
the large-time two-boson propagator is log-normal distributed, and second,
the tetramer is universally tied to the Efimov spectrum.
The latter has been corroborated most accurately for more than just one Efimov trimer (recall \refDeltuva's contribution),
and the former as shown by A.~Nicholson using pertinent lattice configurations.
A.~Nicholson realized that, at unitarity, mean and standard deviation of the log-normal two-boson
propagator are linked, in which case the trimer-tetramer correlation suffices to predict the $2n$-mere.
Although the non-zero overlap between the $n$-th moments of the two-body correlator with $2n$-body states
must sound trivial and not worth spelling out for those engaged in Monte-Carlo simulations, the idea goes
beyond this and would be fascinating to refine and test in parallel to the other approaches which
test the correlation between the three and $n$-body systems via variational techniques
(cf. the presented works by \refCapitani, \refHiyama, and \refKievsky).

In the absence of a repulsive exchange interaction, adding more and more bosons should not lead to discontinuous
structural changes of the ground state. The bosonic effective theories are thus not of the kind which were
presented by \refCapitani. There, the electromagnetic breakup of the nuclear, i.e., fermionic, $^9$Be state
employed the presumed clusterization of the nucleus. The presented results with a $^9$Be
being approximated solely
by a product of point-like $\alpha$'s and a neutron with each pair interaction calibrated to resonant features of the
respective $\alpha$-$\alpha$ and $\alpha$-nucleon interactions and the inclusion of a three-body contact
renormalized to the $^9$Be ground-state energy were convincing that, despite the significant complexity of
dealing numerically and conceptually with the coupling of nuclear currents to multi-pole fields,
the next step of taking the EFT approach seriously by a higher-order calculation--now in the cluster EFT--
is viable and would advance the transition from few- to many-fermi systems through practical and renormalized
effective cluster potentials. In this course, I deem it essential to understand how useful cluster potentials like
$V_{\alpha\alpha}$, $V_{\alpha n}$, $V_{\alpha\alpha n}$ are in the description of amplitudes involving, e.g.,
$\alpha\alpha n n$. To my knowledge, the advantages of the EFT over other effective cluster models have not yet
been exploited, despite the presented technical feasibility.

Compared to that study of an electromagnetic probe interacting with a specific nuclear system, the
universal character of the
laser-atom-dimer and laser-atom-trimer experiments, whose theoretical analysis \refGuan~presented,
is easier to identify.
Presented was an agreement between experimental extractions of dimer and trimer spatial probability
densities obtained from Coulomb explosions with simulations. This consistency was found also when
a second laser pulse was used to measure the atomic clusters as if their constructive two-atom
interaction was tuned to be more or less attractive. Observed and calculated were in that course
certain critical pulse characteristics for which diverging atom-atom scattering lengths marked the
appearance of additional excited bound states. With measurements of the spatial structure and e\&m response
and scattering-length dependence of tetramers available, a discussion of the similarities and differences of
this technique and the employed numerical tools to approximate the spatial atomic wave functions with--what is
suggestive to me--the so-called femtoscopy technique used primarily to extract few-body correlations from
heavy-ion collisions is needed. Especially the differences in three- and more-body correlations as a departure
from unitarity and way to understand non-universality between few-atom and few-nucleon properties are exposed with
those techniques and the accompanying theoretical methods.

Having iterated the importance of higher-order analyses, the contribution of \refHGrie~revisited the refinement
of the nuclear interaction with the perturbative exchange of pions. The conjecture about an unexpected
demotion of partial-wave-mixing vertices was pedagogically-valuable illustrated with a \emph{unitarity window}
through which those scattering phases ``shine'' which parametrize channels of unusually strong interaction while
the other, perturbatively weak channels linger in the so-called \emph{Born corridors}.
As a faithful application of the pionless theory
for $A>3$ will soon be viable at orders where higher partial waves enter, \refHGrie~reminded
those of us
who work also without pions that seemingly surprising features and scalings of operators might enter.

To conclude, I do invite those contributors whose fields I feel even less educated on and thus abstained from
too naive and pretentious words to elaborate on their ideas before summarizing their specific contributions.

\clearpage
\endgroup
