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Perhaps the simplest complex structures are scaling fractals such as
Von Koch's snowflake or its higher-dimensional
relative, romanesco
(or ``fractal'') broccoli. These structures are constructed
from an element with a chosen size $\Lambda_\star^{-1}$
by an operation repeated {\it ad infinitum}, which is characterized by
a scale factor $f>1$.
The end result is that the fractal looks the same
upon an increase in resolution,
\begin{equation}
\Lambda_\star^{-1}\to f^{-n}\Lambda_\star^{-1},
\label{DSI}
\end{equation}
with $n$ a positive integer.
Objects that differ only in the value of
$\Lambda_\star^{-1}$ have the same, universal properties
determined by discrete scale invariance (DSI) \cite{Sornette:1997pb}.
Surprisingly, the structure of atomic nuclei and some other systems,
such as clusters of $^4$He atoms, exhibit DSI
as an approximate symmetry \cite{vanKolck:2017jon}.

DSI emerges at large distances in few- and many-body systems of bosons
and fermions with three or more components, when the two-body system
is close to the unitarity limit. In this limit, the two-body
$S$ matrix $S_2=-1$,
and the $T$ matrix has a zero-energy pole representing
the transition between
a bound and a virtual state.
The unitarity limit is
a good starting point for the description of the dynamics when
all two-body scattering parameters have natural sizes set by
the range $R$ of the interaction, except for
the magnitude of the scattering length $|a_2|\gg R$.
The large scattering length, which sets the size of a
single, shallow $S$-matrix pole, requires a certain amount of fine-tuning
in the underlying interaction. In atomic physics, fine-tuning
can be engineered with an adjustable magnetic field
through the mechanism of Feshbach resonances, but
it also occurs naturally for $^4$He atoms: the dimer is
very large since, taking
the Van der Waals length as a measure of $R$,
$R/a_2\sim 1/20$.
In the nuclear case, the range of the interaction
is set by the mass of the lightest exchanged meson, the pion,
$R\sim 1/m_\pi$.
For the deuteron, $R/a_{2, 1}\sim 1/3$ is not tremendously small,
but for the $^1S_0$ virtual state the disparity is comparable to
atomic $^4$He, $R/a_{2, 0}\sim -1/20$,
It has been conjectured \cite{Beane:2001bc}
that the origin
of the large two-nucleon
scattering lengths is a Feshbach-like mechanism where $m_\pi$ ---which
depends on the quark masses and can be varied independently
of the nonperturbative QCD scale, $M_{\rm QCD}\sim 1$ GeV---
plays the role of the magnetic field.

The long-distance properties of these systems
can be captured by the quantum-mechanical equivalent of a multipole expansion
for the interaction potential, formalized as Short-Range Effective Field Theory
(SREFT) \cite{Hammer:2019poc}.
As any EFT, SREFT includes all interactions allowed by symmetries involving
the degrees of freedom relevant for the momenta $Q\ll R^{-1}$ of interest:
contact interactions involving
an arbitrary number of particles with an increasing number of derivatives.
The singularity of the interactions requires regularization,
which introduces a (sharp or soft) momentum cutoff $\Lambda$.
Regularization smears delta functions over a distance $\sim \Lambda^{-1}$.
Model independence then requires
renormalization, where the interaction strengths,
or ``low-energy constants'' (LECs), depend on $\Lambda$ in such a way
as to render the dependence of observables on the choice of
regulator as small as one wants.
After renormalization, the EFT produces a systematic expansion of observables
in a series of powers of $QR$ and $1/Qa_2$.
The nontrivial aspect of this method is to infer which interactions contribute
at each order, which, in turn, depends on how the LECs scale with $R$.
Needing a rule that can be applied to all LECs and is consistent
with renormalization, one invokes naturalness \cite{vanKolck:2020plz}.
Since the cutoff dependence reflects the sensitivity of virtual
particles to short-range physics, one assumes
that, except perhaps for an occasional fine-tuning,
the finite contribution from a LEC has a size given by setting
$\Lambda\to R^{-1}$
in its cutoff-dependent part.

The unitarity limit is a nontrivial fixed point
of the renormalization group (RG) for the two-body nonderivative delta function
\cite{Weinberg:1991um}.
The renormalization condition $1/a_2=0$ for any $\Lambda$ determines
the LEC.
For particles of mass $m$, it has a universal dependence on
$\Lambda$, $mC_0(\Lambda)\propto \Lambda^{-1}$ \cite{vanKolck:1998bw},
that mitigates the singularity of the interaction.
Kinetic energy and potential then scale with $\Lambda$ in the same
way, and a cutoff-independent binding energy
\begin{equation}
B_2^{(0)}=0
\end{equation}
is obtained.
Because it contains no dimensionful parameter,
the two-body system at leading order (LO) is invariant under continuous
scale transformations \cite{Mehen:1999nd}.

However, renormalization at the three-body level leads to
the anomalous breaking of continuous scale invariance
and to dimensional transmutation,
similarly to what happens in QCD.
As the number of pairs increases faster than the number of particles,
the energy obtained from the two-body nonderivative delta function alone
has an essential dependence $\propto \Lambda^2/m$ \cite{Thomas:1935zz}.
Renormalization can only
be achieved \cite{Bedaque:1998kg, Bedaque:1998km}
if the coefficient of a contact three-body force,
$mD_0(\Lambda, \Lambda_\star)\propto \Lambda^{-4}$, is
fixed to a given three-body energy, say that of the ground state,
\begin{equation}
B_{3,0}^{(0)}=B_{3,0}.
\end{equation}
This introduces a dimensionful parameter $\Lambda_\star\propto \sqrt{mB_{3,0}}$
that fixes the overall scale of the three-body system.
By dimensional analysis,
the dimensionless ratio of LECs can only be
a function of $\Lambda/\Lambda_\star$.
Whereas for local (and thus non-separable) regulators
it can be multivalued \cite{Kirscher:2015yda},
for separable (and thus nonlocal) regulators \cite{Chen:2025iqp},
\begin{equation}
    H_0(\Lambda/\Lambda_\ast) \equiv
    \frac{\Lambda^2 D_0(\Lambda_\star, \Lambda)}{6 m C_0^2(\Lambda)}
    =h_0 \, \frac{\sin(s_0 \ln(\Lambda/\Lambda_\star) - \delta_0)}
     {\sin(s_0 \ln (\Lambda/\Lambda_\star) + \delta_0)},
    \label{H0_0_expr}
\end{equation}
where
$s_0 \simeq 1.00624$, and
$h_0$ and $\delta_0$ are pure numbers that depend on the shape of
the regulator.
The RG flow of $H_0$ admits a complex-conjugate pair of fixed points,
whose separation is controlled by the phase $\delta_0$. It does
not terminate at a real fixed point and acquires a log-periodic
character: $H_0$ is invariant under $\Lambda\to f^{n} \Lambda$, with
$f=\exp(\pi/s_0)$.
This was the first example of a field-theoretical (albeit nonrelativistic)
limit cycle
\cite{Bedaque:1998kg, Bedaque:1998km}, which was later investigated
in simple models \cite{Glazek:2002hq,Glazek:2003ljf}
(see S. G\l azek's contribution to this collection).
The limit cycle implies DSI, Eq. \eqref{DSI}, which then is manifest
as a geometric tower of three-body bound states,
\begin{equation}
\frac{B_{3,n}^{(0)}}{B_{3,0}^{(0)}}=f^{-2n},
\end{equation}
the Efimov spectrum \cite{Efimov:1970zz}. Would-be states with $n<0$ are, by
definition of $n$, outside the regime of validity of SREFT. Excited Efimov
states have been seen around Feshbach resonances \cite{grimm,grimm2}.

Now, the number of triplets grows faster than the number of pairs and
this three-body force stabilizes not only
the three-body system, but also more-body systems
\cite{Platter:2004he, Hammer:2006ct, Bazak:2016wxm, Lin:2023zqw}.
Thus, DSI is an approximate symmetry of
all systems with $A\ge 3$ particles within the regime of validity of SREFT,
with two major consequences:
\begin{itemize}

\item
For each $A$, there are geometric towers of excited states on top of a finite
number of states,
\begin{equation}
\frac{B_{A,i,n}^{(0)}}{B_{A,i,0}^{(0)}}=f^{-2n},
\label{Efimov}
\end{equation}
where the towers are labeled by $i$, with $i=0$ the one containing
the ground state.
For $A=4$ there are two towers \cite{Hammer:2006ct,Lin:2023zqw},
the lowest state of the second tower having been seen
around a Feshbach resonance \cite{Ferlaino:2009zz}.
The number of towers increases with $A$, generating truly complex spectra
\cite{Ste10,Gattobigio:2011ey,vonStecher:2011zz,Gattobigio:2012tk}.
As far as I know,
there is no explanation for the increase in the number of towers
with $A$.

\item
The lowest levels of each tower
for different $A$ are correlated with $B_{3,0}$ through
$\Lambda_\star$,
\begin{equation}
\frac{3B_{A,i,0}^{(0)}}{AB_{3,0}}=\beta_{A,i}
\end{equation}
where $\beta_{A,i}$ are universal numbers that depend only on the
spin-statistics of the particles.

\begin{itemize}
\item
For bosons, $\beta_{4,0}\simeq 3.458$ and $\beta_{4,1}\simeq 0.7517$
are known precisely
\cite{Deltuva:2010xd}. The lowest state of the second tower is so close
to the three-body ground state that it can be treated \cite{Wu:2023mhg}
as a 3+1 two-body system through Halo EFT \cite{Hammer:2019poc}.
For the ground states,
the energy per particle grows fast at first
\cite{Ste10,Nicholson:2012zp,Carlson:2017txq}, very roughly
as $3(A-2)^2/A$, but then saturates \cite{Carlson:2017txq}.
It approaches a bulk value $\beta_{\infty,0}=90\pm 10$,
\begin{equation}
\beta_{A,0}= \beta_{\infty,0} \left(1+\eta A^{-1/3}+\ldots\right),
\end{equation}
with a reduced surface energy $\eta=-1.7\pm 0.3$.
Bosons at unitarity thus form droplets of a quantum liquid
\cite{Carlson:2017txq}.

\item
Four-component fermions have the same values for $\beta_{4,0}$ and
$\beta_{4,1}$ as bosons. For $A>4$, they
tend to break up into four-particle clusters
\cite{Dawkins:2019vcr, Schafer:2020ivj} thanks to the exclusion principle.
Presumably an analogous result holds for $N$-component fermions,
as long as $N\ge 3$.
\end{itemize}
\end{itemize}
Other properties, such as radii, are similarly correlated with $B_{3,0}$
through the only dimensionful parameter, $\Lambda_\star$.

These features are reminiscent of $^4$He atomic clusters and of nuclei.
In both cases, the energy per particle increases rapidly with the number of
particles, with the two-body state much more weakly bound than heavier systems.
This raises the possibility that they can be calculated with SREFT
in a systematic expansion around the unitarity limit.
Of course these systems are not exactly at the unitarity limit,
but deviation from universality can be accounted for subleading orders.
If they are small, these corrections should be amenable to (distorted-wave)
perturbation theory, and in fact renormalization requires it
\cite{Hammer:2019poc}.
One correction of size ${\cal O}(1/Qa_2)$ comes from the finite value
of the two-body scattering length and, depending on its value,
might be paired with other contributions at next-to-leading order (NLO).
At NLO, a correction ${\cal O}(QR)$ to the two-body force
must be included in first-order perturbation theory to incorporate
the effective range \cite{vanKolck:1998bw}.
A correction to the LO three-body force is sufficient to compensate its
effects in the three-body system but, beyond, renormalization
demands a four-body
force \cite{Bazak:2018qnu}.
If the latter is fixed from the four-body ground-state energy
$B_{4,0,0}$, at NLO
the energies per particle for the lowest states of each tower take the form
\begin{equation}
\frac{3B_{A,i,0}^{(1)}}{AB_{3,0}^{(0)}}= \beta_{A,i}
+ \rho_{A,i} \, r_2\sqrt{mB_{3,0}}
+ \beta_{A,i}' \left(\frac{3B_{4,0,0}}{4B_{3,0}}-\beta_{4,0}\right)
+ \alpha_{A,i} \left(a_2 \sqrt{mB_{3,0}}\right)^{-1},
\label{Efimovcorrect}
\end{equation}
where $\rho_{A,i}$, $\beta_{A,i}'$, and $\alpha_{A,i}$ are universal numbers.
Similar relations hold for excited states
and other observables.
Because they are obtained in first-order
perturbation theory, the corrections
depend only on the LO wavefunction and are governed by DSI.
However, the NLO interactions introduce new dimensionful parameters
and states in a tower are no longer related exactly by $f^{-2n}$ as at LO,
Eq. \eqref{Efimov}.
Obviously corrections will be larger for the lowest states, but
as long as they are perturbative these states retain
their universal features ---a point often misunderstood in the literature.
The NLO correlations between the lowest states of the
two four-body towers are discussed in Refs. \cite{Wu:2023mhg,Wu:2026XXX}.
Higher-order corrections account for further details of the dynamics.

Clearly the expansion around unitarity is expected to work only in a
window of momenta, $|a_2|^{-1}\simle Q \simle R^{-1}$.
As we increase $R/|a_2|$, deviations from unitarity increase.
The $^4$He dimer is sufficiently close to the unitarity limit
for an excited trimer to exist \cite{Kunitski:2015qth}.
The deuteron, on the other hand, is relatively less
shallow and the triton has no bound excited state.
At small $Q$ one can no longer
expect convergence, unless we resum the scattering-length contributions
\cite{vanKolck:1998bw}.
This resummation allows for
a description of the remnants of the excited Efimov states,
but injects in the LO an explicit breaking
of DSI that should not be important at the bottom of the
towers.
What improves the description of all energy regions
is to include at LO
small but arbitrary pieces from higher orders in the $QR$ expansion
\cite{vanKolck:2025qof}.
This improvement does not affect the radius of convergence of the EFT
---and thus the uncertainty at each order---
but it does move central values towards data, within error bands
\cite{Contessi:2024vae}.
\\

\noindent
{\bf $^4$He atomic clusters.}
Low-energy few-body data are scarce,
but many sophisticated $^4$He-$^4$He potentials have been constructed
to calculate properties of few- and many-body systems.
They have been used to calibrate and benchmark SREFT.
The properties of $A\le 6$ systems, including the trimer
and tetramer excited states, are well reproduced in SREFT
when scattering-length effects are resummed, whether the action is improved
\cite{Contessi:2023yoz} or not
\cite{Ji:2012nj,Bazak:2016wxm,Bazak:2018qnu,Qin:2020hsc,Lin:2023zqw}.

The trimer ground state and the two lowest tetramer states
are accessible through the unitarity expansion.
Preliminary results \cite{Wu:2026XXX}
suggest that, indeed, the unitarity expansion
converges for these states.
Matching at LO to the $^4$He trimer ground energy gives
good results for the trimer ground-state radius and the
energies of the ground and first-excited tetramers.
Systematic improvement is seen at NLO
for the trimer and tetramer ground-state radii,
as well as for the tetramer excited-state energy,
when the two-body scattering length, two-body effective range,
and tetramer ground-state energy are used as input.
(see Feng Wu's contribution to this collection).
In Ref. \cite{Wu:2023mhg}, it was shown that a Halo EFT constructed
in the unitarity limit provides a convergent expansion for
the radius of the excited tetramer state, which at NLO reaches
good agreement with potential-model results.

We are extending the calculations to the ground energies of a larger number
of particles.
Cluster energies \cite{Pandharipande:1983xez} from phenomenological potentials
are qualitatively similar
to those for unitarity bosons \cite{Carlson:2017txq}.
Preliminary calculations at LO and NLO in SREFT with
resummed scattering length yields convergence and excellent
agreement with potential-model results up to at least
$A=15$ \cite{Madeira:2026XXX}
(see Lucas Madeira's contribution to this collection).
Prospects for the unitarity expansion of $^4$He systems are thus very good.
\\

\noindent
{\bf Nuclei.}
Although the deuteron is not as shallow as the $^4$He
dimer,
the triton energy is, due to $\Lambda_\star$,
correlated with the radius \cite{Platter:2005sj} and
the doublet neutron-deuteron ($nd$) scattering length
(the ``Phillips line'') \cite{Bedaque:1999ve}.
In addition to a $T$-matrix zero just below
threshold, there is a low-energy virtual state in the $nd$ doublet channel.
Below the deuteron break-up, SREFT results with resummed two-body
scattering-length effects \cite{Vanasse:2013sda, Margaryan:2015rzg}
can be matched onto a Halo EFT
with neutron and deuteron as ``elementary'' degrees of freedom
\cite{Rupak:2018gnc}.
In this EFT, the virtual state can be tracked as one artificially dials to the
unitarity limit, and it is seen to
emerge as an Efimov bound state obeying Eq. \eqref{Efimov}
with $n=1$.

A complication that appears for helion and heavier nuclei
is the isospin breaking
that differentiates protons and neutrons. At low energies, it comes mostly
from  the Coulomb interaction between protons, which
becomes nonperturbative at momenta
below around $k_C\equiv \alpha m_N/2$, where $\alpha$ is the fine-structure
constant
and $m_N$ is the nucleon mass.
Since numerically $k_C\sim 1/2|a_{2,0}|$,
one can treat Coulomb in a perturbative expansion in $k_C/Q$
in the same region where
one can expand the $^1S_0$ amplitude in $(a_{2,0}Q)^{-1}$
\cite{Konig:2015aka,Konig:2016iny}.
Because $k_C^2/m$ and $1/ma_{2,0}^{2}$ are small compared to typical nuclear
energies, we expect both effects to enter at NLO or higher,
at least for the ground states of light nuclei.
Still resumming
$a_{2,1}^{-1}$ effects in the two-nucleon $^3S_1$ channel, one obtains
the helion-triton energy splitting and proton-deuteron scattering
in good agreement with experiment.
As expected, the Coulomb interaction can be treated as a
perturbation also in the alpha particle \cite{Mondal:2025wml}.
Isospin-breaking effects from the light-quark mass splitting are
higher order \cite{Konig:2015aka,Konig:2016iny}.

DSI should leave traces in the light nuclear spectrum
because the energy per particle is much smaller for the deuteron
than any other nucleus.
For example, there are correlations through $\Lambda_\star$ between the
triton binding energy, the alpha-particle energy (``Tjon line'')
\cite{Platter:2004zs},
and the neutron-helion/triton scattering lengths
\cite{Kirscher:2010dgl,Kirscher:2011uc,Schafer:2022hzo}.

It was therefore suggested in
Refs. \cite{Konig:2016utl,Konig:2019xxk}
that the full unitarity expansion ---taking both two-nucleon $S$-wave channels
at unitarity in LO---
could provide a basis for a description of nuclei starting from a single
parameter, $\Lambda_\star$. The expansion for the helion
energy converges well \cite{Konig:2016utl,Konig:2019xxk}.
The Tjon line is reproduced:
the energy per nucleon of the alpha particle is $\sim 2.8$
times that of helion, less than 20\% off the universal, LO value, and
the observed value is already nearly recovered when the NLO
corrections from only $a_{2,1}^{-1}$ and $a_{2,0}^{-1}$ are
included \cite{Konig:2016utl}.
At LO the first-excited state is near where it is observed,
in between the proton-triton and neutron-helion thresholds.
Similar success is found for radii \cite{Konig:2019xxk}.

The three- and four-nucleon systems thus also reflect the underlying DSI.
However, technical difficulties appear for larger nuclei, where
perturbation theory is challenged by the LO tendency
of multicomponent fermions to be unstable to break up
into clusters. The problem remains even with a resummation
of two-nucleon scattering-length effects \cite{Contessi:2017rww,Bansal:2017pwn},
even though $S$-wave neutron-alpha scattering works
well \cite{Bagnarol:2023crb}.
A way around this obstacle was found in Ref. \cite{Contessi:2025xue}
using the improvement formulated
in Refs. \cite{Contessi:2023yoz,Contessi:2024vae}.
The ground-state energies of $^6$Li, $^{12}$C, and $^{16}$O are obtained
at NLO within
about 25\% of experiment.
Although an expansion parameter
of $\sim 1/2$ is not particularly small, the expansion seems to converge,
indicating that the physics of nuclei is approximately universal.


\section*{Acknowledgments}

I thank Johannes Kirscher and the participants of
the ECT* workshop ``Universality in strongly-interacting systems:
from QCD to atoms'' for most interesting discussions
on the topics covered here.
Collaboration with many outstanding researchers, in particular
Lorenzo Contessi, Xincheng Lin, Lucas Madeira, and Feng Wu,
has been crucial for the construction of the universal theory outlined above.
This material is based upon work supported in part
by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics,
under award DE-FG02-04ER41338.

\clearpage
\endgroup
